Elastic wave propagation in a cylindrical bore situated

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GLC成分梯度CN_(x)多层膜的微观结构和摩擦学性能

第33卷第6期中国表面工程Vol．33No．62020年12月CHINA SURFACE ENGINEERINGDecember 2020收稿日期：2020-10-13；修回日期：2020-11-29通信作者：郑晓华（1971—），男（汉），副教授，博士；研究方向：材料表面工程、电接触材料；E-mail ：zhengxh@zjut．edu．cn 基金项目：浙江省自然科学基金（LY15E010007）；浙江省重点研发计划（2019C01088）Fund ：Supported by Natural Science Foundation of Zhejiang Province （LY15E010007）and Zhejiang Provincial Key Research and DevelopmentProgram （2019C01088）引用格式：杨芳儿，陆诗慧，杨烁妍，等．GLC /成分梯度CN x 多层膜的微观结构和摩擦学性能［J ］．中国表面工程，2020，33（6）：68-76．YANG F E ，LU S H ，YANG S Y ，et al．Microstructure and tribological properties of magnetron sputtered graphite-like-carbon /compo-sition-gradient CN x multilayer films ［J ］．China Surface Engineering ，2020，33（6）：68-76．doi ：10.11933/j．issn．10079289.20201013001GLC /成分梯度CN x 多层膜的微观结构和摩擦学性能杨芳儿1，陆诗慧1，杨烁妍2，高蔓斌1，郑晓华1（1．浙江工业大学材料科学与工程学院，杭州310014；2．万向钱潮股份有限公司等速驱动轴厂质量部，杭州311200）摘要：类金刚石碳膜通常内应力大、结合力低，而多层膜结构可提高结合力。

弹性波_香港科技大学余同希讲座PPT

advance without changing their shape or magnitude travel with a constant speed

The 1-D longitudinal waves are non-dispersive
6
ELASTIC WAVES (CONT’D)
s f1 x1 ct1 f1 x2 ct2
x1 ct1 x2 ct2
s O u
2012年10月24日
u f1 x ct
t t1 t Biblioteka t2c x2 x1 t2 t1 is the speed of the wave propagation
Let
u f1 x ct f 2 x ct
u cf f1 x ct cf f 2 x ct t 2 u c 2 f1 x ct c 2 f 2 x ct 2 t 2u f1 x ct f 2 x ct 2 x
cL E 0
x1
x2
x
is the speed of the longitudinal wave
f 2 x ct a backward wave
f1 x ct a forward wave,

Both the waves f1 and f2 are related to the wave equation
1.2 VARIOUS TYPES OF ELASTIC WAVES 各种类型的弹性波
2012年10月24日
8
TYPES OF ELASTIC WAVES
Types

超材料翻译

Naturematerials LETTERSPublished online : 18 APRIL 2010| DOI:10.1038/NMAT2747A single-layer wide-angle negative-index metamaterial at visible frequencies在可见光频率的一种单层广角负折射率超材料Metamaterials are materials with artificial electromagnetic properties defined by their sub-wavelength structure rather than their chemical composition.基于亚波长结构而非化学结构，超材料也叫人工电磁材料。

Negative-index materials(NIMs) are a special class of metamaterials characterized by an effective negative index that give rise to such unusual wave behavior as backwards phase propagation and negative refraction.负折射率材料（NIMs）是一类具有有效负折射率的特殊超材料，能够产生逆向传播和负折射的不同寻常的波行为。

These extraordinary properties lead to many interesting functions such as sub-diffraction imaging and invisibility cloaking.这些非凡的性能使得有有趣的功能，如子衍射成像和隐蔽伪装So far ,NIMs have been realized through layering of resonant structures,such as spilt-ring resonators ,and have been demonstrated at microwave to infrared frequencies over a narrow range of angles-of-incidence and polarization.到目前，负折射率材料（NIMs）已经通过谐振结构层实现，例如开环谐振器，而且在较窄范围的红外频率内的入射角和偏振也可以证明。

不规则波理论

55第三章不規則波理論風浪使海面形成一種極為不規則(irregular) 的波形。

從風洞水槽或現地觀測中均可發現水面上的風浪，如照片3.1顯示，大波上面疊有小波，縱橫各方向的波重重疊疊，隨著時間和空間變化，同樣的波形不可能再次發生。

故風浪之波形本身構造複雜，是屬於時間及空間上的一種隨機性(random) 變動量。

風浪既是一種隨機現象，則須以統計的方法來描述其特性。

統計方法中波別分析法(individual wave analysis) 和波譜分析法(spectral analysis) 是目前被採用做為敘述海洋風浪之不規則性最普遍的方法。

利用這兩種方法從不規則波中定義出波高和週期，使其能適用於規則波的波浪理論，以達到各種工程設計應用的目的。

一般將這種統計稱為波浪的短期統計，此外另有波浪的長期統計，又分為波候統計和極值統計。

波候統計是對於長年監測到的資料做歸納、整理、和一般統計分析。

而極值統計是討論重現期的問題，將在第四章中再行敘述。

本章只就短期統計對水面波形之不規則性的問題來討論。

照片3.1實際海面風浪的照片3.1 不規則波的表示方法3.1.1 波別解析法不規則波理論56 單純來看，若視風浪的水面變位為一維的波形變化，如圖3.1所示。

對此不規則波形信號來定義個別波之波高與週期有三種方式。

第一種是零位上切 (zero up cross) 法，所謂上切零點是水位上昇曲線與平均水位線之交點，如圖3.1中小圓圈所示各點。

計算二相鄰上切零點間，水位變動之最高峰與最低谷點間之垂直高差即為波高，二相鄰上切零點的時間長度即為週期。

第二種是以水位下降曲線與平均水位線之交點，如圖3.1中小三角形所示各點，定義出個別波的方法，稱為零位下切 (zero down cross) 法。

另外第三種是無視平均水位的存在，兩相鄰波峰波谷的高差即為波高，兩相鄰波峰之間的時間即為週期，依此定義個別波的方法稱為峰至峰 (crest to crest) 法。

光电子学Ch1

y Wave fronts

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(b) x
2w o O
z Beam axis Intensity (c)
Gaussian

(a) 2w
r
(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross section. (c) Light irradiance ( intensity) vs. radial distance r from beam axis (
E x ( z, t ) Ec e
E (r, t ) E0 cos(t k r 0 )
y
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k
r
E (r ,t ) r z

O
A tr avelling plane EM w ave along a direc tion
?1 999 S.O. K asap, O ptoelectron ics (P ren tice Hall)
x oΒιβλιοθήκη zA plane EM wav e tr av elling alo ng z, h as the same E x (o r B y ) at any po int in a given xy p lan e. All electr ic f ield vecto rs in a given xy p lan e are t heref ore in ph ase. T he xy planes are of infinit e extent in t he x and y directio ns.

Multiple scattering of electro-elastic waves from a buried cavity in a functionally graded piezoe

