1. Types of mechanical waves
Chapter 14: Wave Motion
Types of mechanical waves
Mechanical waves
are disturbances that travel through some material or substance
called medium for the waves.
travel through the medium by displacing particles in the medium
travel in the perpendicular to or along the movement of the
particles or in a combination of both
transverse waves:
waves in a string etc.
longitudinal waves:
sound waves etc.
waves in water etc.

2. Types of mechanical waves (cont’d)
Longitudinal and transverse waves
sound wave = longitudinal wave
C = compression
R = rarefaction
air compressed
air rarefied

3. Types of mechanical waves (cont’d)
Longitudinal-transverse waves

4. Types of mechanical waves (cont’d)
Periodic waves
When particles of the medium in a wave undergo periodic
motion as the wave propagates, the wave is called periodic. l
wavelength A
amplitude
t=0
x=0 x
t=T/4
t=T
period

5. Mathematical description of a wave
Wave function
The wave function describes the displacement of particles
in a wave as a function of time and their positions:
A sinusoidal wave is described by the wave function:
sinusoidal wave moving in
+x direction
angular frequency
velocity of wave, NOT of
particles of the medium
period
wavelength
sinusoidal wave moving in
-x direction
v->-v
phase velocity

6. Mathematical description of a wave (cont’d)
Wave function (cont’d) l
wavelength
t=0 x
x=0
t=T/4
t=T
period

7. Mathematical description of a wave (cont’d)
Wave number and phase velocity
wave number:
phase
The speed of wave is the speed with which we have to
move along a point of a given phase. So for a fixed phase,
phase velocity

8. Mathematical description of a wave (cont’d)
Particle velocity and acceleration in a sinusoidal wave
u in textbook
velocity
acceleration
Also
wave equation

9. Mathematical description of a wave (cont’d)
General solution to the wave equation
wave equation
Solutions:
such as
The most general form of the solution:

10. Speed of a transverse wave
Wave speed on a string
Consider a small segment of string whose
length in the equilibrium position is
The mass of the segment is
The x component of the force (tension) at both
ends have equal in magnitude and opposite in
direction because this is a transverse wave.
The total y component of the forces is:
Newton’s 2nd law
mass
acceleration

11. Speed of a transverse wave (cont’d)
Wave speed on a string (cont’d)
The total y component of the forces is:
wave eq.

12. Energy in wave motion At point a, the force does work on the
Total energy of a short string segment of mass
At point a, the force
does work on the
string segment right of point a.
work done
Power is the rate of work done : a
Pmax

13. Energy in wave motion (cont’d)
Maximum power of a sinusoidal wave on a string:
Average power of a sinusoidal wave on a string
The average of
over a period:
The average power:

14. Wave intensity source: power/unit area
Wave intensity for a three dimensional wave from a point
source:
power/unit area

15. Wave interference, boundary condition, and superposition
The principle of superposition
When two waves overlap, the actual displacement of any
point at any time is obtained by adding the displacement
the point would have if only the first wave were present and
the displacement it would have if only the second wave were
present:

16. Wave interference, boundary condition, and superposition (cont’d)
Constructive interference (positive-positive or negative-negative)
Destructive interference (positive-negative)

17. Wave interference, boundary condition, and superposition (cont’d)
Reflection
incident wave
reflected wave
Free end
For x<xB
At x=xB
Vertical component of the force
at the boundary is zero.

18. Wave interference, boundary condition, and superposition (cont’d)
Reflection (cont’d)
Fixed end
For x<xB
At x=xB
Displacement at the boundary is zero.

19. Wave interference, boundary condition, and superposition (cont’d)
Reflection (cont’d)
At high/low density

20. Wave interference, boundary condition, and superposition (cont’d)
Reflection (cont’d)
At low/high density

21. Standing waves on a string
Superposition of two waves moving in the same direction
Superposition of two waves moving in the opposite direction

22. Standing waves on a string (cont’d)
Superposition of two waves moving in the opposite direction
creates a standing wave when two waves have the same
speed and wavelength.
incident
reflected
N=node, AN=antinode

23. Normal modes of a string
There are infinite numbers of modes of standing waves
funda-mental
first
overtone
second
overtone
third
overtone L
fixed end
fixed end

24. Sound waves Sound is a longitudinal wave in a medium
The simplest sound waves are sinusoidal waves which
have definite frequency, amplitude and wavelength.
The audible range of frequency is between 20 and 20,000 Hz.

