1. Chapter 13
Wave Motion

2. Various waves
Water wave
Sound
Earthquake
Light
Radio wave
Microwave
Vibration
Waves

3. Conditions of mechanical wave
Wave motion is a propagation of oscillation
Mechanical waves governed by Newton’s laws:
1) A wave source (pulse or oscillation)
2) Medium (in which the wave travels)
Wave pattern (not particles) travels in medium 3

4. Particle motion ⊥ direction of wave traveling
Wave types
Transverse wave *
Particle motion ⊥ direction of wave traveling
Longitudinal wave *
Particle motion ∥ direction of wave traveling
Other waves:
Water wave 4

5. Characteristics of wave motion
Source at SHM → medium particles at SHM
And the shape of wave is also sinusoidal
Amplitude A
Wavelength λ
Crest A
Space Periodicity
Frequency f
Period T
Angular frequency ω
Trough
Determined by the source
Time Periodicity 5

6. Velocity of wave pattern → phase velocity
Wave velocity
Velocity of wave pattern → phase velocity
It travels one wavelength in one period
v is different from the velocity of particles
and it depends on the properties of medium
From dynamic analysis of the medium,
We have 6

7. Velocity of different waves
1) Transverse Wave on a String:
FT: tension in the string
μ: mass per unit length
2) Longitudinal Waves in a Solid body
E: elastic modulus
ρ: mass density
3) Longitudinal Waves in a liquid or air
B: bulk modulus
ρ: mass density 7

8. How to obtain a frequency of 2f ?
Playing a guitar
Example1: A guitar string (L=65cm, m=1.3g) is stretched by a tension FT=80N. Determine: a) the velocity of waves on the string; b) the frequency if the wave length is 2L.
Solution: a) b)
How to obtain a frequency of 2f ? 8

9. Communication faster than light?
Push a long rod
Example2: Transmission of force takes time. If you push on a long rod (L=100m, ρ=1g/cm3, E=5×108 N/m2) , a pulse travels in the rod with wave speed. When does the force come to the other end?
Solution:
Sound speed in the rod
Communication faster than light? 9

10. Source at SHM, and shape of wave is sinusoidal
Plane harmonic wave
Wave surface
Ray
Plane wave
Spherical wave
Source at SHM, and shape of wave is sinusoidal
Medium: uniform, isotropy, no absorption
Plane harmonic wave → 1-D wave
Represented by a wave traveling along a line 10

11. Wave motion is a propagation of oscillation
Representation of PHW
Wave motion is a propagation of oscillation
Motional function of any particle: wave function v
x: equilibrium position
y: displacement
v: wave velocity
If we know the motional function of point O:
How about other points, such as point P? 11

12. If the wave travels along negative x-direction
Wave function v
Same A, ω
Phase difference
Point P:
If the wave travels along negative x-direction 12

13. Equivalent forms “-”: travels in +x “+”: travels in -x
4 key factors: A, ω, k, φ
2 variables: x, t
Phase of wave:
Wave number
Phase velocity 13

14. Time and space periodicity
1) If x is determined ( x=x0): T
It describes the motion of a particle at x=x0 as a function of time
2) If t is determined ( t=t0):
It describes the shape of wave at t=t0 14

15. Traveling wave on string
Example3: A wave traveling on a string is shown by
y=0.1cos(6x + 80t + π/3) (SI). Determine (a) the wavelength, frequency, wave velocity and amplitude;
(b) maximum speed of particles in the string.
Solution: a) b)
comparing with v 15

16. Motional function of P:
Oscillation and wave
Example4: A wave travels along -x with v=10m/s and A=0.04m, point P moves from trough to crest when t=0→2s. a) motion of P; b) wave function. x 5m P v y o
Solution: a)
Period T = 4s
Initial phase of P:
Motional function of P:
b) Wave function 16

17. Shapes of wave
Example5: A wave traveling toward +x, its shapes at t=0 and t=0.5s are shown in the graph (T>1s). Solve: a) wave function; b) motional function of point P.
Solution: a) o 1 2 3 4
x(m)
y(m)
0.2
t=0.5
t=0 P
A=0.2, =4m
T= /v=2s，=
 =/2
b) Point P: x=2 17

18. The wave equation
2-variable function
Wave equation
Comparing with 18

19. * Transverse Wave on a String
Consider the small amplitude case, so that each particle can be assumed to move only vertically. 19