Multiple scattering of electro-elastic waves from a buried cavity in a functionally graded piezoelectric material layerXue-Qian Fang *Department of Engineering Mechanics,Shijiazhuang Railway Institute,Bei erhuandong Road,Shijiazhuang,Hebei 050043,PR Chinaa r t i c l e i n f o Article history:Received 19March 2008Received in revised form 9May 2008Available online 26June 2008Keywords:Functionally graded piezoelectric material layerMultiple scattering of electro-elastic waves Dynamic stress concentration factor Circular cavity Image methoda b s t r a c tThis paper presents a theoretical method to investigate the multiple scattering of electro-elastic waves and the dynamic stress around a buried cavity in a functionally graded piezoelectric material layer bonded to a homogeneous piezoelectric material.The analyti-cal solutions of wave ﬁelds are expressed by employing wave function expansion method,and the expanded mode coefﬁcients are determined by satisfying the boundary conditions around the cavity.The image method is used to satisfy the mechanical and electrically short conditions at the free surface of the structure.According to the analytical expression of this problem,the numerical solutions of the dynamic stress concentration factor around the cavity are presented.The effects of the piezoelectric property,the position of the cavity in the layer,the incident wave number and the material properties on the dynamic stress around the cavity are analyzed.Analyses show that the piezoelectric property has great effect on the dynamic stress in the region of higher frequencies,and the effect increases with the decrease of the thickness of FGPM layer.If the material properties of the homoge-neous piezoelectric material are greater than those at the surface of the structure,the dynamic stress resulting from the piezoelectric property is greater.The effect material properties at the two boundaries of FGPM layer on the distribution of dynamic stress around the cavity is also examined.Ó2008Elsevier Ltd.All rights reserved.1.IntroductionFunctionally graded piezoelectric materials (FGPMs)are the new generation of composites and important area of mate-rials science research.In recent years,FGPMs are widely applied in smart materials and structures,so the theoretical inves-tigation on FGPMs has received considerable attention in the literatures.The study of elastic wave propagation through FGPMs has many important applications.Through analysis,we can predict the response of composite materials to various types of loading,and obtain the high strength and toughness of materials.The problem is also a theoretical background of the non-destructive analysis of FGPM microstructures by using ultrasonic technique.During the serving of composite structures,many failures induced by various loading have been found in these materials.The discontinuities,such as holes,cracks and inclusions in composite structures,are the major reason for these failures.If the discontinuities exist in composite structures,it is deﬁnitely vital to determine them and analyze their effects.With the ad-vent of FGPMs,the fracture mechanics under various loading conditions in piezoelectric materials has received much atten-tion in recent years.In the past decade,considerable amounts of analytical,numerical and experimental work about the stress in piezoelectric materials have been done to improve the reliability of structures.0020-7683/$-see front matter Ó2008Elsevier Ltd.All rights reserved.doi:10.1016/j.ijsolstr.2008.06.014*Tel.:+8645186410268;fax:+8645189264194.E-mail addresses:fangxueqian@ ,fangxueqian@International Journal of Solids and Structures 45(2008)5716–5729Contents lists available at ScienceDirectInternational Journal of Solids and Structuresjournal homepage:www.elsevier.c o m /l o c a t e /i j s o l s t rUp to present time,analyses about the stress problem in FGPMs mainly focus on the behavior under the static loading.Soh et al.(2000)analyzed the behavior of a bi-piezoelectric ceramic layer with a central interfacial crack subjected to anti-plane shear and in-plane electric ing integral transform method,Wang and Noda (2001)discussed the fracture behavior of a cracked smart actuator made of piezoelectric materials with functionally grade material properties.By means of singular integral equation technique,Wang (2003)investigated the mode III crack problem in FGPMs,and both a single crack and a series of collinear cracks were considered.Li and Weng (2002)studied the problem of a ﬁnite crack in a strip of FGPMs under an anti-plane mechanical loading and in-plane electric loading.The non-local theory was also applied to obtain the behavior of two collinear cracks in FGPMs under anti-plane shear loading for permeable electric boundary conditions (Zhou and Wu,2006).By making using of the Gauss–Chebyshev integration technique,Chue and Ou (2005)investigated the singular electromechanical ﬁeld near the crack tips of an internal crack in FGPMs.Due to the increasing demand of an understanding of dynamic processes in piezoelectric composites,it is highly desirable to study the stress in FGPMs in a fully dynamic framework.However,considering the complexity of wave scattering resulting from the non-homogeneous property of FGPMs and the complexity of multiple scattering from the scatterer and the bound-ary,relatively little work has been done regarding on the wave propagation in FGPMs.Recently,Chen et al.(2003)have con-sidered the electromechanical impact response of FGPMs with a crack using integral transform technique.Ma et al.(2004)have investigated the stress and electric displacement intensity factors of two collinear cracks subjected to anti-plane shear waves in FGPMs.Most recently,Fang et al.(2007)studied the dynamic stress from a circular cavity buried in a semi-inﬁnite functionally graded piezoelectric material,and both the displacement ﬁeld and the piezoelectric ﬁeld were considered.To the author’s knowledge,the multiple scattering of electro-elastic waves and dynamic stress around a cavity in a functionally graded piezoelectric material layer are still unavailable in the literatures.The objective of this paper is to investigate the analytical solutions of the electro-elastic ﬁeld and dynamic stress around a cavity embedded in a functionally graded piezoelectric material layer bonded to a homogeneous piezoelectric material.The incidence of anti-plane shear waves at the surface of the structure is applied,and both the displacement ﬁeld and piezoelec-tric ﬁeld in functionally graded piezoelectric materials are presented.The mechanical and electrically short conditions at the free surface are considered,and the image method is used to satisfy the free boundary conditions of the structure.The wave ﬁelds and electric potentials are expanded by using wave function expansion method (Pao and Mow,1973).The expanded mode coefﬁcients are determined by satisfying the boundary conditions around the cavity.Addition theorem for Bessel func-tions is used to accomplish the translation between different coordinate systems.The analytical solution of the dynamic stress concentration factor around the cavity is presented,and the numerical solutions are graphically illustrated.The effects of the piezoelectric property,the incident wave number and the position of the cavity in the FGPM layer on the dynamic stress concentration factors around the cavity are also analyzed.2.Wave motion equations in FGPMs and their solutionsConsider a FGPM layer bonded to a homogeneous piezoelectric material,as depicted in Fig.1.The mechanical and elec-trically short conditions at the free surface are considered.c 144;e 115;e 111;q 1are the elastic stiffness,piezoelectric constant,dielectric constant and density of materials at the surface of FGPM layer,and c 244;e 215;e 211;q 2those of the homogeneous pie-zoelectric material.The material properties in the FGPM layer vary smoothly along the x -direction.Let a circular cavity lie in the functionally graded piezoelectric material layer.The distance between the center of the cavity and the upper edge of the layer is h 1,and that between the center of the cavity and the lower edge of the FGPM layer is h 2.xor ′θ′x ′θraDistribution of material X.-Q.Fang /International Journal of Solids and Structures 45(2008)5716–57295717All materials exhibit transversely isotropic behavior and are polarized in the z-direction.Let an anti-plane shear wave with frequency x hit the surface of the FGPM layer in the positive x-direction.In this case,the mechanically and electrically coupled constitutive equations can be written asr zx¼c44ðxÞo uo x þe15ðxÞo/o x;r zy¼c44ðxÞo uo yþe15ðxÞo/o y;ð1ÞD x¼e15ðxÞo uo xÀe11ðxÞo/o x;D y¼e15ðxÞo uo yÀe11ðxÞo/o y;ð2Þwhere r zj,u,D j and/(j=x,y)are the shear stress,anti-plane displacement,in-plane electric displacement and electric po-tential,respectively;c44(x)is the elastic stiffness of graded materials measured in a constant electricﬁeld,e11(x)is the dielec-tric constant of graded materials measured in constant strain and e15(x)is the piezoelectric constant of graded materials.The anti-plane governing equation and Maxwell’s equation in FGPMs are described aso r zx o x þo r zyo y¼qðxÞo2uo t2;ð3Þo D x o x þo D yo y¼0;ð4ÞSubstituting Eqs.(1)and(2)into Eqs.(3)and(4),the following equations can be obtained:o c44ðxÞo uþc44ðxÞo2u2þo e15ðxÞo/þe15ðxÞo2/2þc44ðxÞo2u2þe15ðxÞo2/2¼qðxÞo2uo t;ð5Þo e15ðxÞo x o uo xþe15ðxÞo2uo x2Ào e11ðxÞo xo/o xÀe11ðxÞo2/o x2þe15ðxÞo2uo y2Àe11ðxÞo2/o y2¼0:ð6ÞFor convenience,it is assumed that all material properties vary continuously and have the same exponential function distri-bution along the x-direction in the layer,i.e.c44ðxÞ¼c044expð2b xÞ;e15ðxÞ¼e015expð2b xÞ;e11ðxÞ¼e011expð2b xÞ;qðxÞ¼q0expð2b xÞ;ð7ÞAccording to the continuous condition of the material properties in the layer and at the position of x=h2,the constantsb;c044;e015;e011and q0can be calculated asb¼12ðh1þh2Þlnc244c144;ð8Þc044¼c144expð2b h1Þ;e015¼e115expð2b h1Þ;e011¼e111expð2b h1Þ;q0¼q1expð2b h1Þ:ð9ÞIn the above formulations,it is assumed that the ratio c244=c144is equal to e215=e115;e211=e111,and q2/q1.Though the variations areunrealistic,it would allow us to comprehend the effect of material properties of FGPMs on the dynamic stress around the cavity and can provide references for the non-destruction detection in FGPMs.Substituting Eq.(7)into Eqs.(5)and(6),the following equations are obtained:2b c044o uo xþc044r2uþ2b e015o/o xþe015r2/¼2b q0o2uo t2;ð10Þ2b e015o uo xþe015r2u¼2be011o/o xþe011r2/:ð11ÞHere,r2=o2/o x2+o2/o y2is the two-dimensional Laplace operator in the variables x and y.Assume that another electro-elasticﬁeld w is expressed asw¼/Àk1u;ð12Þwhere k1¼e015=e011.From Eqs.(10)and(11),the following equations can be obtained:r2uþ2b o u¼1cSHo2uo t;ð13Þ2b o wo xþr2w¼0;ð14Þwhere c SH¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃl e=q0pwith l e¼c044þ½ðe015Þ2=e011 being the wave speed of electro-elastic waves.5718X.-Q.Fang/International Journal of Solids and Structures45(2008)5716–5729The steady solution of this problem is investigated.Assuming that u =u 0U e Ài x t ,Eq.(13)can be changed intor 2U þ2bo U o xþk 2U ¼0;ð15Þwhere x is the frequency of the incident waves and k =x /c SH is the wave number of incident waves.To solve Eq.(15),the solution can be proposed asU ¼exp ðÀb x Þw ðx ;y Þ;ð16Þwhere w (x ,y )is the function introduced for derivation.Substituting Eq.(16)into Eq.(15),one can see that the function w (x ,y )should satisfy the following equation:r 2w þj 2w ¼0;ð17ÞHere,j ¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðk 2Àb 2Þq .According to Eqs.(15)–(17),one can see that there exist elastic waves with the form of u =u 0U e Ài x t =u 0exp(Àb x )e i(j x Àx t ),which denotes the propagating wave with its amplitude of vibration attenuating in the x -direction.Similarly,the solution of w in Eq.(14)has the following form:w ¼w 0exp ðÀb x Þe i ði b x Àx t Þ:ð18ÞNote that all ﬁeld quantities have the same time variation e Ài x t ,which is suppressed in all subsequent representations for notational convenience.According to Eqs.(14)and (17),the general solutions of the scattered ﬁeld of electro-elastic waves resulting from the cav-ity in FGPMs can be described,using wave function expansion method (Pao and Mow,1973),asu s¼exp ðÀb r cos h ÞX 1n ¼À1a n H ð1Þn ðj r Þe i n h ;ð19Þw s ¼exp ðÀb r cos h ÞX 1n ¼À1b n H ð1Þn ði b r Þe i n h ;ð20Þwhere (r ,h )is the corresponding cylindrical coordinate system shown in Fig.1,H ð1Þn ðÁÞis the n th Hankel function of the ﬁrst kind,and a n and b n determined by satisfying the boundary conditions are the mode coefﬁcients of the scat-tered waves.Note that Hankel function H ð1Þn ðÁÞdenotes the outgoing wave and satisﬁes the radiation condition at inﬁnity.The solution of the scattered–reﬂected waves has the same form as that of the scattered waves (Fang et al.,2006).3.The multiple scattering of electro-elastic waves and the total wave ﬁeldConsider the electro-elastic waves propagating along the positive x -direction in the FGPM structure.In the local coordi-nate system (r ,h )of the real cavity,the incident waves can be expanded as Pao and Mow (1973)u ði Þ1¼u 0exp ½Àb x þh 1ði j Àb Þ ei j x¼u 0exp ½Àb r cos h þh 1ði j Àb ÞX 1n ¼À1i nJ n ðj r Þe i n h ;ð21Þwhere u 0is the amplitude of the incident waves,j is the wave number of the propagating waves and J n (Á)is the n th Besselfunction of the ﬁrst kind.Similarly,the incident ﬁeld w (i )is expressed asw ði Þ1¼k 1u 0exp ðÀb x À2b h 1ÞeÀb x¼k 1u 0exp ðÀb r cos h À2b h 1ÞX 1n ¼À1i nJ n ði b r Þe i n h :ð22ÞIn the local coordinate system (r ,h )of the real cavity,the scattered ﬁeld can be described asu ðs Þ1¼exp ðÀb r cos h ÞX 1l ¼1X 1n ¼À1A l n 1H ð1Þn ðj r Þe i n h;ð23Þw ðs Þ1¼k 1exp ðÀb r cos h ÞX 1l ¼1X 1n ¼À1B l n 1H ð1Þn ði b r Þe i n h :ð24Þwhere l denotes the scattering time between the real and image cavities,and A l n and B l n ðl ¼1;2;...;1Þdetermined by sat-isfying the boundary conditions are the mode coefﬁcients of the l th scattering resulting from the real cavity.When the electro-elastic wave propagates in the FGPM layer,it is scattered by the circular cavity at ﬁrst.Then,the out-going scattered wave from the cavity is reﬂected on the straight surface (x =Àh 1),and the reﬂected waves arise.The reﬂected waves are scattered by the cavity again.This complex phenomenon is shown in Fig.1.X.-Q.Fang /International Journal of Solids and Structures 45(2008)5716–57295719To satisfy the mechanical and electrically short conditions at the free surface,the image method is applied.The reﬂected waves at the edge of FGPM layer are described by the scattered waves resulting from the virtual image cavity.The distance between the virtual image cavity and the straight boundary is also h1.The magnitudes of the incident waves and scattered waves of the real and image cavities are the same,however,the directions of them are opposite.So,the boundary conditions at the free surface can be satisﬁed.For the image cavity,the waves propagate in the negative x0-direction and can be expressed asuðiÞ2¼u0exp½b x0þh1ði jÀbÞ eÀi j x0¼u0exp½b r0cos hþh1ði jÀbÞX1n¼À1iÀn Jnðj r0Þe i n h0;ð25ÞwðiÞ2¼k1u0expðb x0À2b h1Þe b x0¼k1u0expðb r0cos hÀ2b h1ÞX1n¼À1iÀn Jnði b r0Þe i n h0:ð26ÞLikewise,in the local coordinate system(r0,h0),the scatteredﬁelds resulting from the image cavity can be described asuðsÞ2¼expðb r0cos h0ÞX1l¼1X1n¼À1A ln2Hð1Þnðj r0Þe i n h0;ð27ÞwðsÞ2¼k1expðb r0cos h0ÞX1l¼1X1n¼À1B ln2Hð1Þnði b r0Þe i n h0;ð28Þwhere A ln2and B ln2ðl¼1;2;...;1Þdetermined by satisfying the boundary conditions are the mode coefﬁcients of the l thscat-tering resulting from the image cavity.Thus,the totalﬁeld of elastic waves in the material is taken to be the superposition of the incidentﬁeld,the scatteredﬁeld and the reﬂectedﬁeld at the surface of materials,namely,u t¼uðiÞ1þuðsÞ1þuðsÞ2:ð29ÞSo,the total electric potential in the material is expressed as/t¼k1u tþwðiÞ1þwðsÞ1þwðsÞ2:ð30ÞIn the cavity,the elastic waveﬁeld vanishes and only the electricﬁeld exists.The electric potential in the real cavity is the standing wave and is expressed as/I 1¼k1X1l¼1X1n¼À1C ln1Jnði b rÞe i n h:ð31ÞSimilarly,the electric potential in the image cavity is written as/I 2¼k1X1l¼1X1n¼À1C ln2Jnði b r0Þe i n h0:ð32ÞTo make computation tractable,the expression of elasticﬁelds and electricﬁelds in the local coordinate system(r0,h0)can be translated into another local coordinate system(r,h).According to addition theorem for Bessel functions(Stratton,1941),the following relation can be derived:Hð1Þn ðpr0Þe i n h0¼X1m¼À1ðÀ1ÞmÀn Hð1ÞmÀnð2ph1ÞJ mðprÞe i m h:ð33ÞSimilarly,Hð1Þn ðprÞe i n h¼X1m¼À1Hð1ÞmÀnð2ph1ÞJ mðpr0Þe i m h0:ð34ÞSo,the following translation of coordinate systems can be obtained:expðb r0cos h0ÞX1n¼À1Hð1Þnðpr0Þe i n h0¼exp½bð2h1þr cos hÞ ÂX1n¼À1X1m¼À1ðÀ1ÞmÀn Hð1ÞmÀnð2ph1ÞJ mðprÞe i m h;ð35Þwhere r0¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃr2þ4h21þ4rh1cos hqand cos h0¼ððr0Þ2þ4h21Àr2Þ=4h1r0.expðÀb r cos hÞX1n¼À1Hð1ÞnðprÞe i n h¼exp½bð2h1Àr0cos h0ÞX1n¼À1X1m¼À1Hð1ÞmÀnð2ph1ÞJ mðpr0Þe i m h0;ð36Þwhere r¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃðr0Þ2þ4h21À4r0h1cos h0q,and cos h¼Àðr2þ4h21Àðr0Þ2Þ=4h1r.5720X.-Q.Fang/International Journal of Solids and Structures45(2008)5716–57294.Boundary conditions around the cavityWithout loss of generality,the case that the cavity is free of traction is investigated.For the cavity,the boundary condi-tions around it are that the radial shear stress is equal to zero,and the electric potential and normal electric displacement are continuous.They can be expressed asr rz jr¼a ¼o u to rr¼aþk2o/to rr¼a¼0;ð37ÞD jr¼a ¼e15o u to rr¼aÀe11o/to rr¼a¼Àe0o/Io rr¼a;k3oðwðiÞ1þwðsÞ1þwðsÞ2Þo rr¼a¼o/Io rr¼a:ð38Þ/t jr¼a ¼/I jr¼a:ð39ÞHere,k2¼e015=c044and k3¼e011=e0.Note that e0=8.85Â10À12F/m is the dielectric constant of vacuum.5.Determination of scattering mode coefﬁcients and dynamic stress concentration factorMultiple scattering of waves takes place between the real and image cavities.By satisfying the boundary conditions around the cavities,the mode coefﬁcients of electro-elastic waves are determined.Substituting Eqs.(29)–(32)into Eqs.(37)–(39),multiplying by eÀi s h at both sides of Eqs.(37)–(39),and then integrating fromÀp to p,the following recurrence formulae can be obtained.When l=1,the relations among every mode coefﬁcient of the scattered waves are written asð1þk1k2ÞA1s1f b a cos h Hð1Þsðj aÞÀ½sHð1Þsðj aÞÀj aHð1Þsþ1ðj aÞ gþk1k2B1s1f b a cos h Hð1Þsði b aÞÀ½sHð1Þsði b aÞÀi b aHð1Þsþ1ði b aÞ g¼Àð1þk1k2Þi s exp½h1ði jÀbÞ f b a cos h J sðj aÞÀ½sJ sðj aÞÀj aJ sþ1ðj aÞ gÀk1k2i s expðÀ2b h1Þf b a cos h J sði b aÞÀ½sJ sði b aÞÀi b aJ sþ1ði b aÞ g:ð40Þð1þk1k2ÞA1s2f b a cos h0Hð1Þsðj aÞþ½sHð1Þsðj aÞÀj aHð1Þsþ1ðj aÞ gþk1k2B1s2f b a cos h0Hð1Þsði b aÞþ½sHð1Þsði b aÞÀi b aHð1Þsþ1ði b aÞ g¼Àð1þk1k2ÞiÀs exp½h1ði jÀbÞ f b a cos h0J sðj aÞþ½sJ sðj aÞÀj aJ sþ1ðj aÞ gÀk1k2iÀs expðÀ2b h1Þf b a cos h J sði b aÞþ½sJ sði b aÞÀj aJ sþ1ði b aÞ g:ð41Þk1k3B1s1f b a cos h Hð1Þsði b aÞÀ½sHð1Þsði b aÞÀi b aHð1Þsþ1ði b aÞ gÀk1C1s1f b a cos h J sði b aÞÀ½sJ sði b aÞÀi b aJ sþ1ði b aÞ g¼Àk1k3i s expðÀ2b h1Þf b a cos h J sði b aÞÀ½sJ sði b aÞÀi b aJ sþ1ði b aÞ g:ð42Þk1k3B1s2f b a cos h0Hð1Þsði b aÞþ½sHð1Þsði b aÞÀi b aHð1Þsþ1ði b aÞ gÀk1C1s2f b a cos h0J sði b aÞþ½sJ sði b aÞÀi b aJ sþ1ði b aÞ g¼Àk1k3iÀs expðÀ2b h1Þf b a cos h0J sði b aÞþ½sJ sði b aÞÀi b aJ sþ1ði b aÞ g:ð43ÞA1 s1Hð1Þsðj aÞþB1s1Hð1Þsði b aÞÀC1s1J sði b aÞ¼Ài s expðÀ2b h1Þ½J sðj aÞþJ sði b aÞ :ð44ÞA1 s2Hð1Þsðj aÞþB1s2Hð1Þsði b aÞÀC1s2J sði b aÞ¼ÀiÀs expðÀ2b h1Þ½J sðj aÞþJ sði b aÞ :ð45ÞWhen l=2,3,...,1,the relations among every mode coefﬁcient of the scattered waves are written asA l s1ð1þk1k2ÞexpðÀb a cos hÞfÀb a cos h Hð1Þsðj aÞþ½sHð1Þsðj aÞÀj aHð1Þsþ1ðj aÞ gþB ls1k1k2expðÀb a cos hÞfÀb a cos h Hð1Þsði b aÞþ½sHð1Þsði b aÞÀi b aHð1Þsþ1ði b aÞ g¼ÀA lÀ1s2ð1þk1k2Þexp½bð2h1þa cos hÞ b a cos hX1m¼À1ðÀ1ÞsÀm Hð1ÞsÀmð2j h1ÞJ sðj aÞ(þX1m¼À1ðÀ1ÞsÀm Hð1ÞsÀmð2j h1Þ½sJ sðj aÞÀj aJ sþ1ðj aÞ)X.-Q.Fang/International Journal of Solids and Structures45(2008)5716–57295721Àk 1k 2B l À1s 2exp ½b ð2h 1þa cos h Þ b a cos hX1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2i b h 1ÞJ s ði b a Þ(þX1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2i b h 1Þ½sJ s ði b a ÞÀi b aJ s þ1ði b a Þ );ðl ¼2;3;...;1Þ:ð46Þð1þk 1k 2ÞA l s 2exp ðb a cos h 0Þf b a cos h 0H ð1Þs ðj a Þþ½sH ð1Þs ðj a ÞÀj aH ð1Þs þ1ðj a Þ g þk 1k 2B l s 2exp ðb a cos h 0Þf b a cos h 0H ð1Þs ði b a Þþ½sH ð1Þs ði b a ÞÀi b aH ð1Þs þ1ði b a Þ g¼Àð1þk 1k 2ÞA l À1s 1exp ½b ð2h 1Àa cos h 0Þ Àb a cos h 0X 1m ¼À1H ð1Þs Àm ð2j h 1ÞJ s ðj a Þ(þX 1m ¼À1H ð1Þs Àm ð2j h 1Þ½sJ s ðj a ÞÀj aJ s þ1ðj a Þ)Àk 1k 2B l À1s 1exp ½b ð2h 1Àa cos h 0Þ b a cos hX 1m ¼À1H ð1Þs Àm ð2i b h 1ÞJ s ði b a Þ(þX1m ¼À1H ð1Þs Àm ð2i b h 1Þ½sJ s ði b a ÞÀi b aJ s þ1ði b a Þ );ðl ¼2;3; (1)ð47Þk 1k 3B l s 1exp ðÀb a cos h ÞfÀb a cos h H ð1Þs ði b a Þþ½sH ð1Þs ði b a ÞÀi b aH ð1Þs þ1ði b a Þ gÀk 1C 1s 1½sJ s ði b a ÞÀi b aJ s þ1ði b a Þ ¼Àk 1k 3B l À1s 2exp ½b ð2h 1þa cos h Þ b a cos hX1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2i b h 1ÞJ s ði b a Þ(þX1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2i b h 1Þ½sJ s ði b a ÞÀi b aJ s þ1ði b a Þ );ðl ¼2;3;...;1Þ:ð48Þk 1k 3B l s 2exp ðb a cos h 0Þf b a cos h 0H ð1Þs ði b a Þþ½sH ð1Þs ði b a ÞÀi b aH ð1Þs þ1ði b a Þ gÀk 1C 1s 2½sJ s ði b a ÞÀi b aJ s þ1ði b a ÞÀk 1k 3B l À1s 1exp ½b ð2h 1Àa cos h 0Þ b a cos h 0X 1m ¼À1H ð1Þs Àm ð2i b h 1ÞJ n ði b a Þ(þX 1m ¼À1H ð1Þð2i b h 1Þ½sJ s ði b a ÞÀi b aJ s þ1ði b a Þs Àm);ðl ¼2;3;...;1Þ;ð49Þexp ðÀb a cos h Þf A l s 1H ð1Þs ðj a ÞþB l s 1H ð1Þs ði b a Þg ÀC l n 1J n ði b a Þ¼Àexp ½b ð2h 1þa cos h ÞÂA l À1s 2X 1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2j h 1ÞJ s ðj a ÞþB l À1s 2X 1m ¼À1ðÀ1Þs Àm H ð1Þs Àm ð2i b h 1ÞJ s ði b a Þ();ðl ¼2;3;...;1Þ:ð50Þexp ðb a cos h 0Þf A l s 2H ð1Þs ðj a ÞþB l s 2H ð1Þs ði b a Þg ÀC l s 2J s ði b a Þ¼Àexp ½b ð2h 1Àa cos h 0ÞÂA l À1s 1X 1m ¼À1H ð1Þs Àm ð2j h 1ÞJ s ðj a ÞþB l À1s 1X 1m ¼À1H ð1Þs Àm ð2i b h 1ÞJ s ði b a Þ();ðl ¼2;3;...;1Þ:ð51ÞEqs.(40)–(51)are the algebra equations determining the mode coefﬁcients A l n 1;A l n 2;B l n 1;B l n 2;C l n 1and C l n 2of the scatteredand refracted waves.5722X.-Q.Fang /International Journal of Solids and Structures 45(2008)5716–5729。

