wave 1

For a given point x=x0
y ( x 0 , t ) A sin ( k x 0 t )
——oscillation equation (振动方程)
Period, angular frequency and frequency
y ( x 0 , t ) A s in ( k x 0 t ) A s in [ k x 0 ( t T ) ]
period：
T f ω
ω k
f T ω 2πf 2π T
Wave speed： v
λ T
Example 1:
The equation of a transverse wave traveling along a very long string is
y 6 . 0 sin ( 0 . 0 2 x 4 . 0 t )
Simple Harmonic Wave along a string
· · · · · · · · · ·· · · · · · · · · · · · · · ·t = 0 ·· ·· · · · · · · · ·· · · t = T/4 · · · · · · · · ·· · · · · ·· · · · · · · · · · · · · ·· · · · · t = T/2 · ·· · · · · · · · · · · ·t = 3T/4 · · · · · · ·· · ·· · · · · · · ·· · ·· · · ···· t = T ·· ·· · ·· · ·
y ( x , t ) A sin ( k x t )
Left-traveling wave (逆行波)
y ( x , t ) A s in ( k x t ) kx t x v k ( x x ) (t t ) t k
Chapter 16
Waves－ I
Types of waves :
1. Mechanical waves water wave, sound wave, seismic wave… (a) they can exist only within medium (b) elements of medium obey the Newton’s Law 2. Electromagnetic waves can travel in vacuum
F 2 s in 2
2
l
a
v
R
l R

l R
l
v
2
v

Elastic property
R
Inertial property
Exercise : The equation of a transverse wave on a string is
u m a x A 2 4 cm /s= 7 5 .4 cm /s
(g)
y ( 3 .5 c m , 0 .2 6 s ) 6 . 0 s in ( 0 . 0 2 3 . 5 4 . 0 0 . 2 6 ) 2 .0 c m
Wave velocity and wave equation Mass of element m l Linear density Acceleration Restoring force
Phase velocity
v

k
d dk
Group velocity v g
The quantities of wave
Spatial period

Wavelength ：
λ
Wave number： k
k
2π λ
1
Temporal period
frequency： Angular frequency：
shapeofsimpleharmonicwaveyxtsinakxt?????yxtcosatkx?????orforagiventimett000yxtsinakxt?????waveshape波形曲线yxtsinsintaakkxt?0t??wavelengthandwavenumber00sinakxt??????????2?2??k?k???wavenumber00sinyxtakxt?????foragivenpointxx0oscillationequation振动方程isinaakkxperiodangularfrequencyandfrequency2?2?tt?????000sinyxttakxtt???????????12??fft??kxtxvkxxtttk???????????????????????phaseandwavevelocityyxtsinakxtkxtx?????????x?k???????????lefttravelingwave逆行波righttravelingwave正行波yxtsinakxt?????vkxttt???????????phasevelocitygroupvelocityvk??gd?vdk?thequantitiesofwavespatialperiodkwavelength
dK dt 1 2
A cos ( kx t )
2 2 2
dx dt

1 2
A co s ( k x t )v
2 2 2
Pk
dK dt

1 2
A c o s ( k x t )v
2 2 2
The average transport rate of kinetic energy
0
4
8
12
16
20
24
1、The elements of medium oscillate parallel to the y axis, the wave travels along x axis.
—质元未“随波逐流”， 波的传播不是物质的 传播。 2、The motion states of elements repeat that of the oscillation source with the increase of time along the string . 绳子上的质元依次出现与振源相同的运动状态 （简谐振动）。 —波是振动状态的传播。 3、 The motion states of elements is determined by phase. — 波是相位的传播，波速即同相位值 的传播速度。
-1
(b) (c)

2 k
100cm
-1
4 .0 s
-1
f

2
0 .2 s
(d)
v
dx dt


k
2 0 0 c m /s
(e)
in negative direction
(f) The velocity of the particle is
u y t A cos( kx t )
Energy and Power of a wave traveling along a string
dm dx
u
y t
1 2
A cos( kx t )
dm u
2 2
dK 1 2
d x A co s ( k x t )
2 2
The transport rate of kinetic energy
y n et ( x , t ) y 1 ( x , t ) y 2 ( x , t )
Interference of waves
y 1 ( x , t ) A s in ( k x t ) y 2 ( x , t ) A s in ( k x t ) y n et ( x , t ) y1 ( x , t ) y 2 ( x , t ) 2 A c o s ( / 2 ) s in ( k x t / 2 )
where x and y are expressed in centimeters and t is in seconds. Determine (a) the amplitude, (b) the wavelength, (c) the frequency, (d) the speed, (e) the direction of propagation of the wave, and (f) the maximum transverse speed of a particle in the string. (g) What is the transverse displacement at x=3.5 cm when t=0.26s?
Solution:
Compare the wave function
y 6 . 0 sin ( 0 . 0 2 x 4 . 0 t )
with the standard one
y ( x , t ) A sin ( k x t )
(a)
A 6 .0 c m k 0 .0 2 c m
The oscillation travels along some direction.
Transverse wave: The direction of oscillation is perpendicular to the direction of travel.
Longitudinal wave: The direction of oscillation is parallel to the direction of travel.

——wave shape (波形曲线)
Wavelength and wave number
y ( x , t 0 ) A s in ( k x t 0 ) A s in [ k ( x ) t 0 ]
k 2 k 2

Wave number
2 0 2 2 2 2 y n et 1 y n et x v t 2 0 2 2 2 2 x v t y2 1 y2 2 0 2 2 x v t y1
2
1 y1
2
Principle of superposition Two or more waves can transverse the same space independent of one another. When they overlap the displacement of the resultant (net) wave is the algebraic sum of the displacements of all the waves.
Shape of Simple Harmonic Wave
y ( x , t ) A sin ( k x t )
or
y ( x , t ) A co s( t k x )
For a given time t=t0
y ( x , t 0 ) A sin ( k x t 0 )
The Wave Equation (波动方程)
y
2
x
2

1 y
2
v
2
t
2
0
v

Wave equation is a second-order linear, homogeneous , differential equation
y n et ( x , t ) y 1 ( x , t ) y 2 ( x , t )
y 2 . 0 m m sin [( 2 0 m
1
) x (600s
1
)t ]
The tension in the string is 15 N. (a)What is the wave speed? (b) Find the linear density of this string in grams per meter.
T 2
2 T
f 1 T , 2 f
Phase and wave velocity
kx t x v k ( x x ) (t t ) t k
Right-traveling wave (正行波)
3. Matter waves (probability waves)
all particles such as electrons, neutron, protons, photons, etc. have wave-particle duality
Mechanical wave:
The elements of medium oscillate around their own equilibrium positions.
Pk 1 T
T

0
Pk d t
1 4
A v
2 2
The average kinetic energy equals the average potential energy, therefore
Pk P p
Average power P Pk P p
1 2
A v
2 2