20. Energy transported by waves
Wave motion is also a propagation of energy.
Energy is transferred as vibrational energy from particle to particle in the medium. dx
dm= Sdx v S
For a SHM:
For infinitesimal:
Energy transported by wave 20

21. Average power: (rate of energy transferred)
Intensity
dx=vdt v S
Average power: (rate of energy transferred)
Intensity of a wave: average power transferred across unit area ⊥ the direction of energy flow 21

22. Energy of spherical wave
Example6: The intensity of earthquake wave is 2.2106 W/m2 at a distance of 100 km from the source. What was the intensity when it passed a point only 4.0 km from the source?
Solution: It is a spherical wave
Total power: 22

23. Principle of superposition
In a superposition, the actual displacement is the vector or algebraic sum of the separate displacements.
1) Amplitude not too large
Hooke’s law still holds
2) After passing by, continue to move independently. *
3) Composite wave
Fourier’ theorem 23

24. 2) same oscillation direction coherent waves
Interference
1) same frequency
2) same oscillation direction
coherent waves
Stable picture that does not vary with time
Constructive & destructive interference 24

25. out of phase, destructive
In phase & out of phase
where S2 S1 r1 r2 P
in phase, constructive
out of phase, destructive 25

26. Interference from two sources
Example7: Point P stays at rest in the interference. The motion of S1 is y1s=Acos(2πt+0.5π), what is the motion of S2? (S1P=1.5λ, S2P=1.2λ)
Solution: Oscillation from S1 to P:
Destructive interference 26

27. What about area from S1 to S2?
Interference on a line
Example8: Two coherent waves from S1 and S2 are traveling on a line. Same A, , determine the positions of constructive interference.
Solution:
Left of S1:
constructive
Right of S2:
destructive
What about area from S1 to S2? S1 S2
3/4 a x x c x b
standing wave 27

28. A special case of interference
Standing waves
A special case of interference
same A, but travel in opposite directions
Standing wave
node
antinode
A standing wave is not a traveling wave
It has a certain wave-pattern,
but it does not move (to left or right) 28

29. Mathematical representation
Two traveling waves
Superposition:
This is the equation of standing wave
1) x and t are separated, not a traveling wave
2) Amplitude varying oscillation 29

30. 3) Nodes: minimum amplitude = 0
Nodes & antinodes
3) Nodes: minimum amplitude = 0
Antinodes: maximum amplitude = 2A
Distance between neighboring nodes → λ/2
A method to measure the wavelength 30

31. cos k x > 0 → phase of particle:
Phase and energy
4) Nodes: cos k x = 0
cos k x > 0 → phase of particle:
cos k x < 0 → phase of particle:
Particles between neighboring nodes: in phase
Particles in different sides of a node: out of phase
5) A standing wave does not transfer energy
local flow of energy, average rate is 0 31

32. Solution: a) Wave function of y2 can be written as:
Make a standing wave
Example9: Waves y1 and y2 make a standing wave, where and there is a node at x=2. Determine: a) wave function of y2 ; b) equation of standing wave; c) position of antinodes.
Solution: a) Wave function of y2 can be written as:
A node at x=2: 32

33. b) Equation of standing wave:
c) Position of antinodes: maximum amplitude
where n = 0, 1, 2, …
How about nodes?

34. Reflection and transmission
1) wave from light section to heavy section x
standing wave
transmission
node
reflection
phase change of 
( fixed end )
2) wave from heavy section to light section x
antinode
no phase change
( free end ) 34

35. Standing waves on a string
On a string: incoming wave + reflection wave
Fixed ends → nodes
natural frequencies
harmonics
n=1: fundamental frequency
n>1: overtones 35

36. Solution: Reflection wave
Reflect at a free end
Example10: A wave y1=Acos(ωt -kx) is traveling on a string, and it is reflected at a free end (x=L). Determine the standing wave on the string.
x=L
Solution: Reflection wave
No phase change at a free end (antinode): 36

37. Reflect at a fixed end
Question: A wave is traveling on a string, and it is reflected at a fix end (x=5.2). Determine the standing wave on the string.
x=5.2 37

38. *Refraction & Diffraction Transmission with an angle → refraction
Waves bend around an obstacle → diffraction
wavelength ~
size of obstacle 38