基于神经网络算法的复合材料板声发射源定位

经网络方法定位在中国的研究还比较少，本文使用一种相对成熟的神经网络算法一一误差反向传播算法
５０
导弹与航天运载技术
（ｃｒｐｇｔｎＢ）来实现复合材料板上的声发ＢａｋＰｏａａｉ，Ｐｏ
１神经网络原理定位的相关理论．２
作为神经网络训练的训练函数。隐函数个数的选取经过多次尝试后采用１，选取方法见文献［】０８。
１试验设计．３
试验所用的碳纤维编织板为４０ｍｉ ×２０ｍｍ，４ｌ４ｌ在上面画出一个３０ｍｍ×１０ｍｍ的区域，将４个传５６
拥有很多其他无损检测手段所不具备的优点，如实时
动态检测性、被动检测性等。理论上来讲，复合材料
在加载中出现的声发射信号可以包含关于复合材料损
拟合各向波速的工作，在形状复杂的试件表面难以实施；并且在应力波传播路径更加复杂的情况（如含多
值、振铃计数等；）算法的实现和计算坐标与实际坐ｄ标的对比。
１传感器的选择．１声发射信号的频率范围涵盖次声频、声频直到超
声频，从几个赫兹到几兆赫兹。通常声发射试验所采用的断铅信号的频率主要集中在１ＭＨｚ范围以内。
Ａｂｔａｔｅｅｂｒ－ａｑａｄｒｉｉｇｆｎｔｏｓａｐｉｄｔｏａｅｔｅａｏｓｉｍｉｓｏｏｒｅｏｏｏｉｌｔ，ｎｓｒｃ：ＬｖｎｅｇＭｒｕｒｔａｎｎｃｉｎｗａｐｌｅｏｌｃｔｈｃｕｔｃｅｓｉｎｓｕｃｎｃｍｐｓｔｐａｅｉｔｕｅｗｈｃｅａｒｖｎｉｆｗａｅｗａｈｓｎａｈｎｕｅｔｒＡｆｅｒｉｉｇｔｅｐｓｔｏｆｔｅＡＥｓｕｃｓｗｅｌｍａｐｄｔｉｈｔｒｉｉｇｔｍｅｏｖｓｃｏｅｓｔｅｉｐｔｖｃｏ￣ｔｒｔａｎｎ，ｈｏｉｎｏｏｒｅｉｌｈｉｈｐｅｏｔｅａｒｖｎｉｆｅｃｅｓｒｈｒｉｉｇｔｍｅｏａｈｓｎｏ．Ｂｅｉｅ，ａｍｏｅｐｅｉｅｗａｎａｔｉｉｇｔｅａｒｖｎｉ，ｗｈｃａｅｅｓｌｐｌｄｔｓｄｓｒｒｃｓｙｉｔｎｎｈｒｉｉｇｔａｍｅｉｈｃｎｂａｉａｐｉｏｙｅｅｇｎｅｉｇｉｉｃｓｅｅｅｉｓｅｄｏｅｃｎｅｔｏａｙｏｒｓｉｇｆｘｄｔｒｓｏｄｗｈｃｙｒｓｔｎｓｍｅｄｖａｉｎｏｎｉｅｒｎ，ｓｄｓｕｓｄｈｒｎｔａｆｈｏｖｎｉｎｌｔｗａｆｃｏｓｎｅｅｈｌ，ｉｈｍａｅｕｌｉｏｅｉｔｆｉｈｏ

Method for retrieving effective properties of locally resonant acoustic metamaterials

II. RETRIEVAL METHOD FOR THE We extend a method for retrieving effective material properties of electromagnetic materials31,32 to acoustic metamaterials. In this retrieval method, the effective refractive index n and impedance z are obtained from reﬂection and transmission coefﬁcients for a plane wave normally incident on the slab. The effective mass density and sound speed are then calculated from n and z. This technique allows the use of experimental measurements to obtain metamaterial effective properties. A schematic illustrating the retrieval method is shown in Fig. 1. The metamaterial ͓Fig. 1͑a͔͒ is replaced by a homogeneous ﬂuid slab of material ͓Fig. 1͑b͔͒ which provides the same amplitude and phase of reﬂection and transmission coefﬁcients. Effective properties are obtained by using an inverse technique ͓Fig. 1͑c͔͒, which is discussed in this section. Let us consider the reﬂection R and transmission T coefﬁcients for a plane wave incident on a liquid acoustic layer with density ␳2 and sound speed c2 placed between two different media with densities ␳1, ␳3 and sound speeds c1, c3 ͑Ref. 33͒: R= ͑Z1 + Z2͒͑Z2 − Z3͒e−2i␾ + ͑Z1 − Z2͒͑Z2 + Z3͒ , ͑Z1 + Z2͒͑Z2 − Z3͒e−2i␾ + ͑Z1 − Z2͒͑Z2 − Z3͒ ͑1͒

计算地震学

1.2 Equations of motion 1.3 Constitutive relations 1.4 Boundary-value problems for a self-gravitating and hydrostatically pre-stressed Earth
R. Wang
Introduction – 1
R. Wang
Introduction – 5
Question: There are 9 unknowns (3 displacement components + 6 stress components), but only 3 differential equations of motion. Other 6 equations or constraints are needed, that are given by the material law.
7th day:
Student report and discussion (2 hrs.)
R. Wang
Introduction – 0
1. Introduction to Geo-mechanical Boundary-value Problems
1.1 Displacement, strain and stress
1.4 Boundary-value problems for a self-gravitating and hydrostatically pre-stressed Earth Initial equilibrium
po(x) = o(x) o(x) 2 o(x) = –4G o(x) x O r uL(x,t) where o = density, o = gravitational potential (go = o), po = hydrostatic pressure, G = 6.6710-11 m3kg-1sec-2 (gravitational constant). The index o denotes the unperturbed initial state. Description of displacement of a continuum u = uL(x,t) = r(x,t) – x = uE(r,t) = r – x(r,t) --- Lagrangeian (material) description --- Eulerian (local) description uE(r,t) x(r,t) r(x,t)

基于石墨烯的波束可重构太赫兹天线

通过对天线的基本性能和可调特性系统地模拟和优化数值结果表明通过改变加在石墨烯导向器的偏压天线的主辐射波瓣角方位角可以在30150的范围内进行扫描并且具有非常快的调制速度和非常低的回波损耗
第 39 卷第 1 期 2020 年 2 月
文章编号：1001-9014（2020）01-0039-08
红外与毫米波学报
sion line，a metallic reflector，a metallic half-round micro patch，and three groups of monolayer-graphene-patch directors. Because the conductivities of the graphene patches can be electrically tuned，the radiation direction of the QYU antenna can be dynamically controlled by adjusting the bias voltages applied to the graphene patches. The basic properties and the tunable behaviors of the antenna are systemically simulated and optimized. It has very fast modulation speed and very low return loss. In particular，the numerical results show that the main radia⁃ tion lobe angle φ（azimuth angle）of the QYU antenna can be swept from 30° to 150° in θ =60°（zenith angle） plane with different configurations of bias voltages applied to the graphene-patch directors. The proposed antenna is very suitable for THz beam reconfigurable applications，such as phased array radar systems. Key words：beam reconfigurable antenna，graphene，patch antenna，Yagi-Uda antenna PACS：84. 40. Ba， 87. 50. U-

