声学基础课件(许肖梅)fundamentals of acoustics 07-10共23页

p p 0 y z
2 p t 2
c02
2 p x2
The solution of the wave equation
We already know the nature of its solutions
• We introduce two new variable quantities :
x c0 t; x c0 t
2 p t 2
c02
2 p x2
2p 0
p
f 2 ( )
pf1()f2()
pf1(tcx0)f2(tcx0)
pf1(tcx0)f2(tcx0)
• The sum of these two functions is the complete general solution of the wave equation.
• If all the acoustic variables are functions of only one spatial coordinate, the phase of any variable is a constant on any plane , such a wave is called a plane wave.
The complex form of the harmonic
solution for the acoustic pressure of
a plane wave is :
j(tx)
j(tx)
pA1e c0 A2e c0
pA 1 ei( t k)xA 2 ei( t k)x
Where the wave number k is defined by
• The product 0 c 0 Often has grater significance as a characteristic property of the medium than does either p0 or c0 individually.
• For this reason, 0 c 0 is called the characteristic impedance of the medium.
0c0415Pa.s/m
At a temperature of 200c and one atmospheric pressure ,resulting in a characteristic impedance of water is 1.5*106Pa.s/m
At a temperature of 200c and atmospheric pressure the density of air is 1.21kg/m3 and the speed of sound is 343m/s ,giving the standard characteristic impedance of air.
The Relationship Between The Velocity and Pressure
From the equation of motion
u1pxdtu0
u
1
p x
dt
pA 1ej(t kx)pm ejΒιβλιοθήκη Baidut kx)
u
p e m j(tkx) 0c0
u
umej(tkx)
p
0c0
Displacement is
• The simplest solutions to the wave equation (3-4) are those that depend on only one of the three spatial coordinates.
2t2pc22p
(34)
We may as well call that one x. the equation (3-4) reduces to follow equation
Za ra jxa
Where ra is called the acoustic resistance and xa the acoustic reactance of the medium for the particular wave being considered.
• The MKS unit of acoustic impedance is Pa.s/m
k 2 c0
A1,A2 are two arbitrary constants.
If there is not reflected waves, A2=0, so we obtain :
pA 1 ej(t kx)p m ej( t kx)
Where the amplitudes pm is a constant, it does not change with the distance.
udt u e m j(tkx)
j
umej(kx02)ejt m ej(tkx02)
The Acoustic Impedance and The Characteristic Impedance of The Medium
• The ratio of acoustic pressure in a medium to the associated particle speed is the acoustic impedance :
Za
p u
For plane waves this ratio is :
Za 0c0
Although the acoustic impedance of the medium is a real quantity for plane wave, this is not true for standing plane waves or for diverging wave. In general, Za will be found to be complex
• The function f1(t-x/c0) represents a wave traveling in the right at a constant speed c0
• Similarly ,f2(t+x/c0) represents a wave moving in the –x direction with speed c0.