25. Sound waves (cont’d) Sinusoidal sound wave function: Change of volume:
Sound wave (sinusoidal wave)
Sinusoidal sound wave function:
Change of volume:
undisturbed
cyl. of air
disturbed
cyl. of air
Pressure:
pressure
bulk modulus S Dx x
x+Dx

26. Pressure amplitude and ear
Pressure amplitude for a sinusoidal sound wave
Pressure:
Pressure amplitude:
Ear

27. Perception of sound waves
Fourier’s theorem and frequency spectrum
Fourier’s theorem:
Any periodic function of period T can be written as
fundamental freq.
where
Implication of Fourier’s theorem:

28. Perception of sound waves
Timbre or tone color or tone quality
Frequency spectrum
noise
music
piano
piano

29. Speed of sound waves (ref. only)
velocity of wave
The speed of sound waves in a fluid in a pipe
movable piston
longitudinal momentum carried
by the fluid in motion
fluid in
equilibrium
original volume of the fluid in
motion
velocity
of fluid
change in volume of the fluid
in motion
bulk modulus B:
-pressure change/frac. vol. change
change in pressure in the fluid
in motion
fluid in motion
fluid at rest
boundary moves at speed of wave

30. Speed of sound waves (ref. only) (cont’d)
The speed of sound waves in a fluid in a pipe (cont’d)
longitudinal impulse = change in momentum
speed of a longitudinal
wave in a fluid
The speed of sound waves in a solid bar/rod
Young’s modulus

31. Speed of sound waves (cont’d)
The speed of sound waves in gases
bulk modulus of a gas
ratio of heat capacities
equilibrium pressure of gas
In textbook
- P in textbook
(background
pressure).
- r density
speed of a longitudinal
wave in a fluid
gas constant J/(mol K)
temperature in Kelvin
molar mass

32. Sound level (Decibel scale)
As the sensitivity of the ear covers a broad range of intensities,
it is best to use logarithmic scale:
Definition of sound intensity:
( unit decibel or dB)
Sound intensity in dB
Intensity (W/m2)
Military jet plane at 30 m
140
102
Threshold of pain
120 1
Whisper 20
10-10
Hearing thres. (100Hz)
10-12

33. Standing sound waves
Sound wave in a pipe with two open ends

34. Standing sound waves
Standing sound wave in a pipe with two open ends

35. Standing sound waves
Sound wave in a pipe with one closed and one open end

36. Standing sound waves Displacement
Standing wave in a pipe with two closed ends
Displacement

37. Normal modes
Normal modes in a pipe with two open ends
2nd normal mode

38. Normal modes (stopped pipe)
Normal modes in a pipe with an open and a closed end
(stopped pipe)

39. Resonance
Resonance
When we apply a periodically varying force to a system that can
oscillate, the system is forced to oscillate with a frequency equal
to the frequency of the applied force (driving frequency): forced
oscillation. When the applied frequency is close to a characteristic
frequency of the system, a phenomenon called resonance occurs.
Resonance also occurs when a
periodically varying force is applied
to a system with normal modes.
When the frequency of the applied
force is close to one of normal
modes of the system, resonance
occurs.

40. Interference of waves destructive constructive d2 d1
Two sound waves interfere each other
destructive
constructive d2 d1

41. Beats Two waves with different frequency create a beat
Two interfering sound waves can make beat
Two waves with different
frequency create a beat
because of interference
between them. The beat
frequency is the difference
of the two frequencies.

42. Beats (cont’d) Suppose the two waves have frequencies and
Two interfering sound waves can make beat (cont’d)
Suppose the two waves have frequencies
and
For simplicity, consider two sinusoidal waves of equal intensity:
Then the resulting combined wave will be:
As human ears does not distinguish negative and positive amplitude,
they hear two max. or min. intensity per cycle, so 2 x (1/2)|fa-fb|=
|fa-fb| is the beat frequency fbeat.

43. Doppler effect Source at rest Source at rest Listener moving left
Moving listener
Source at rest
Listener moving left
Source at rest
Listener moving right

44. Doppler effect (cont’d)
Moving listener (cont’d)
The wavelength of the sound wave does not change whether
the listener is moving or not.
The time that two subsequent wave crests pass the listener
changes when the listener is moving, which effectively changes
the velocity of sound.
freq. listener hears
freq. source generates
velocity of sound at source
- for a listener moving away from
+ for a listener moving towards
the source.
velocity of listener

45. Doppler effect (cont’d)
Moving source
When the source moves

46. Doppler effect (cont’d)
Moving source (cont’d)
The wave velocity relative to the wave medium does not
change even when the source is moving.
The wavelength, however, changes when the source is moving.
This is because, when the source generates the next crest, the
the distance between the previous and next crest i.e. the wave-
length changed by the speed of the source.
The source at rest When the source is moving
+ for a receding source
- for a approaching source

47. Doppler effect (cont’d)
Moving source and listener
- for a listener moving away from
+ for a listener moving towards
the source.
+ for a receding source
- for a approaching source
The signs of vL and vS are measured
in the direction from the listener L to the
source S.
Effect of change of source speed

48. Doppler effect (cont’d)
Example 1
A police siren emits a sinusoidal wave with frequency fs=300 Hz.
The speed of sound is 340 m/s. a) Find the wavelength of the waves
if the siren is at rest in the air, b) if the siren is moving at 30 m/s, find
the wavelengths of the waves ahead of and behind the source. a)
b) In front of the siren:
Behind the siren:

49. Doppler effect (cont’d)
Example 2
If a listener l is at rest and the siren in Example 1 is moving away
from L at 30 m/s, what frequency does the listener hear?
Example 3
If the siren is at rest and the listener is moving toward the left at 30
m/s, what frequency does the listener hear?