Improved accuracy for the Helmholtz equation in unbounded domains

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt.J.Numer.Meth.Engng2004;59:1963–1988(DOI:10.1002/nme.882)Improved accuracy for the Helmholtz equationin unbounded domainsEli Turkel1,∗,†,‡,Charbel Farhat2and Ulrich Hetmaniuk2,§1Department of Mathematics,Tel Aviv University,Tel Aviv,Israel2Department of Aerospace Sciences,University of Colorado at Boulder,Boulder,Colorado,U.S.A.SUMMARYBased on properties of the Helmholtz equation,we derive a new equation for an auxiliary variable.This reduces much of the oscillations of the solution leading to more accurate numerical approximations to the original putations conﬁrm the improved accuracy of the new models in both two and three dimensions.This also improves the accuracy when one wants the solution at neighbouring wavenumbers by using an expansion in k.We examine the accuracy for both waveguide and scattering problems as a function of k,h and the forcing mode l.The use of local absorbing boundary conditions is also examined as well as the location of the outer surface as functions of k.Connections with parabolic approximations are analysed.Copyright᭧2004John Wiley&Sons,Ltd.KEY WORDS:Helmholtz equation;preconditioning;unbounded domain;absorbing boundary conditions1.INTRODUCTIONPreconditioning is a technique where the equation is multiplied by an approximation to the solution operator.This(hopefully)yields a new equation with better conditioning.In classical preconditioning the purpose is to improve the convergence rate of an iterative solver(see, however,Reference[1]).In this study,we use an extension of preconditioning to improve the accuracy of a numerical approximation.The solutions of the Helmholtz equation are difﬁcult to calculate because they are highly oscillatory[2].By removing much of this oscillatory behaviour through the preconditioning the remaining problem is easier to treat on a coarse mesh.We stress that this preconditioning is physically motivated and so depends on the nature of the problem to be solved.We consider both waveguide and scattering problems.The preconditioning willCorrespondence to:Eli Turkel,Department of Mathematics,Tel Aviv University,Ramat Aviv Tel Aviv69978, Israel.†E-mail:turkel@math.tau.ac.il‡Work done while visiting the University of Colorado at Boulder.§Present Address:Computational and Mathematics Algorithms Department,Sandia National Laboratory,Albu-querque,New Mexico,U.S.A.Contract/grant sponsor:Ofﬁce of Naval Research;contract/grant number:N-00014-01-0356Received3February2003 Copyright᭧2004John Wiley&Sons,Ltd.Accepted29April20031964 E.TURKEL,C.FARHAT AND U.HETMANIUKbe different for each of these cases and also depends on the number of dimensions.For any given preconditioning one can construct an example that will not be improved or even made worse by the preconditioner.However,these counterexamples will be of the class for which the preconditioner is not appropriate.We compare the accuracy of linearﬁnite elements for the approximation of both the Helmholtz equation and the new formulation.We also check the effect on calculating the solution at neighbouring wave numbers.We consider the accuracy in a waveguide not only as a function of k and h but also the mode l of the forcing boundary function.We will also check this dependence for scattering problems.We also study the effect of local absorbing boundary conditions in both two and three dimensions.In addition,we evaluate the effect of the position of the outer boundary has on the accuracy of the approximation.2.ONE-DIMENSIONAL EQUATIONThe Helmholtz equation in one dimension reduces to the ordinary differential equationu xx+k2u=0(1) This has solutions of the form u=A e i kx+B e−i kx.We assume the waves move from left to right.The Sommerfeld radiation condition eliminates waves entering the domain from inﬁnity and so B=0.Thus,we are led to deﬁning a new variable u=v e i kx.Thus,for this simple problem the only outgoing solution is v=constant.v solves the equationv xx+2i kv x=0(2) The Sommerfeld radiation conditionu (x)−i ku(x)=o(1),x→∞is replaced byv (x)=o(1),x→∞Note that(2)has a solution of the form v=A+B e−2i kx and so has oscillatory behaviour. B=0only because of the Sommerfeld radiation condition.Hence,the improved smoothness of v is due to the combination of the equation and the farﬁeld boundary condition and so is a global property.Hence,an analysis of the behaviour of methods,e.g.multigrid,for this problem must use a global analysis and not just a local modal analysis that ignores the far ﬁeld boundary condition.3.W A VEGUIDE PROBLEMWe consider the two-dimensional problemu xx+u yy+k2u=0,0<x<∞,0<y<1(3) with boundary conditions u(0,y)=f(y),u y(x,0)=u y(x,1)=0and u outgoing at inﬁnity.The Neumann conditions at the top and bottom can easily be replaced by other boundary conditions. Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–1988HELMHOLTZ EQUATION IN UNBOUNDED DOMAINS 1965We use Neumann conditions on the side walls because they are a natural boundary condition for the ﬁnite element method.The extension to three-dimensional Cartesian co-ordinates is straightforward.By separation of variables the solution to (3)isu(x,y)= l<k a l e i √k 2−l 2x cos (ly)+ l>kb l e −√l 2−k 2x cos (ly),l is a positive integer multiple of (4)The ﬁrst sum represents travelling waves and the second sum evanescent waves.By linearity it is sufﬁcient to consider the inﬂow condition u(0,y)=cos (ly).We assume that the main propagation of waves is in the positive x direction and that the waves in the y direction are principally standing waves.As in the one-dimensional case we introduce the new variable v byu =v e i kx(5)The equation for v isv xx +v yy +2i kv x =0,0<x<∞,0<y<1(6)with boundary conditions v(0,y)=f (y),v y (x,0)=v y (x,1)=0and v e i kx outgoing at inﬁnity.The outgoing condition is imposed at a ﬁnite location x =L and every y byv x (L,y)−i ( k 2−l 2−k)v(L,y)=0for each travelling mode and a similar condition for the evanescent modes.If l =0this reduces to a homogeneous Neumann condition as in one dimension.Hence,for the lowest mode l =0the solution v is constant.For a general travelling mode l we have v =e i (√1−l 2/k 2−1)kx cos (ly).If l/k is small then v is a much smoother function that u .For evanescent modes the new variable v does not help and one should use instead u =v e −kx .Since the evanescent modes decay exponentially therefore,in many problems that contain both travelling and evanescent modes it is more important to treat the travelling modes correctly than the evanescent modes.If we eliminate the higher derivative v xx in (6)because it is small compared with 2i kv x then we recover the parabolic equation given by [3,4],2i kv x +v yy =0,0<x<∞,0<y<1(7)Equation (6)has the disadvantage that one is solving an elliptic equation rather than a parabolic equation which can be marched in x .However,it has the advantage that no terms are ignored and so is more accurate than the parabolic approximation.It is also more accurate (on the numerical level)than the original Helmholtz equation since we are solving for the smooth por-tion of the solution.Each mode of (6)depends on e i (√k 2−l 2−k)x cos (ly).So v xx is proportional to (√k 2−l 2−k)2which is small when l/k and k are small.The parabolic equation (7)has the Sommerfeld radiation condition built in.This is an advantage since we do not need to implement an absorbing boundary condition at the exit.However,it cannot handle backscat-tering.The full Equation (6)accounts for backscattering though it will be less accurate since the wavenumber for the backscattered wave is essentially doubled.Copyright ᭧2004John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng 2004;59:1963–19881966 E.TURKEL,C.FARHAT AND U.HETMANIUKThe solution to (7)is v =e −i (l 2/2k)x cos (ly)compared withe i (√1−(l −1)kx cos (ly)for (6).Expanding the square root,we see that the parabolized equation has a phase error,for the l th mode,given by ( 1−l 2/k 2−1)k +l 2/2k −l 4/8k 3.The parabolic approximation will be discussed in more detail in the last section.3.1.Perfectly matched layers (PML)For the Helmholtz equation one needs to impose a boundary condition at each surface.At the exit to the waveguide one needs to simulate the Sommerfeld radiation condition that no waves enter the physical domain from inﬁnity.A popular way is to add an artiﬁcial layer and to solve the PML equations within this layer.This was ﬁrst proposed by Berenger [5]for the time-dependent Maxwell equations.This can also be applied to the Helmholtz equation (see for example Reference [6]).This is given by *1*u +S *2u +Sk 2u =0,S(x)=1+ (x)i k (8)=0inside the physical waveguide so we recover the Helmholtz equation.In the artiﬁcial layer is an increasing function of x starting at zero to match the interior and reaching max at the ually is chosen as a polynomial in the distance from the end of the waveguide.We can generalize our previous approach by takingu =v ei k x 0S (9)As before this removes the oscillatory behaviour.v satisﬁes the equation **x1S *v *x +S *2v *y 2+2i kS *v *x =0(10)3.2.Bands of kIn Reference [7]Djellouli,Farhat and Tezaur present a method to solve the Helmholtz equation for a wave number k + k given the calculated solution at wave number k .The basic idea is to expand u(k + k)in a Taylor series in k .The higher derivatives of u with respect to k obey a Helmholtz equation with a forcing term that depends on lower order derivatives.Assuming an LU decomposition of the Helmholtz operator has been formed these subsequent Helmholtz equations are solved very efﬁciently.This Taylor series does not converge very well due to numerical inaccuracies in approximating the Helmholtz equation for the higher order derivatives.This series can be accelerated by using a Padéseries or the epsilon algorithm.We will study the effect of using (5)and (6)on improving the convergence and accuracy of these expansions.In Reference [7]they were mainly concerned with the efﬁciency of ﬁnding solutions for a band of wavenumbers given the solution at a given wavenumber.It is well known (see e.g.[8–10])that the accuracy of second-order (in L 2)ﬁnite element method depends on k 3h 2.Hence,the solution for high wavenumbers can be prohibitively expensive since it requires an extremely ﬁne mesh.We will investigate the use of (5)on the pollution error.We emphasize that this is not done for all possible solutions of the Helmholtz equation but only for those that are physically relevant,e.g.the waveguide and scattering problems.Copyright ᭧2004John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng 2004;59:1963–1988HELMHOLTZ EQUATION IN UNBOUNDED DOMAINS1967 Consider the two-dimensional waveguide problem for a single mode,i.e.Equation(3)with f(y)=cos(ly),l an integer multiple of .Deﬁne u(n)=(1/n!)(*n u/*k n).Then u(n)satisﬁes the equation[7]u(n)xx+u(n)yy+k2u(n)=−2ku(n−1)−u(n−2),n 1u(−1)=0(11) with boundary conditionsu(n)(0,y)=0,u(n)y(x,0)=u(n)y(x,1)=0and forﬁnite x and every yu(n)x(x,y)−ik2−l2u(n)(x,y)=inm=11m!*m*k m(k2−l2)u(n−m)(x,y)Similarly,we deﬁne v(n)=(1/n!)(*n v/*k n).This satisﬁesv(n)xx+v(n)yy+2i kv(n)x=−2i v(n−1)x,n 1(12) with boundary conditionsv(n)(0,y)=0,v(n)y(x,0)=v(n)y(x,1)=0and forﬁnite x and every yv(n)x(x,y)−i(k2−l2−k)v(n)(x,y)=inm=11m!*m*k m(k2−l2−k)v(n−m)(x,y)To avoid the large sums in the Sommerfeld radiation condition for the higher modes we keep only two terms in the sum,i.e.those that depend on u(n−1)and u(n−2)and similarly for v.3.3.ImplementationWe now consider three ways to calculate u(k+ k).Each has advantages and disadvantages. Theﬁrst way is to solve(6)by one’s favourite method.This requires an approximation for theﬁrst derivative.We note that i v x is a self-adjoint operator.Hence,if done appropriately the approximation to the equation is Hermitian.However,the Sommerfeld radiation condition is complex symmetric and so the total problem of equation plus boundary condition has no symmetry.As we will see theﬁrst BGT approximation to the Sommerfeld condition for scattering leads to a problem which is Hermitian symmetric for the Dirichlet boundary valued problem.This alternative has the major advantage that we expect the basic approximation for v to be much more accurate than that for u(see Section5).When we consider the solution for nearby wavenumbers we have u(k+ k)=v(k+ k)e i(k+ k)x. Hence,once we use the expansion toﬁnd v(k+ k)we immediately know u(k+ k).In Reference[7]they solved the waveguide problem only for the lowest mode l=0,and so u=e i kx and v=1.This implies that all the higher derivatives of v are zero and so the Taylor series collapses to theﬁrst term which gives the exact answer.To avoid this triviality,we shall show results for higher modes.Nevertheless,as long as l/k>1we expect accurate results and a fast convergence.Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–19881968 E.TURKEL,C.FARHAT AND U.HETMANIUKNevertheless,this method has several disadvantages.First,it requires new coding.Second,the symmetry may be lost due to either the boundary conditions or difﬁculties with the ap-proximation to the ﬁrst derivative.Hence,we present an alternative to the direct solution of (6).The second way is to solve the original equation for u and its derivatives.We then use (5)to calculate v and its derivatives in order to get a Taylor series for v(k + k).This has the advantage that the approximation to the partial differential equations need not be changed.This implies that basic approximation for u is not being improved as it is when solving (6).It has the further disadvantage that the Taylor series for v(k + k)is polluted by the larger discretization errors in the derivatives of u .As we shall see in the results section this option gives much better results than solving (11)and a Taylor series for u but not as good as solving (12).We assume we are given u and u (n)=(1/n !)(*n u/*k n )and wish to ﬁnd v (n)=(1/n !)(*n /*k n )(u e −i kx ).One can evaluate this directly by the binomial formula.This becomes complicated for high derivatives.In addition directly evaluating the factorial is extremely inaccurate for large n .Instead,we use a recursion formula to ﬁnd v (n).We shall do this in two stages.First we ﬁnd the n th derivative of v and then account for the factorial.We assume we are givenv(x ;k)=F (x,k)u(x ;k)and G(x,k)=*F *kF We will consider in this study three special casesCartesian :F (x,k)=e −i kx ,G(x,k)=−i x Cylindrical :F (r,k)=1H (1)0(kr),G(r,k)=H (1)1(kr)H (1)0(kr)r −i r +12k (13)Spherical :F (r,k)=kr e −i kr ,G(r,k)=−i r +1k We also consider three simpliﬁcations of these Cartesian :F (x,k)=e −i kx ,G(x,k)=−i xCylindrical :F (r,k)=√r e −i kr ,G(r,k)=−i r (14)Spherical :F (r,k)=r e −i kr ,G(r,k)=−i r Note that for (14)G is independent of k .In the following,we ignore the derivatives of G with respect to k either because (14)is used or else we use the more accurate (13)but ignore the higher derivatives as being negligible for large k .The Cartesian case is the same in both sets.We then haveu(k + k)=v(k + k)F (x,k + k)=F (x,k)F (x,k + k)N n =0( k)n 1F (x,k)n !*n v *k nCopyright ᭧2004John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng 2004;59:1963–1988HELMHOLTZ EQUATION IN UNBOUNDED DOMAINS1969 Deﬁne w n by w0=u and*v *k =F(x,k)G(x,k)w0+*w0*k=F(x,k)w1*n v *k n =F(x,k)G(x,k)w n−1+*w n−1*k=F(x,k)w n(15)Sow1=G(x,k)w0+*w01 n!*n w1*k n=G(x,k)1n!*n w0*k n+(n+1)1(n+1)!*n+1w0*k n+1(16)More generally deﬁnew(n)0=1n!*n u *k nw(n)p=1n!*n w p*k n,n=0,...,N−p,p=1,...,N(17)Then from(16)and(17)we havew(n)p=G(x,k)w(n)p−1+(n+1)w(n+1)p−1,n=0,...,N−p,p=1,...,NFinally,deﬁne v(n)p=w (n)p/p!.Thenv(n)p=G(x,k)v(n)p−1+(n+1)v(n+1)p−1,n=0,...,N−p,p=1,...,N(18)and1 p!*p v*k p=F(x,k)v(0)p(19)Note that from(18)v(0)p is computed recursively starting with v(p).Sou(k+ k)=F(x,k)F(x,k+ k)Nn=0( k)n1F(x,k)n!*n v*k n=F(x,k)F(x,k+ k)Nn=0( k)n v(0)n.(20)The third alternative is to utilize the difference formula for v to construct a new difference formula for u.We shall illustrate this for the one-dimensional equation.A standard second-order difference(or linearﬁnite element)approximation to(2)isv j+1−2v j+v j−1+i kh(v j+1−v j−1)=0Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–19881970 E.TURKEL,C.FARHAT AND U.HETMANIUKSubstituting v j=e−i jkh u j and canceling the common factor of e−i jkh we gete−i kh u j+1−2u j+e i kh u j−1+i kh(e−i kh u j+1−e i kh u j−1)=0oru j+1−2u j+u j−1+[e−i kh(1+i kh)−1]u j+1+[e i kh(1−i kh)−1]u j−1=0The basic approximation to the differential equation without considering the boundariesis again Hermitian symmetric.A Taylor series expansion shows that this is a second-orderapproximation to the Helmholtz equation.By construction the formula is exact for u=e i kx.However,it is not clear if this can be represented as a Petrov–Galerkin method since we haveonly manipulated u and v and not the basis functions.As with theﬁrst approach this requiresrecoding the approximation to the partial differential equation.The Sommerfeld condition willbe simpliﬁed as it was for theﬁrst approach.However,now a Neumann boundary conditionat the inlet or scatterer presents no difﬁculty as it can be either a natural boundary conditionwithin aﬁnite element approach or any standard approximation to the normal derivative fortheﬁnite difference approach.3.4.Dispersion analysisWe have already shown that(6)is exact for theﬁrst mode,i.e.for one-dimensional wavepropagation.We next investigate the behaviour of the various schemes for higher modes.Ratherthan have the modes excited by the boundary conditions it is easier to analyse the differentialequation with a forcing term.In this case,the speed of the numerical approximation isﬁxed bythe forcing mode and the only error can be the amplitude of the wave not in its phase.Whenwe consider the sum of waves the concept of phase is not clear.Consider u=A1e i k1x+A2e i k2x by varying the amplitudes of A1and A2one change the phase of u between k1and k2.Hence,amplitude and phase are distinct entities only for a single wave but not for a sum of waves.We are interested in the numerical approximation toU xx+k2U=(k2−m2)e i mx(21) Boundary conditions are added so that the solution is U=A e i mx with A=1.For a travelling wave we assume k>m.By comparison with(4)we choose m2=k2−l2where l is the mode number of the inlet boundary condition.We begin with a second-orderﬁnite difference approximation to the forced Helmholtz equa-tion.u j+1−2u j+u j−1+k2h2u j=h2(k2−m2)e i mjh(22) The solution is of the form u=A h e i mjh.Substituting this into(22)we getA h=(k2−m2)h2k2h2−2(1−cos(mh))=l2k2−2h2(1−cos(√k2−l2h))expanding the cosine for small kh we haveA h∼(k2−m2)h2(k2−m2)h2+m4h4/12=11+(k2−l2)2h2/12l2Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–1988HELMHOLTZ EQUATION IN UNBOUNDED DOMAINS1971 At l=k we have A h=1and there is no error.As l decreases A h decreases and the error in the amplitude increases.We next consider the equation for v=u e−i kx.v j+1−2v j+v j−1+i kh(v j+1−v j−1)=−h2(m2+2km)e i mjh(23)Now m=√k2−l2−k.The solution is again of the form v=A h e i mjh.Substituting this into(23)we getA h=(m2+2km)h22(1−cos(mh))+2kh sin(mh)expanding the sine and cosine we haveA−A h∼1−(m2+2km)h2(m2+2km)h2−m3h4(m+4k)/12=1−11−m3(m+4k)h212(m2+2km)m3(m+4k)h212(m2+2km)=−m3(m+4k)h212l2Now when l=k then m=0and A h=A.The original algorithm is less accurate at the low modes compared with high modes while the v equation is most accurate at low modes.4.SCATTERINGWe next consider the three-dimensional scattering around some body.The two-dimensional problem will be described later.DeﬁneB( , )v=1sin **sin*v*+1sin2*2v* 2(24)Then the Helmholtz equation in spherical co-ordinates is given by*r2*u+1sin*sin*u+1sin*2u+r2k2u=0B1u=*u*r−i k−1ru=O1r3as r→∞(25)We have expressed the Sommerfeld radiation condition in terms of theﬁrst BGT approxima-tion[11].Deﬁning a new variable s=kr then both the Helmholtz equation and the Sommerfeld radiation condition are independent of k in the s co-ordinate,i.e.k directly affects the solution only through the r variable.Another way that k can enter the solution is through the boundary condition on the scatterer for the scattered wave.So we deﬁne a new variable given by the fundamental solution(see also Reference[12])u=v e i krkr(26) Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–19881972 E.TURKEL,C.FARHAT AND U.HETMANIUKWe stress that this is based on spherical waves rather than the plane waves used for the wave guide.For general conﬁgurations one can replace the spherical distance r by an ellipsoidal or spheroidal co-ordinate(see e.g.Reference[13]).Li et al.[14]consider a similar transformation without the denominator in the context of inﬁnite elements.However,to fully account for the fundamental solution and not just the oscillatory portion it is more accurate to include the denominator1/kr in the transformation.As an example of this change of variables consider a plane wave u inc(x)=e i kx cos impacting on a sphere(either hard or soft)of radius a.The exact solution for the scattered wave along the z-axis is given by[15,16]u=−∞n=0i n(2n+1)a n h(1)n(kr)P n(cos )(27)for a soft sphere a n=j n(ka)/h(1)n(ka)for a hard sphere a n=j n(ka)/h(1) n(ka).The spherical Hankel functions j n(kr)and h n(kr)both have the forme i kr krnm=0b nm(kr)mv satisﬁes the equation(see(24))v+2i k−1r*v*r=*2v*r2+2i k*v*r+1r2B( , )v=0˜B1v=*v=0at outer boundary(28)For the function v we get the functional formv=1k ∞n=0b n(kr)nF n( , )=b0kF0( , )+···F n independent of k(29)i.e.v has no oscillatory part that depends explicitly on k just terms that decay in r.The number of signiﬁcant terms depends linearly on the wavenumber k.Hence,we hope that a numerical approximation to this problem for v should require a grid that only depends on the points per wavelength.For other bodies there might be some dependence greater than linear in k in the and directions in the nearﬁeld but not the farﬁeld.Outside a sphere surroundingthe scatterer one can prove[16]u(x)=∞n=0nm=−na nm h(1)n(kr)Y m n(ˆx)converges absolutely and uniformly on compact subsets.The approximation to the equation is Hermitian whenﬁnite differences are used.The properties of aﬁnite element approximation depends on the details of the integration formulae.Hence,for a Dirichlet condition at the scatterer the total problem for v can be Hermitian symmetric.Instead of theﬁrst BGT radiation Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–1988condition,we can improve this to BGT-2B2u=*2u*r2+4r*u*r+2r2u=0˜B2v=*2v*r2+2r*v*r=0(30)The second derivative with respect to r is eliminated by using the differential -ing(28)we can rewrite this as*v=−12r(1−i kr)B( , )v(31)This destroys the Hermitian symmetry of the matrix.The deﬁnition of the farﬁeld pattern isu(x)=e i krru∞xr+O1rSince v=e i kr/kr this translates tokv(x)=u∞ x+O1Using(29)we can improve this tokv+kr *v*r=u∞xr+O1k2r2kv+kr *v*r+k2**rr2*v*r=kv+2kr*v*r+kr22*2v*r2=u∞xr+O1k3r34.1.Two-dimensional scatteringIn polar co-ordinates the Helmholtz equation is given by1 r **rr*u*r+1r2*2u* 2+k2u=0(32)andu(x)=e i kr√ru∞xr+O1rIn two dimensions(26)is replaced byu(r, )=H(1)0(kr)v(r, )(33) Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–1988Then we havev−2k H(1)1(kr)H(1)0(kr)*v*r=0(34)However,−2k H(1)1(kr)H(1)0(kr)+1r=2i k1+18k2r2+O1k4r4So we can approximate(34)asv+2i k−1r+18k2r2v=*2v*r2+2i k1+18k2r2*v*r+1r2*2v* 2=0We will drop the1/8k2r2term for all the results and so solve*2v *r2+2i k*v*r+1r2*2v* 2=0(35)In two dimensions theﬁrst two BGT farﬁeld boundary conditions are(see References[11,17])B1u=*u*r−i k−12ru=0B2u=2i k−1*u+1*2u*+2k2+3i k−34ru=0(36)We can approximate(33)byu(r, )=e i kr√krv(r, )(37) Then(36)becomes˜B1v=*v*r=0˜B2v=2i k−1r*v*r+1r2*2v* 2−14r2v=0(38)We can get a more accurate farﬁeld boundary condition by using an expansion in terms of Hankel functions rather than inverse powers of r.This is equivalent(through second-order) with a local approximation of the DtN operator[18].The farﬁeld generalization of(36)isB1u=*u*r−kH 0(kr)H0(kr)u=0B2u=*u*r−kH 0(kr)H0(kr)u+H 0(kr)H0(kr)−H 1(kr)H1(kr)*2u* 2=0(39)Copyright᭧2004John Wiley&Sons,Ltd.Int.J.Numer.Meth.Engng2004;59:1963–1988where H 0(kr)=−H 1(kr)and H 1(kr)=0.5(H 0(kr)−H 2(kr)).Translating this to v deﬁned by (33)we get˜B1v =*v *r=0˜B 2v =*v *r−kH 0(kr)H 0(kr)−H 1(kr)H 1(kr) *2v * 2=0(40)For three dimensions there exist similar formulae in terms of spherical Hankel functions.Thompson and Pinsky [19]and later Harari and Djellouli [20]have shown that through second order these are identical to the BGT far ﬁeld boundary conditions (30).5.RESULTS5.1.WaveguideWe solve (3)by a linear ﬁnite element method.This is equivalent to13(u i,j +1+u i,j +1+u i +1,j −1+u i +1,j +u i +1,j +1+u i −1,j −1+u i −1,j +u i −1,j +1−8u i,j )+k 2h 236(u i +1,j −1+u i +1,j +1+u i −1,j −1+u i −1,j +1+4u i,j +1+4u i,j +1+4u i +1,j +4u i −1,j +16u i,j )=0The accuracy of a linear ﬁnite element method for the Helmholtz equation depends on k 3h 2.However,for the wave guide problem there is also the mode number l ,and the solution behavesas e i k √1−(l 2/k 2)x sin (ly).It is not clear from ﬁnite element estimates how the accuracy depends on l ,see however,[21].In Reference [22]it is shown how one can improve the accuracy by replacing the k 2h 2that appears in the linear element algorithm by 2(1−cos (kh)).We shall refer to this as the stabilized method.We also solve for v =u e −i kx and translate back to u .We consider 0<x,y<1with homogeneous Neumann conditions at top and bottom,u =cos (l y)at x =0and a radiation condition,*u/*x +i ku =0at x =1.In Table I we display the absolute L 2error for the wave guide with various values of k,h,l .Comparing k =6,N =20,l =1with k =24,N =160,l =1and l =4we see that the error is a function of k ,h and l and not just k 3h 2.This is most obvious for the stabilized method and the transformation to v variables which are exact for l =0and so very accurate when l/k >1.For l ﬁxed and increasing k ,and constant kh the error decreases!For the case where a general function f (y)is given at the inlet it will depend on the Fourier expansion of f .If the Fourier components of f are band limited we expect the accuracy to increase as we increase k relative to the largest signiﬁcant component in the Fourier expansion of f .However,if f has many signiﬁcant modes with high l then as we increase k and N more Fourier components will become signiﬁcant and the error gets larger as we increase k .For linear elements the error for l =1is larger than for l =4as is shown by the dispersion analysis.We see that stabilization helps but (5)and (6)is by far the best.Copyright ᭧2004John Wiley &Sons,Ltd.Int.J.Numer.Meth.Engng 2004;59:1963–1988。

A numerical scheme for axisymmetric elastic waves in solids

*This paper is dedicated * Corresponding author. to professor Alan Jeffrey, University of Newcastle Upon Tyne, on the occasion of his 65th birthday
0165-2125/95/$09.50 @ 1995 Elsevier Science B.V. All rights reserved SSDIO165-2125(94)00046-8
116
X. Lin, J. Balltnann/Wave
Motion 21 (1995) 115-126
Nevertheless, the existing numerical schemes dealing with hyperbolic systems with a source term seem not yet as well-developed as those for the systems without a source term. A widely used method is the time splitting technique which alternately solves a system of conservation laws without any source term and a system of ordinary differential equations modeling the source effect. However, it seems that this technique can produce misleading results, see Westenberger and Ballmann [ 91. For one-dimensional problems, some promising efforts were made by Glimm et al. [ lo], Glaz and Liu [ 111 and Roe [ 121. In this paper, we first propose an explicit finite difference scheme for the numerical integration of hyperbolic PDEs with a source term. Then, this scheme is applied to solve axisymmetric problems of elastic waves, e.g. the wave propagation in a half space due to an impact on a circular area of the boundary, the dynamic stress intensity factor at the tip of a penny-shaped crack and the focussing of stress waves caused by boundary effects. As far as possible the results will be compared with those of other authors in order to show the efficiency of the numerical scheme or to achieve a clearer explanation for the considered problem. 2. The governing equations and the numerical scheme Let r and z be the cylindrical coordinates and t the time. The elastodynamic equations for isotropic, linear elastic solids undergoing torsionless axisymmetric deformations can be written in the following form: au 5 aa, 87 =ar+az+~, au ar +a$+& au au cy$+ff-jy+;’ au ar aa, --+x+;, 57ar

基于伴随仿真的偏振复用超构透镜

Abstract: Polarization imaging technology has important application value in target detection, biomedicine, and other fields, but traditional polarization imaging systems suffer from complex structures, large volume,
基于伴随仿真的偏振复用超构透镜刘永健张飞谢婷蒲明博赵泽宇李雄马晓亮沈同圣罗先刚 Polarization-multiplexed metalens enabled by adjoint optimization LIU Yong-jian, ZHANG Fei, XIE Ting, PU Ming-bo, ZHAO Ze-yu, LI Xiong, MA Xiao-liang, SHEN Tong-sheng, LUO Xian-gang
引用本文: 刘永健,张飞,谢婷,蒲明博,赵泽宇,李雄,马晓亮,沈同圣,罗先刚. 基于伴随仿真的偏振复用超构透镜[J]. 中国光学, 2021, 14(4): 754-763. doi: 10.37188/CO.2021-0035 LIU Yong-jian, ZHANG Fei, XIE Ting, PU Ming-bo, ZHAO Ze-yu, LI Xiong, MA Xiao-liang, SHEN Tong-sheng, LUO Xiangang. Polarization-multiplexed metalens enabled by adjoint optimization[J]. Chinese Optics, 2021, 14(4): 754-763. doi: 10.37188/CO.2021-0035