50. Doppler effect (cont’d)
Example 4
If the siren is moving away from the listener with a speed of 45 m/s
relative to the air and the listener is moving toward the siren with a
speed of 15 m/s relative to the air, what frequency does the listener
hear?
Example 5
The police car with its 300-MHz siren is moving toward a warehouse
at 30 m/s, intending to crash through the door. What frequency does
the driver of the police car hear reflected from the warehouse?
Freq. reaching
the warehouse
Freq. heard by
the driver

51. Exercises Problem 1 A transverse wave on a rope is given by:
(a) Find the amplitude, period, frequency, wavelength, and speed of
propagation. (b) Sketch the shape of the rope at the following values
of t : s, and s. (c) Is the wave traveling in the +x or –x
direction? (d) The mass per unit length of the rope is kg/m.
Find the tension. (e) Find the average power of this wave.
Solution
A=0.75 cm, l=2/0.400 = 5.00 cm, f=125 Hz, T=1/f= s and
v=lf=6.25 m/s.
(b) Homework
(c) To stay with a wave front as t increases, x decreases. Therefore the
wave is moving in –x direction.
(d) , the tension is
(e)

52. Exercises
Problem 2
A triangular wave pulse on a taut string travels in the positive +x direction
with speed v. The tension in the string is F and the linear mass density of
the string is m. At t=0 the shape of the pulse I given by
Draw the pulse at t=0. (b) Determine the wave function y(x,t) at all
times t. (c) Find the instantaneous power in the wave. Show that the
power is zero except for –L < (x-vt) < L and that in this interval the
power is constant. Find the value of this constant.
Solution y
(a) h L -L x

53. Exercises Problem 2 (cont’d) Solution
(b) The wave moves in the +x direction with speed v, so in the experession
for y(x,0) replace x with –vt:
(c)
Thus the instantaneous power is zero except for –L < (x-vt) < L where
It has the constant value Fv(h/L)2.

54. Exercises
Problem 3
The sound from a trumpet radiates uniformly in all directions in air. At a
distance of 5.00 m from the trumpet the sound intensity level is 52.0 dB.
At what distance is the sound intensity level 30.0 dB?
Solution
The distance is proportional to the reciprocal of the square root of the
intensity and hence to 10 raised to half of the sound intensity levels
divided by 10:

55. Exercises
Problem 4
An organ pipe has two successive harmonics with frequencies 1,372 and
1,764 Hz. (a) Is this an open or stopped pipe? (b) What two harmonics are
these? (c) What is the length of the pipe?
Solution
For an open pipe, the difference between successive frequencies is
the fundamental, in this case 392 Hz, and all frequencies are integer
multiples of this frequency. If this is not the case, the pipe cannot be
an open pipe. For a stopped pipe, the difference between the successive
frequencies is twice the fundamental, and each frequency is an odd
integer multiple of the fundamental. In this case, f1 = 196 Hz, and
1372 Hz = 7f1 , 1764 Hz = 9f1 . So this is a stopped pipe.
(b) n=7 for 1,372 Hz, n=9 for 1,764 Hz.
(c) so

56. Exercises Problem 5 Two identical loudspeakers are located at
points A and B, 2.00 m apart. The loud-
speakers are driven by the same amplifier
and produce sound waves with a frequency
of 784 Hz. Take the speed of sound in air to
be 344 m/s. A small microphone is moved out
from Point B along a line perpendicular to the
line connecting A and B. (a) At what distances
from B will there be destructive interference?
(b) At what distances from B will there be
constructive interference? (c) If the frequency
is made low enough, there will be no positions
along the line BC at which destructive
interference occurs. How low must the
frequency be for this to be the case? A
2.00 m B C x

57. Exercises Problem 5 Solution
(a) If the separation of the speakers is denoted by h, the condition for
destructive interference is
where is an odd multiple of one-half. Adding x to both sides, squaring,
canceling the x2 term from both sides and solving for x gives:
Using and h from the given data yields:
(b) Repeating the above argument for integral values for , constructive
interference occurs at 4.34 m, 1.84 m, 0.86 m, 0.26 m.
(c) If , there will be destructive interference at speaker B.
If , the path difference can never be as large as
The minimum frequency is then v/(2h)=(344 m/s)/(4.0 m)=86 Hz.

58. Exercises
Problem 6
A 2.00 MHz sound wave travels through a pregnant woman’s abdomen
and is reflected from fetal heart wall of her unborn baby. The heart wall is
moving toward the sound receiver as the heart beats. The reflected sound
is then mixed with the transmitted sound, and 5 beats per second are
detected. The speed of sound in body tissue is 1,500 m/s. Calculate the
speed of the fetal heart wall at the instance this measurement is made.
Solution
Let f0=2.00 MHz be the frequency of the generated wave. The frequency
with which the heart wall receives this wave is fH=[(v+vH)/v]f0, and this is
also the frequency with which the heart wall re-emits the wave. The detected
frequency of this reflected wave is f’=[v/(v-vH )]fH, with the minus sign indicating
that the heart wall, acting now as a source of waves, is moving toward the
receiver. Now combining f’=[(v+vH)/(v-vH)]f0, and the beat frequency is:
Solving for vH ,