Correction for the in

CWP-668Correction for the inﬂuence of velocity lenses on nonhyperbolic moveout inversion for VTI mediaMamoru Takanashi 1,2&Ilya Tsvankin 11Center for Wave Phenomena,Geophysics Department,Colorado School of Mines,Golden,Colorado 804012Japan Oil,Gas and Metals National Corporation,Chiba,JapanABSTRACTNonhyperbolic moveout analysis plays an increasingly important role in velocity model building because it provides valuable information for anisotropic param-eter estimation.However,lateral heterogeneity associated with stratigraphic lenses such as channels and reefs can signiﬁcantly distort the moveout parame-ters,even when the structure is relatively simple.Here,we discuss nonhyperbolic moveout inversion for 2D models that include a low-velocity isotropic lens embedded in a VTI (transversely isotropic with a ver-tical symmetry axis)medium.Synthetic tests demonstrate that a lens can cause substantial,laterally varying errors in the normal-moveout velocity (V nmo )and the anellipticity parameter η.The area inﬂuenced by the lens can be identiﬁed using the residual moveout after the nonhyperbolic moveout correction and the dependence of errors in V nmo and ηon spreadlength.To remove lens-induced traveltime distortions from prestack data,we propose an algorithm that involves estimation of the incidence angle of the ray passing through the lens for each recorded ing the velocity-independent layer-stripping method of Dewangan and Tsvankin,we compute the lens-induced traveltime shift from the zero-oﬀset time distortion (i.e.,from “pull-up”or “push-down”anomalies).Synthetic tests demonstrate that this algorithm substantially reduces the errors in the eﬀective and interval parameters V nmo and η.The corrected traces and reconstructed “background”values of V nmo and ηare suitable for anisotropic time imaging and producing a high-quality stack.Key words:P-waves,anisotropy,transverse isotropy,velocity analysis,lat-eral heterogeneity,velocity lenses,nonhyperbolic moveout inversion,traveltime shifts1INTRODUCTIONKinematics of P-wave propagation in VTI (transversely isotropic with a vertical symmetry axis)media are gov-erned by the vertical velocity V 0and the Thomsen pa-rameters and δ(Tsvankin &Thomsen,1994).P-wave reﬂection traveltime in laterally homogeneous VTI me-dia above a horizontal or dipping reﬂector depends only on the normal moveout velocity V nmo and the anel-lipticity parameter η(Alkhalifah &Tsvankin,1995;Tsvankin,2005):V nmo =V 0√1+2δ,(1)η=−δ1+2δ.(2)The parameters V nmo and η,which control all P-wavetime-processing steps,can be obtained from nonhyper-bolic moveout or dip-dependent NMO velocity.In par-ticular,the nonhyperbolic moveout equation introduced by Alkhalifah &Tsvankin (1995)and its extension for layered media (Alkhalifah,1997;Grechka &Tsvankin,1998;Tsvankin,2005)have been widely used for esti-mating V nmo and ηand building anisotropic velocity models.Nonhyperbolic moveout analysis is performed un-der the assumption that the overburden is laterally ho-mogeneous on the scale of spreadlength.However,even384M.Takanashi&I.Tsvankingentle structures often contain small-thickness lenses (such as channels and carbonate reefs),whose width is smaller than the spreadlength(Armstrong et al.,2001; Fujimoto et al.,2007;Takanashi et al.,2008;Jenner, 2009;see Figure1).For isotropic media,lateral hetero-geneity of this type has been recognized as one of the sources of the diﬀerence between the moveout and true medium velocities(Al-Chalabi,1979;Lynn&Claer-bout,1982;Toldi,1989;Blias,2009).Such lens-induced errors in V nmo lead to misties between seismic and well data(Fujimoto et al.,2007).Although the moveout parameters(especiallyη) were shown to be sensitive to correlated traveltime er-rors(Grechka&Tsvankin,1998),overburden hetero-geneity is seldom taken into account in nonhyperbolic moveout inversion.Grechka(1998)shows analytically that a constant lateral velocity gradient does not distort the estimates of V nmo andη,if anisotropy and lateral heterogeneity are weak.The second and fourth hori-zontal velocity derivatives,however,can cause errors in V nmo andη.Still,Grechka’s(1998)results are limited to a single horizontal layer and cannot be directly applied to models with thin lenses.Recently,isotropic traveltime tomography has been used to estimate the velocity inside the lens and re-move the lens-induced velocity errors(Fujimoto et al., 2007;Fruehn et al.,2008).These case studies show the importance of integrating seismic and geologic in-formation and understanding the relationship between the overburden heterogeneity and velocity errors.In principle,laterally varying anisotropy parameters can be estimated from anisotropic reﬂection tomography (e.g.Woodward et al.,2008).However,if the lens lo-cation is unknown,lens-induced traveltimes shifts can hinder accurate parameter estimation on the scale of spreadlength.Here,we study the inﬂuence of velocity lenses on nonhyperbolic moveout inversion for2D VTI models.To analyze lens-induced distortions of reﬂection data,we performﬁnite-diﬀerence modeling and apply moveout inversion using the Alkhalifah-Tsvankin(1995)nonhy-perbolic equation.We show that even a relatively thin velocity lens may cause pronounced errors in the move-out parameters V nmo andηand describe several crite-ria that can help identify range of common-midpoint (CMP)locations,for which reﬂected rays cross the lens.To remove lens-induced traveltime shifts,we pro-pose a correction algorithm designed for gently dip-ping anisotropic layers.Synthetic tests demonstrate that this algorithm suppresses lens-related distortions on the stacked section and substantially reduces errors in the eﬀective and interval parameters V nmo andη.Figure1.Time-migrated section from the central North Sea (after Armstrong et al.,2001).Amplitude anomalies at the bottom of the channel-like structures(arrows)and pull-up anomalies below the structures(inside the rectangles)indi-cate the presence of lateral heterogeneity associated with the channelﬁlls.Pull-up and push-down anomalies caused by high and low velocities,respectively,in channels or carbon-ate reefs are also observed in other hydrocarbon-producing regions,such as the Middle East and Northwest Australia. 2DISTORTIONS CAUSED BY VELOCITY LENSESTo generate synthetic data,we performﬁnite-diﬀerence simulations(using Seismic Unix code suea2df;Juhlin, 1995)and ray tracing for2D models that include a low-velocity isotropic lens inside a VTI layer.The parame-ters V nmo andηare estimated from nonhyperbolic move-out inversion based on the Alkhalifah-Tsvankin(1995) equation:t2=t20+x2V2nmo−2ηx4V2nmo[t2V2nmo+(1+2η)x2],(3) where t is the P-wave traveltime as a function of the oﬀset x and t0is the zero-oﬀset time.Equation3can be applied to layered VTI media with the eﬀective pa-rameters given by(Tsvankin,2005):V2nmo(N)=1NXi=1(V(i)nmo)2t(i),(4)η(N)=11nmo0"NXi=1(V(i)nmo)4(1+8η(i))t(i)#−1ﬀ,(5)where t(i),V(i)nmo,andη(i)are the interval values,and N is the number of layers.Although the Alkhalifah-Tsvankin equation pro-vides a good approximation for P-wave moveout in VTI media,the estimates ofηare sensitive to correlated trav-eltime errors because of the tradeoﬀbetweenηand V nmoCorrection for lenses in nonhyperbolic inversion385Figure 2.Single VTI layer with an isotropic lens.The lens velocity is3km/s;the background parameters are V0=4km/s,δ=0.07and =0.16.Points A,B and C correspond to CMP locations discussed in the text.The test is performed for a spreadlength of4km;the target depth is 2km.(Grechka&Tsvankin,1998).In our model,such errors are caused by an isotropic velocity lens in the overbur-den.2.1Single-layer modelFirst,we consider a rectangular lens embedded in a ho-mogeneous VTI layer(Figure2).The section in Fig-ure3is computed by aﬁnite-diﬀerence algorithm for common-midpoint(CMP)gathers outside the lens(lo-cation A)and at the center of the lens(location B). Whereas the lens does not distort traveltimes at loca-tion A,it causes a near-oﬀset time delay of17ms and waveform distortions(related to the inﬂuence of the side and edges of the lens)in the mid-oﬀset range at location B.Using equation3,weﬁnd the best-ﬁt V nmo andηfor the target reﬂector from a2D semblance scan;for com-parison,we also perform conventional hyperbolic move-out inversion(Figure4).The NMO velocity estimated from the nonhyperbolic equation at location A is close to the analytic value.At location B,however,V nmo is about10%greater,although the exact eﬀective NMO velocity should decrease by2%due to the low veloc-ity inside the lens.At a CMP location near the edge of the lens(location C),V nmo is7%smaller than the exact value.Interestingly,nonhyperbolic moveout inver-sion produces an error in V nmo,which is two times larger than that obtained from hyperbolic moveout analysis.The reason for the lens-induced distortion in V nmo is described in Al-Chalabi(1979)and Biondi(2006) (Figures5a,b).Near-oﬀset rays at location B pass twice through the lens,while far-oﬀset rays miss the lens com-pletely.Since the lens has a lower velocity,this leads to a smaller traveltime diﬀerence between the near-and far-oﬀset traces and,therefore,a higher NMO velocity.In contrast,for location C,the lens is missed bynear-oﬀset(a)(b)parison of CMP gathers for the model from Figure2computed(a)outside the lens(location A)and (b)above the center of the lens(location B)with aﬁnite-diﬀerence algorithm.The traveltime shifts at near oﬀsets(ar-rows)at location B cause signiﬁcant errors in the parameters V nmo andη.The lens-related waveform distortion at location B is contoured by the ellipse.rays and the traveltime diﬀerence between the near and far oﬀsets becomes larger,which reduces V nmo(Figure 5b).The nonhyperbolic inversion gives a closer approxi-mation to the actual traveltime due to the contribution of the additional parameterη.Hence,the best-ﬁt non-hyperbolic moveout curve at location B reproduces the increase in the near-oﬀset traveltime,which causes a pronounced deviation of the estimated V nmo from the exact value(Figure5c).Note that the hyperbolic cor-rection distorts the velocity V nmo outside the lens due to the inﬂuence of nonhyperbolic moveout.The laterally varyingη-curve resembles the re-versed version of the V nmo-curve.While in the absence386M.Takanashi &I.Tsvankin(a)(b)Figure teral variation of the inverted (a)V nmo and (b)η(bold solid lines)along the line using a spreadlength of 4km (the spreadlength-to-depth ratio X/D =2).The dashed line on plot (a)is the NMO velocity obtained from hyper-bolic moveout analysis for the same spreadlength.The exact eﬀective V nmo (equation 4)and η(equation 5)are marked by thin solid lines.of the lens the eﬀective ηat location B should almost coincide with the background η=0.08,the estimated η=–0.07is much smaller.The understated value of ηis explained by the need to compensate for the over-stated estimate of V nmo in reproducing traveltimes at moderate and large oﬀsets (Grechka &Tsvankin,1998;Tsvankin,2005).The magnitude of the variation (the diﬀerence between the largest and smallest values)in ηalong the line is close to 0.3.2.2Dependence of distortions on the lens parametersUsing ray-traced synthetic data,we investigate the de-pendence of the inverted moveout parameters on the ve-locity,width,and depth of the lens.The replacement of ﬁnite diﬀerences with ray tracing does not signiﬁcantly change the inversion results.As expected,the magnitude of the errors in Vnmo(a)(b)(c)Figure 5.(a)Schematic picture of near-and far-oﬀset ray-paths from a horizontal reﬂector beneath a low-velocity lens at three CMP locations (modiﬁed from Biondi,2006).(b)The inﬂuence of the lens on the moveout curves.The ray-paths and moveout curves at locations A,B and C are shown by dotted,solid and dashed lines,respectively.(c)Schematic estimated moveout curves (dotted lines)obtained from hy-perbolic (left)and nonhyperbolic (right)inversion at location B.The actual moveouts at locations A and B are shown by thin and bold solid lines,respectively.and ηis proportional to the velocity contrast between the lens and the background (Figure 6a).When the spreadlength is ﬁxed,the time distortions depend on the ratio W/L ,where W is the width of the lens and L is the maximum horizontal distance between the in-cident and reﬂected rays at the lens depth (Figure 5a).Note that L decreases with increasing lens depth.For the model used in the test,the distortions in V nmo and ηare largest when the width of the lens is 0.5km (or W/L =0.25)(Figure 6b).On the other hand,the error in V nmo estimated from hyperbolic moveout inversion has a ﬂat maximum for the width ranging from 0.5km to 1.5km.Correction for lenses in nonhyperbolic inversion387(a)(b)(c)Figure6.Dependence of the magnitude of the variation in V nmo(left)andη(right)on(a)the velocity contrast deﬁned as (V lens−V back)/V back,where V lens is the lens velocity and V back is the background velocity,(b)the width and(c)the depth of the lens.V nmo is obtained from nonhyperbolic(solid lines)and hyperbolic(dashed lines)moveout inversion.The spreadlength is4km(oﬀset-to-depth ratio X/D=2).Since W/L at the surface is0.25(W=1km,spreadlength is4km)in this model,a shallower lenscauses larger errors in V nmo andη(Figure6c).For adepth of0.25km(W/L =0.29),the errors are close tothe largest distortions for the test in Figure6b.2.3Identifying lens-induced distortionsIdentifying the range of CMP locations inﬂuenced bythe lens is critical for avoiding the use of distorted pa-rameters.It is clear from the above results that largevariations of V nmo andηon the scale of spreadlengthare strong indications of the ing the single-layerlens model,we suggest two additional indicators of the388M.Takanashi &I.Tsvankin(a)(b)Figure 7.Semblance value for moveout-corrected gathers before (dashed line)and after (solid line)applying trim stat-ics.The data were contaminated by random noise with the signal-to-noise ratio equal to (a)10and (b)5.lens –residual moveout after application of nonhyper-bolic moveout correction and the dependence of V nmo and ηon spreadlength.The moveout curve distorted by the lens cannot be completely ﬂattened by the nonhyperbolic move-out equation.To estimate the magnitude of the resid-ual moveout,one can use so-called trim statics (Ursen-bach &Bancroft,2001).Trim statics involves cross-correlation between a near-oﬀset trace and all oﬀset traces,which helps evaluate the statics shifts needed to eliminate the residual moveout.Due to the presence of residual moveout in the area inﬂuenced by the lens,application of trim statics increases the semblance (Fig-ure 7).Still,the semblance value after trim statics at location B is lower than that at location A because of the lens-induced waveform distortions.Trim statics,however,may not perform well when the data contain random or coherent noise (Ursenbach &Bancroft,2001).If the signal-to-noise (S/N)ratio is less than ﬁve,trim statics increases the semblance by aligning noise components in the statics-corrected gather (Figure 7).Thus,trim statics can be used to de-lineate the area inﬂuenced by the lens only for relatively high S/Nratios.(a)(b)Figure 8.Dependence of (a)V nmo and (b)ηon the spreadlength-to-depth ratio (X/D ).Another possible lens indicator is the variation of the moveout parameters with spreadlength.As shown in Figure 8,the shape of the V nmo -and η-curves is highly sensitive to the spreadlength-to-depth ratio (X/D ).In contrast,the estimated moveout parameters at location A outside the lens are weakly dependent on spreadlength.2.4Layered modelThe conclusions drawn above remain valid for a more realistic,layered model containing a parabola-shaped lens,which causes a maximum time distortion (or push-down anomaly)of 18ms (Figure 9).We generate syn-thetic data with ﬁnite-diﬀerences and apply nonhyper-bolic moveout inversion for the two interfaces (A and B)below the lens (Figure 10).For a spreadlength of 4km,the maximum distor-tion (the maximum deviation from the exact value)in V nmo reaches approximately 9%for interface A and 11%for interface B (Figure 10),while the distortion in ηCorrection for lenses in nonhyperbolic inversion389(a)(b)Figure teral variation of estimated V nmo (left)and η(right)for the model from Figure 9for (a)interface A and (b)interface B.The dashed lines correspond to a spreadlength of 4km,solid lines [only on plot (b)]to a spreadlength 6km.The thin solid lines mark the exactparameters.Figure yered model with a parabola-shaped lens.The ﬁrst layer is isotropic and vertically heterogeneous;V 0changes from 1.5km/s at the surface to 2.5km/s at the 1km depth.The second layer is homogeneous VTI with V 0= 3.5km/s,δ=0.07and =0.16and contains an isotropic lens with V 0=2.7km/s.The maximum thickness of the lens is 100m.The third layer is homogeneous VTI with V 0=4.2km/s,δ=0.05and =0.1.reaches 0.15and 0.33,respectively.The larger errors for interface B are related to its lower ratio X/D .When we use a spreadlength of 6km (X/D =2),the distortions in V nmo and ηfor interface B decrease to 5%and 0.08,respectively.As is the case for a ho-mogeneous background medium,the moveout-corrected gather exhibits residual moveout in the area inﬂuenced by the lens (Figure 11).Thus,the presence of residual moveout and the dependence of the moveout parame-ters on the spreadlength can serve as lens indicators for layered media as well.3CORRECTION ALGORITHMIt is clear from the modeling results that even a thin lens can cause signiﬁcant errors in the parameters V nmo and η.Another serious lens-induced distortion is the push-down anomaly on the stacked time section (Figure 12a).Although the time anomaly becomes smaller if the stack is produced using the background moveout parameters estimated away from the lens,the stacked event then has a smaller power because of a larger residual moveout390M.Takanashi &I.TsvankinFigure 11.Moveout-corrected gathers computed using the best-ﬁt parameters V nmo and η.Residual moveout is ob-served inside the area marked by the dashedline.(a)(b)Figure 12.Stacked section generated using (a)the best-ﬁt moveout parameters and (b)the background parameters.(Figure 12b).Clearly,it is desirable to produce an accu-rate stacked section without reducing the stack power.Here,we introduce two methods for correcting P-wave data from layered VTI media for the inﬂuence of the lens.One of them is designed to mitigate the distor-tions on the stacked section using trim statics.The othermethod makes it possible to remove the traveltime dis-tortions from each recorded trace and,therefore,obtain both accurate moveout parameters and a high-quality stack.3.1Trim staticsBy eliminating residual moveout,trim statics makes all traces kinematically equivalent to the zero-oﬀset trace (Figure 13a).Thus,trim statics increases stack power and generates a stack that kinematically reproduces the zero-oﬀset section (compare Figure 13b with Fig-ure 12a).To remove the zero-oﬀset time distortion,we as-sume that the zero-oﬀset raypath is not inﬂuenced by the lens and remains vertical for all horizontal inter-faces.Then the distortion of t 0should be the same at interfaces A and B.This assumption allows us to use the estimated push-down at interface A for correcting the time distortions for both interfaces.The resulting stacked section is kinematically correct and has a high stack power (Figure 13c).However,as discussed above,trim statics works only for high S/N ratios and cannot be used to estimate the background values of V nmo and η.3.2Prestack traveltime shifts 3.2.1MethodThe correction algorithm discussed here is designed for a horizontally layered overburden containing the lens,but the target reﬂector can be dipping or curved.Unlike the statics correction,this technique involves computa-tion of traveltime shifts as functions of oﬀset and target depth (Figure 14a).As the input data we use the zero-oﬀset time shifts (“pull-up”or “push-down”anomalies,∆t 0)for the horizontal reﬂector immediately below the lens.The lens-related perturbation of the raypath is assumed to be negligible,so that the ray in the layer containing the lens can be considered straight.Then the ray crossing the lens can be reconstructed using the velocity-independent layer-stripping method (VILS)of Dewangan &Tsvankin (2006).VILS builds the interval traveltime-oﬀset function by performing kinematic downward continuation of the waveﬁeld without knowledge of the velocity model.Each layer in the overburden is supposed to be laterally ho-mogeneous with a horizontal symmetry plane,so that the raypath of any reﬂection event is symmetric with respect to the reﬂection point.The bottom of the tar-get layer,however,can be curved and the layer itself can be heterogeneous.Wang &Tsvankin (2009)show that VILS provides more robust estimates of the interval moveout parameters in VTI and orthorhombic models than Dix-type equations.VILS can be applied to our model under the as-Correction for lenses in nonhyperbolic inversion391(a)(b)(c)Figure 13.(a)Moveout-corrected gather after application of trim statics;(b)the stacked section after trim statics,and (c)the stacked section from plot (b)after removing the push-down anomaly at interface A.sumption that the raypath in the overburden is not distorted by the lens.The idea of VILS is to identify reﬂections from the top and bottom of a certain layer that share the same upgoing and downgoing ray seg-ments.This is accomplished by matching time slopes on common-receiver and common-source gathers.Ap-plication of VILS to the reﬂections from the target and top of the layer containing the lens yields the horizontal coordinates x T 1and x R 1(Figure 14a).Likewise,the co-ordinates x R 2and x T 2are estimated by combining the target event with the reﬂection from the bottom of the layer containing the lens.Under the straight-ray assumption,we ﬁnd the hor-izontal coordinates of the crossing points and the ray angles (Figures 14a,b):x T L =x T 1+zT L (x T 2−x T 1),(6)x RL =x R 1−z RL (x R 1−x R 2),(7)cos θT L =zp (x T 2−x T 1)2+z 2,(8)cos θRL =zp (x R 1−x R 2)2+z 2,(9)where z is the thickness of the layer with the lens,and z T L and zRL are the distances from the lens to the top of the layer at locations x T L and x RL ,respectively.If the lens produces a suﬃciently strong reﬂection and the layer is vertically homogeneous,the ratio z /z can be estimated from the corresponding zero-oﬀset traveltimes (t /t ).In the layered model,we can clearly identify the lens reﬂection at t =1.15s on the stacked section (Figure 15a).This indicates that the horizontal coordinates and the ray angles can be estimated without complete information about the velocity and anisotropy parameters.Then the total lens-related traveltime shift for the target event (∆t ta )can be computed as∆t ta =12…∆t 0(x T L ,0)cos θT L +∆t 0(x RL ,0)cos θRL «,(10)where ∆t 0(x T L ,0)and ∆t 0(x RL ,0)are the zero-oﬀsettime distortions below the lens at locations x T L and392M.Takanashi &I.Tsvankin(a)(b)Figure 14.(a)Ray diagram of the correction algorithm.The horizontal coordinates x T 1,x T 2and x R 1,x R 2are deter-mined from the velocity-independent layer-stripping method.(b)Upgoing ray segment crossing the ing the valuesof z RL and z ,we can compute the horizontal location of the crossing point (x RL )and the ray angle (θRL ).x RL ,respectively.Both ∆t 0(x T L ,0)and ∆t 0(x RL ,0)can be estimated from the near-oﬀset stack.The ray angles θT L and θRL do not have to be the same,which makes the algorithm suitable for dipping or curved tar-get reﬂectors.After the correction,the kinematics of the prestack data should be close to the reﬂection traveltime de-scribed by the background values of V nmo and η.The in-terval parameters V (i )nmo and η(i )can be computed using the layer-stripped data corrected for the lens-induced time shifts.The removal of the time distortions also helps generate an accurate stacked section.3.2.2Synthetic testThe prestack correction algorithm is tested here on the layered model from Figure 9.First,we need to estimate the three required input quantities:∆t 0,the ratio z /z ,and the thickness z of the layer containing the lens.The values of ∆t 0(Figure 15b)and z /z (t /t )are obtained from the near-oﬀset stacked section (Figure 15a).For purposes of this test,the thickness of the layer contain-ing the lens is assumed to be known.Application of traveltime shifts computed from equation 10eliminates the time-varyingpush-down(a)(b)Figure 15.(a)Near-oﬀset stacked section obtained for the oﬀset range from 0to 200m,and (b)the magnitude of the push-down anomaly (solid line)estimated by picking the maximum amplitude along interface A.The dotted line in (b)is the exact ∆t 0.anomaly and increased the S/N ratio of the stacked section (Figure 16a).Also,the correction signiﬁcantly reduces the residual moveout in the moveout-corrected gathers (Figure 16b)and the errors in the eﬀective parameters V nmo and η(Figure 17a).For interface B,the distortion in V nmo decreases from 5%to less than 1%,and in ηfrom 0.08to 0.02.Figure 17b shows that the correction algorithm also produces much more ac-curate interval parameters V nmo and ηestimated from the layer-stripped data.The remaining errors are largely caused by the straight-ray assumption for the layer con-taining the lens.It is important to evaluate the sensitivity of the pa-rameter estimation to errors in the input data.Exten-sive testing shows that when the error in ∆t 0is smaller than 25%,the moveout-corrected gather is almost ﬂat.To test the sensitivity to the ratio z /z ,we move the lens down by 100m and 200m,which corresponds to 10%and 20%errors in z /z .Although distortions in the moveout-corrected gather become noticeable when the error reaches 20%,the magnitude of the residual move-out is still much smaller than that before the correction.(a)(b)Figure 16.(a)Stacked section and (b)moveout-corrected gathers obtained after applying prestack traveltime shifts that compensate for the inﬂuence of the lens.Finally,a thickness error up to 20%proves to have lit-tle impact on the output of the correction algorithm.An accurate stacked section can be generated even for somewhat larger errors in these input quantities.4DISCUSSIONThe correction algorithm requires knowledge of the zero-oﬀset time anomaly ∆t 0,the ratio z /z and the thickness z of the lens-containing layer.In the synthetic test,∆t 0was accurately estimated from the push-down anomaly on the near-oﬀset stacked section,and the ratio z /z was obtained from the corresponding time ratio t /t using the reﬂection from the lens (Figure 15).Since depth uncertainty seldom exceeds 20%in practice,errors in z are not expected to have a signiﬁcant impact on the correction results.The suggested approach should be applicable to many ﬁeld data sets.For example,the time section from the central North Sea in Figure 1contains channel-like structures and pull-up anomalies (marked area in Figure 1),which indicate the presence of high-velocity chan-nel ﬁlls (Armstrong et al.,2001).The lens reﬂections are suﬃciently strong for estimating the ratio t /t (and,therefore,z /z ),and the pull-up time anomaly can be accurately measured as well.Our algorithm can also be applied to layered media with multiple lenses,if it is possible to estimate the val-ues of ∆t 0and z /z for each lens separately.Then the total traveltime shifts are obtained by summing the in-dividual lens-induced time distortions.However,the al-gorithm will produce distorted time shifts when a layer contains multiple lenses or the lens reﬂections cannot be identiﬁed.Also,the algorithm assumes a laterally homogeneous overburden and straight rays in the layers containing the lenses.Therefore,the correction may be-come inaccurate when the overburden includes dipping interfaces or has a strong velocity contrast between the lens and the background.5CONCLUSIONSWe demonstrated that a relatively thin velocity lens may cause signiﬁcant,laterally varying distortions in the moveout parameters V nmo and ηestimated from nonhyperbolic moveout analysis.The magnitude of the distortion depends on the width and depth of the lens and is proportional to the velocity contrast between the lens and the background.The error in V nmo is larger after nonhyperbolic moveout inversion compared with the conventional hyperbolic algorithm applied for the same spreadlength,particularly when the lens is narrow or is located in a shallow layer.Hence,although non-hyperbolic moveout analysis produces smaller residual moveout and higher stacking power than the hyperbolic equation,it does not guarantee a more accurate estima-tion of NMO velocity in the presence of lateral hetero-geneity.Identifying the area inﬂuenced by the lens is crit-ical for avoiding use of distorted moveout parameters.We showed that the residual moveout can serve as a lens indicator because the lens-induced distortion cannot be completely removed by nonhyperbolic moveout inver-sion.The presence of residual moveout can be identiﬁed from the increase in semblance after application of trim statics,provided the signal-to-noise ratio is suﬃciently high.A lens also manifests itself by making the moveout parameters strongly dependent on spreadlength and the lateral coordinate.To correct for lens-induced traveltime shifts on prestack data,we developed an algorithm based on velocity-independent layer stripping (VILS).Synthetic tests conﬁrmed that the algorithm successfully removes lens-induced distortions on the stacked section and sub-stantially reduces the errors in the eﬀective and interval parameters V nmo and η.The correction requires esti-mates of the zero-oﬀset time distortion ∆t 0,the thick-ness z of the layer containing the lens and the ratio z /z ,where z is the distance between the lens and the top of。

Elastic Wave Propagation for the Interpretation of Laser Acoustic Measurements

Center of Mechanics, ETH Zurich, 8092 Zurich, Switzerland Email: juerg.bryner@imes.mavt.ethz.ch
Abstract— The wave propagation in anisotropic 3-D structures with arbitrary geometries is numerically calculated in order to interpret laser acoustic measurements in microstructures. The laser acoustic Pump-Probe technique generates bulk waves in structures in a thermo elastic way. Here, the wave propagation for various geometries and materials is investigated. In the ﬁrst parsotropic, axisymmetric structures is simulated with a 2-D ﬁnite difference formulation. The numerical results are veriﬁed with measurements of macroscopic specimens. In a second step, the simulations are extended to 3-D structures with anisotropic material properties. The implemented code allows the calculation of the wave propagation for different orientations of the material axes (orientation of the orthotropic axes relative to the geometry of the structure). Limits of the presented algorithms are discussed and future directions of the on-going research project are presented.

Dynamic material property characterization by using SHPB technique

Nuclear Engineering and Design 226(2003)119–125Dynamic material property characterization by using splitHopkinson pressure bar (SHPB)techniqueO.S.Lee a ,∗,M.S.Kim baSchool of Mechanical Engineering,Failure Control Center,InHa University,#253Yonghyun-dong,Nam-ku,Incheon 402-751,South Koreab Department of Mechanical Engineering,Graduated School,InHa University,Incheon 402-751,South KoreaReceived 22November 2001;received in revised form 3April 2003;accepted 24June 2003AbstractMechanical properties of the materials used for nuclear power plants,transportations and industrial machinery under high strain rate loading conditions such as seismic loading are required to provide appropriate safety assessment to these mechanical structures.The split Hopkinson pressure bar (SHPB)technique with a special experimental apparatus can be used to obtain the material behavior under high strain rate loading conditions.In this paper,dynamic deformation behaviors of the aluminum alloys (Al2024-T4,Al6061-T6and Al7075-T6)under both high strain rate compressive and tensile loading and PMMA under high strain rate compressive loading are determined using the SHPB technique.©2003Elsevier B.V .All rights reserved.1.IntroductionRecently,we may ﬁnd many cases in that mechan-ical materials are used under extreme conditions such as forging and rolling characterized by high stresses and high strain rate loading.In order to design struc-tures used under extreme loading conditions,we need to know mechanical deformation behaviors of the ma-terial under high strain rate in detail.It is not easy,however,to get the mechanical properties under the high strain rate loading condition.The compressive and tensile tests under high strain rate loading should be distinguished from those of low strain rate tests.The effect of inertia is not negligible in the test under high strain rate loading.The manifes-tation of inertia in a dynamic test is mainly threefold.∗Corresponding author.Tel.:+82-32-860-7315;fax:+82-32-868-1716.E-mail address:leeos@inha.ac.kr (O.S.Lee).First,it induces a radial component to the stress that may not be negligible in some conditions.Second,in-ertia is responsible for the heterogeneity of deforma-tion present in the specimen at the beginning of the test.And third,inertia affects the elongation stability.In this study a speciﬁc experimental method,the split Hopkinson pressure bar (SHPB)technique (Bragow,1994)has been used to determine the dy-namic material properties of commercial aluminum alloys and PMMA under the impact compressive and tensile loading conditions with strain-rate of the order of 103per second.2.Theory2.1.One-dimensional elastic wave propagation in a barPochhanmmer and Chree solved the longitudinal and radial inertia effect (Pochhammer,1876;Chree,0029-5493/$–see front matter ©2003Elsevier B.V .All rights reserved.doi:10.1016/S0029-5493(03)00189-4120O.S.Lee,M.S.Kim /Nuclear Engineering and Design 226(2003)119–1251889;Lee et al.,1998)on a specimen perfectly con-tacted with the bars in SHPB experiment.By this result,the specimen geometry in Hopkinson bar ex-periment could be designed to remove inertia effect (Hopkinson,1941).If the stress wave were a cosine wave of wavelength,λ,the longitudinal deformation and stress by the stress wave might be constant,when R/λ 1(Davies,1948).2.2.Uniform deformation of the specimenIt is difﬁcult to analyze the deformation of a speci-men due to the effects of plastic wave propagation and friction,while the elastic wave propagation in the bar may be expected.The inﬂuence of friction is reduced by spreading a viscous lubrication cream evenly.Even though the specimen deforms uniformly,errors may be generated by the longitudinal and radial inertia caused by the sudden particle acceleration in high strain rate.2.3.The stress–strain rate determination by SHPB pressive testIn conventional SHPB technique,the specimen is located in between incident and transmitted bars.Gen-eral compressive elastic wave propagation behavior in SHPB is shown in Fig.1(a).When the striker bar im-pacts the incident bar,rectangular stress pulse is gen-erated and travels along the incident bar until it hits the specimen.Part of the incident stress pulse reﬂects from the bar/specimen interface because of the ma-terial impedance mismatch,and part of it transmits through the specimen.The transmitted pulseemittedFig.1.A schematic diagram of specimen and elastic stress wave propagation for the compressive test (a)and the tensile test (b).from the specimen travels along the transmitted bar until it hits the end of the bar.The stress,strain and strain rate in the specimen can be obtained in terms of the recorded strains of the two bars as follows (Lee and Kim,2000a;Follansbee,1985):σspecimen =E AA s εT (1)εspecimen =−2C 0L εR d t(2)˙εspecimen =d ε(t)d t =−2CLεR (t)(3)2.3.2.Tensile testIn the tension test using SHPB technique,specimen is located in between incident and transmitted bars.General tensile elastic wave propagation behavior in SHPB is shown in Fig.1(b).The compressive stress pulse generated in the in-cident bar by the impact of striker bar travels along the specimen and the split ring (see Fig.4).The com-pressive stress pulse propagates until it arrives the end of the transmitted bar.The compressive stress pulse arrived the end of the transmitted bar reﬂects by the shape of tensile stress pulse.The tensile stress pulse is recorded at strain gage B.Part of tensile stress pulse reached at the specimen propagates to the incident bar,and the rest of the wave reﬂects to the transmitted bar.It is important to locate the strain gages where no in-terference between the tensile stress wave (εT )and the spurious wave generated at the incident bar/split ring interface.The spurious wave has bad effect on the re-sults because it applies pretension to the specimen.O.S.Lee,M.S.Kim /Nuclear Engineering and Design 226(2003)119–125121But this phenomenon is unavoidable in high strain rate tests.The split ring located between incident bar and transmitted bar has no effects on the tensile loading because it does not mechanically jointed to the two bars.The snug ﬁt between split ring and incident and transmitted bars is important to keep one-dimensional wave propagation condition.3.Experiment3.1.Loading apparatus and striker barIn this study,the incident,transmitted and striker bars are made of STB2whose yield strength is 490MPa and the modulus of elasticity is 225GPa,SHPB apparatus used for this study is shown in Fig.2.The length and the diameter of the striker are 300and 16mm,respectively.The diameters of incident bar and transmitted bar have the same dimension as the striker bar.3.2.Incident and transmitted barspressive testThe smaller the diameter of the pressure bars,the higher strain rate in the specimen will be gained.The bar length has to be twice of wavelength of stress pulse in the bars,so the ratio of the length to the diameter of the bars was designed to be 100.These two bars are made of the same material of striker bar and havetheFig.2.SHPB experimental setup at Fracture Control Research Center of InHa University.identical diameter with striker bar.To obtain perfect contact of the incident and the transmitted bar surfaces,the ends of the bars are ﬁnely grinded.3.2.2.Tensile testThe incident bar and transmitted bar are screwed to ﬁx the specimen in them.When we set the spec-imen into the incident and transmitted bars,we turn the specimen in one direction.So,one of the bars is machined right-hand screw and the other is left-hand screw.The cross-sections are heat treated to prevent deformation by continuous impacts.3.3.Straight-line guider and stopperOne of the most important things of the apparatus is the straight-line guide so that the stress pulse can propagate in one dimension.After ﬁne grinding an I-beam,the bar is setup on the beam by using the ﬁne bearingsystem.Fig.3.Geometry of compressive specimen (dimension:mm).Fig.4.Specimen geometry and setting between incident and trans-mitted bars (dimension:mm).122O.S.Lee,M.S.Kim /Nuclear Engineering and Design 226(2003)119–125Fig.5.The compressive (a)and tensile (b)strain signals recorded at Oscilloscope.3.4.Velocity and wave measurement system To measure the velocity,three photo sensors are located at the distance of 50mm in the end of the gun barrel.When the striker cuts the light of the photo sensors,and oscilloscope,Nicolet 410,gets electric signals.By the strain gages bonded on the middle of the bars,the stress pulse can be obtained.s t r e s s (M P a )strain100200300400500600700800s t r e s s (M P a )strains t r e s s (M Pa )strain2004006008001000s t r e s s (M P a )strain(a)(b)(c)(d)Fig.6.Dynamic compressive stress–strain curve for aluminum alloys and PMMA:(a)Al2024-T4,(b)Al6061-T6,(c)Al7075-T6,(d)PMMA.3.5.Specimen preparationpressiveThe geometry of specimen should meet the condi-tion to minimize the effect of inertia.So the specimens used for this study have 5mm thickness and 10mm diameter.The geometry of specimen is shown in Fig.3.O.S.Lee,M.S.Kim /Nuclear Engineering and Design 226(2003)119–1251233.5.2.TensileThe whole length,diameter and gage length of spec-imen is 34,4and 12mm,respectively.The both ends of specimen manufacture in a screw shape in order to be ﬁxed in incident and transmitted bars (Lee and Kim,2000b;Lee et al.,2000;Bragow,1994).The ratio of cross-sectional area of split ring to pressure bar cross-sectional area is 3:4.The ratio of cross-sectional area of split ring to specimen cross-sectional area is 12:1.Geometry of split ring and collar is shown in Fig.4.4.Results and discussionThe typical compressive and tensile signal outputs form strain gages attached on incident and transmitted bars,respectively,are shown in Fig.5.It is noted thats t r e s s (M P a )strains t r e s s (M P a )strain(a)(b)s t r e s s (M P a )strains t r e s s (M P a )strain(c)(d)Fig.7.Dynamic tensile stress–strain curve for aluminum alloys and PMMA:(a)Al2024-T4,(b)Al6061-T6,(c)Al7075-T6,(d)PMMA.the superposed wave of the reﬂected and transmitted wave are almost the same as the incident ing Eqs.(1)and (2),the relationship between stress and strain under high strain rate compres-sive and tensile loading conditions can be obtained.The typical results for various aluminum alloys and PMMA are shown in Figs.6and 7.The effects of strain rate on the relationship of stress–strain in vary-ing aluminum alloys and PMMA are found to be pronounced for both compressive and tensile loading cases as shown in Figs.6and 7.It is interesting to note that the mechanical deformation behaviors of aluminum alloys under high strain rate of compressive and tensile loading conditions quite differ from each other.Furthermore,the relationships between the log strain rates and the maximum stresses estimated from the experimental results are shown in Fig.8.AsO.S.Lee,M.S.Kim/Nuclear Engineering and Design226(2003)119–125125mentioned before,the relationship between maximum stresses and log strain rate shows quite different phe-nomena according to compressive and tensile loading conditions.It is speculated that this phenomena are originated from the experimental set ups which are different for the loading conditions.And also it may be thought that it is the material characteristics under different loading conditions.We need to investigate this point of view further in detail near future.It can be found that the sensitivity{(σdyn−σstat)/σstat}for tensile strength of Al2024-T4, Al6061-T6and Al7075-T6are more sensitive than compressive strength.And,it is also found that ten-sile strength under high strain-rate loading increased in bilinear.5.ConclusionThe dynamic deformation behaviors of Al2024-T4, Al6061-T6,Al7075-T6and PMMA under both com-pressive and tensile loading conditions are estimated by using SHPB techniques and the following experi-mental results are obtained.1.The relationship between compressive/tensilestrength and strain rate are bilinear.2.The sensitivity{(σdyn−σstat)/σstat}of Al2024-T4,Al6061-T6and Al7075-T6are286,320and178% at compressive strength and63,90,45%at ten-sile strength,respectively.And the sensitivity of PMMA is690%at compressive strength and394% at tensile strength.AcknowledgementsThis research is supported by2001InHa University research fund.ReferencesBragow, A.M.L.,1994.Methodological aspects of studying dynamic material properties using the lolsky method.Int.J.Impact Energy16,321–330.Chree,C.,1889.The equations of an isotropic elastic solid in polar and cylindrical coordinates,their solutions and applications.Cambridge Phil.Soc.Trans.14,250.Davies,R.M.,1948.A critical study of the Hopkinson pressure bar.Phil.Tran.A.240,375.Follansbee,P.S.,1985.The Hopkinson Bar’.In:Metals Handbook, 9th ed.Mechanical Testing.American Society for Metals.vol.8,pp.198–203.Hopkinson, B.,1941.A method of measuring the pressure produced in the detonation of explosives or by the impact of bullets.Phil.Trans.A.213,437.Lee,O.S.,Lee,S.S.,Chung,J.H.,Kang,H.S.,1998.Dynamic deformation under bar experiment.KSME Int.J.12(6),1143–1149.Lee,O.S.,Lee,J.Y.,Kim,G.H.,Hwang,H.S.,2000.High strain-rate deformation of composite materials using a split Hopkinson bar technique.Key Eng.Mater.183–187(Part1),307–312. Lee,O.S.,Kim,G.H.,2000a.Thickness effects on mechanical behavior of a composite material(1001P)and polycarbonate in split Hopkinson pressure bar technique.J.Mater.Sci.Lett.19,1805–1808.Lee,O.S.,Kim,G.H.,2000b.Determination of deformation behavior of the Al6061-T6under high strain rate tensile loading using SHPB technique.Trans.KSME(A)24(12),3033–3039 (in Korean).Pochhammer,L.,1876.On the propagation velocities of small oscillations in an unimited isotropic circular cylinder.J.Reine Angewandte Math.81,324.。

聚合物非对称脊波导结构参数与有效折射率

Ｔｈｓｌｉｖｌａｌｏｆｒｎｅｉｒｓｆｐｌａｉｎｏｅｏｇｎｃｐｌｍｅｓｍｍｅｒｄｅｗａｅｕｄ．ｅｒｕｔｓａｕｂｅｆｒｒｅｅｃｔｍｐｉｔｆｈｒａｉｏｙｒｙｅｅｎｅｏａｃｏｔａｔｉｒｇｖｇｉｅｃｉＫｅｒｓｏｇｎｃｐｌｍｅ；ｓｍｍｅｒｃｒｄｅｗａｅｕｄ；ｆｃｉｅｉｄｘｓｒｃｕａａａｔｒｔｅｓｏｉｅｗｉｔｆｙｗｏｄ：ｒａｉｏｙｒａｙｔｉｉｇｖｇｉｅｅｅｔｎｅ；ｔｕｔｒｌｒｍｅｅ；ｈｈｒｓｄｄｈｏｖｐｔｔｅｃｒｌｔｈｏｅｐａｅ
ｈｔ：／ｔｐ／ｗｗｗ．ｄｃａ．ｎｇｇ。ｃｃ
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光电工程
Ｏ
２１０２年８月
脊宽以及芯层平板短侧宽度情况下有效折射率的变化规律，为应用有机聚合物非对称脊波导设计聚合物电
光调制器奠定基础。
方法。而在后一种方法中，有效折射率法是一种相对简单且应用广泛的研究方法，它用简单的平板光波导替代复杂的二维光波导结构，避免了复杂的波传输方程求解Ｌ。该方法不仅可用于矩形波导结构，还可用８Ｊ
于非矩形波导结构。
图２给出了有效折射率法将脊波导等效为平板波导的示意图。
ｅｅｔｖｅｉｘｃｅｓｄｔｅｕｃｉｆｔｅｓｏｒｉｅｗｉｈｈｅｃｒａｅｗｈｎｔｅｏｎｔｏｎｓｗｅｅｃｏｔｎｔｆｃｉｎｄｅｄｅｒａｅｗｉｈｒｄｔｏｎｏｈｈｔｓｄｄｔｏｆｔｏｅｐｌｔｅｏｈｒｃｄｉｉｒｎｓａ．光场．脊波导按照脊两边芯层宽度是否对称，可以分为对称和非对称两种结构。图１ａ给出了芯层厚度ｈ１５（）＝．岬、脊高（ｈ＝．Ｈ－）０５岬时，称脊波导结构及其光场分布。１ｂ给出了同样条件下，对图（）短边芯层宽度２岬时的非对称脊波导结构及其光场分布，图中显示脊两边的芯层宽度是不相等的，且芯层的不对称性导致了

ELASTIC WAVE FILTER, MULTIPLEXER, DUPLEXER, HIGH-F

专利名称：ELASTIC WAVE FILTER, MULTIPLEXER,DUPLEXER, HIGH-FREQUENCY FRONT-ENDCIRCUIT, AND COMMUNICATION DEVICE 发明人：TAKAMINE, Yuichi,高峰　裕一申请号：JP2016/068688申请日：20160623公开号：WO2016/208677A1公开日：20161229专利内容由知识产权出版社提供专利附图：摘要：This transmission-side filter (10) has a functional electrode, which is formed upon a piezoelectric substrate (50), and is provided with five series resonators (101-105),which are connected to each other in series between a transmission-input terminal (11) and an antenna terminal (31), and with parallel resonators (151-154). The frequency differences Δf2, Δf3, and Δf4, which are the differences between the antiresonant frequencies and the resonant frequencies of the series resonators (102-104) that are not the series resonator (101) that is connected closest to the transmission-input terminal (11) and are not the series resonator (105) that is connected closest to the antenna terminal (31), are smaller than the frequency differences Δf1 and Δf5 for series resonators (101 and 105). The antiresonant frequencies of series resonators (102-104) are lower than the antiresonant frequencies of series resonators (101 and 105).申请人：MURATA MANUFACTURING CO., LTD.,株式会社村田製作所地址：〒6178555 JP,〒6178555 JP国籍：JP,JP代理人：YOSHIKAWA, Shuichi et al.,吉川　修一更多信息请下载全文后查看。

Elastic wave device and electronic component

专利名称：Elastic wave device and electroniccomponent发明人：Morio Onoe申请号：US12653929申请日：20091221公开号：US20100164325A1公开日：20100701专利内容由知识产权出版社提供专利附图：摘要：To provide an elastic wave device that is small sized and in which a frequency fluctuation due to a change with time hardly occurs, and an electronic component using the above elastic wave device. A trapping energy mode portion provided in an elasticwave waveguide made of an elastic body material excites a second elastic wave being an elastic wave in an energy trapping mode by a specific frequency component included in a first elastic wave being an elastic wave in a zero-order propagation mode propagated from a first propagation mode portion and a cutoff portion provided in a peripheral region of the trapping energy mode portion has a cutoff frequency being a frequency higher than that of the second elastic wave. A second propagation mode portion mode-converts the second elastic wave leaked through the cutoff portion to a third elastic wave being the elastic wave in the zero-order propagation mode to propagate the third elastic wave.申请人：Morio Onoe地址：Setagaya-ku JP国籍：JP更多信息请下载全文后查看。

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S K T O M A R and RAJNEESH KUMAR* Department of Applied Mathematics, Guru Jambheshwar University, Hisar 125 001, India * Department of Mathematics, Kurukshetra University, Kurukshetra 136 119, India MS received 27 January 1999
Eringen and his coworker [4, 9] developed the theory of simple microelastic solids in which the 'micro' deformations and rotations of the material particles contained in a microvolume element with respect to its centriod are taken into account in an average sense. Material affected by such micromotion and micro deformations are called micromorphic materials. Later, Eringen [5] developed theory for a subclass of micromorphic materials which are called 'micropolar solids'. These solids represent the materials that are made up of dipole atoms or dumbbell type molecules and are subjected to surface and body couples. The deformation in these materials is characterized not only by classical translational degrees of freedom represented by the deformation vector field ff(s t), but also by the rotation vector ~7(s t). In fact, micropolar elastic solids are special class of microelastic solids in which micro-stretch is ignored compared to micro-rotation. Eringen [6] further extended his work to include the effect of axial stretch during the rotation of molecules and presented a theory of 'micropolar elastic solids with stretch'. These materials can be thought of being composed of a large number of short springs which can possess average inertia moments and can deform in axial direction. The deformation in these solids is characterized by three quantities, i.e., the intrinsic axial stretch ~b*(~, t), which is a scalar quantity, in addition to the deformation field vectors ff(~, t), and ~(.~, t) of micropolar elasticity. The problem of wave propagation along the surface of a cylindrical borehole in an elastic medium of infinite extent had been first studied by Blot [1]. He calculated the borehole guided wave, the pseudo Rayleigh wave, and the Stoneley wave dispersion curves. Banerji and Sengupta [2, 3] studied the corresponding problem in micropolar 425 Surface waves; cylindrical bore; micropolarity; stretch; phase velocity;
Abstract. Propagation of surface elastic waves in a cylindrical bore through a micropolar elastic medium with stretch is analysed in two cases. In the first case, the cylindrical bore is considered empty while in the second case, the bore is filled with homogeneous inviscid liquid. In both the problems, period equations are obtained in closed form. The problem of Banerji and Sengupta [2,3] has been reduced as a special case. Numerical calculations have been performed for a particular model and results obtained are presented graphically. It is noticed that the effect of micropolarity on dispersion curve is significant while the effect of micro-stretch on dispersion curve is not appreciable. Keywords. dispersion. I. Introduction
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S K mar and Rajneesh Kumar
elastic medium. It is believed that the soil is very close to micromorphic medium and the problems of long well or borehole are useful in exploration of chemicals, oils, water etc. beneath the earth surface through wave propagation technique. Since the micropolar elastic material with stretch is very close to the micromorphic media e.g. soil. Our research studies the motivation of this type of physical situation. In order to determine the dispersive nature of the elastic waves in the bore hole, we have considered the problem of wave propagation in a cylindrical bore through micropolar elastic solid with stretch. The frequency equation of axial symmetric waves propagating at the surface of the cylindrical bore is obtained in two cases. In Case I, the cylindrical bore passing through the micropolar elastic medium with stretch and of infinite extent is assumed to be empty, while in the Case II, it is assumed to be filled with homogeneous inviscid liquid.
2. Propagation of waves in cylindrical empty bore Case I
Consider a cylindrical empty bore of circular cross section having diameter 2a, in a homogeneous, isotropic, micropolar elastic medium with stretch and of infinite extent. The waves of axial symmetry and pure sinusoidal along the axial direction are considered. Taking cylindrical polar co-ordinates (r, 0, z) with z-axis pointing upward. In the absence of body forces and body moments, the governing equations (ref. [6], eqs (6.3) to (6.5), pp. 12) in the medium considered are: (# + K ) V 2 ~ + (A + Iz)V(V . ~) + K V • ~ = p Ot2 , 7V2a7 + (a +/3)V(V 9o7) - 2Ko7 + KV x ~ = pj Ot2 ,