1. Waves – Chapter 14

2. Transverse Wave Pulse
A wave is a disturbance that propagates from one place to another.
The easiest type of wave to visualize is a transverse wave, where the displacement of the medium is perpendicular to the direction of motion of the wave.
Very cool explanations and animations (Dan Russel, Kettering):

3. Transverse Harmonic Wave
Time = 0
Time = T

4. Transverse Harmonic Wave
Wavelength λ: distance over which wave repeats
Period T: time for one wavelength to pass a given point
Frequency f :
Speed v :

5. Harmonic Wave Functions
Since the wave has the same pattern at x + λ as it does at x, at any moment in time the wave must be of the form:
a wiggle,
in space
Since the wave has the same pattern at t=0 as it does at t=T, at fixed position the wave must also be of the form:
Each point
in SHM
Together, the wave equation:
Together: a wiggle moving with time

6. Harmonic Wave Functions
at fixed x, y travels in simple harmonic motion with period T
at fixed t, changes with x with wavelength λ
This implies that the position of the peak changes with time as:

7. Out to Sea
a) 1 second
b) 2 seconds
c) 4 seconds
d) 8 seconds
e) 16 seconds
A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ? t
t + Δt
Answer: b

8. Out to Sea
a) 1 second
b) 2 seconds
c) 4 seconds
d) 8 seconds
e) 16 seconds
A boat is moored in a fixed location, and waves make it move up and down. If the spacing between wave crests is 20 m and the speed of the waves is 5 m/s, how long does it take the boat to go from the top of a crest to the bottom of a trough ?
We know that v = f λ = λ / T, hence T = λ / v. If λ = 20 m and v = 5 m/s, then T = 4 secs.
The time to go from a crest to a trough is only T/2 (half a period), so it takes 2 secs !! t
t + Δt

9. Types of Waves
In a longitudinal wave, the displacement is along the direction of wave motion.

10. Sound Waves
Sound waves are longitudinal waves, similar to the waves on a Slinky:
Here, the wave is a series of compressions and stretches.

11. Speed of Wave on a String
The speed of a wave is determined by the properties of the material through which is propagates
For a string, the wave speed is determined by:
the tension in the string, and
the mass per unit length of the string.
As the tension in the string increases, the speed of waves on the string increases.
A larger mass per unit length results in a slower wave speed.

12. Speed of Wave on a String
speed of a wave on a string:
The speed increases when the tension increases, and when the mass per length decreases.

13. When a wave reaches the end of a string, it will be reflected.
Wave Reflection
When a wave reaches the end of a string, it will be reflected.
If the end is fixed, the reflected wave will be inverted.
If the end of the string is free to move transversely, the wave will be reflected without inversion.

14. Waves of any shape can be decomposed into harmonic waves with different frequencies. In mathematics this is called Fourier decomposition.
So once you understand harmonic waves, you can analyze any wave.

15. Sound Waves
In a sound wave, the density and pressure of the air (or other medium carrying the sound) are the quantities that oscillate.

16. Speed of Sound
The speed of sound is different in different materials; in general, the stiffer a material is, the faster sound travels through it... the denser a material, the slower sound travels through it.

17. Sound Waves
“Sound waves” generally means mechanical compression waves, and can have any frequency. The human ear can hear sound between about 20 Hz and 20,000 Hz.
Sounds with frequencies greater than 20,000 Hz are called ultrasonic; sounds with frequencies less than 20 Hz are called infrasonic.
Ultrasonic waves are familiar from medical applications; elephants and whales communicate, in part, by infrasonic waves.

18. Sound Bite I
When a sound wave passes from air into water, what properties of the wave will change?
a) the frequency f
b) the wavelength 
c) the speed of the wave
d) both f and 
e) both vwave and 
Answer: e

19. Sound Bite I
When a sound wave passes from air into water, what properties of the wave will change?
a) the frequency f
b) the wavelength 
c) the speed of the wave
d) both f and 
e) both vwave and 
Wave speed must change (different medium).
Frequency does not change (determined by the source).
Now, v = f and because v has changed and f is constant then  must also change.
Follow-up: Does the wave speed increase or decrease in water?

20. Sound Intensity
The intensity of a sound is the amount of energy that passes through a given area in a given time.

21. Sound Intensity II
You hear a fire truck with a certain intensity, and you are about 1 mile away. Another person hears the same fire truck with an intensity that is about 10 times less. Roughly, how far is the other person from the fire truck?
a) about the same distance
b) about 3 miles
c) about 10 miles
d) about 30 miles
e) about 100 miles
Answer: b

22. Sound Intensity II
You hear a fire truck with a certain intensity, and you are about 1 mile away. Another person hears the same fire truck with an intensity that is about 10 times less. Roughly, how far is the other person from the fire truck?
a) about the same distance
b) about 3 miles
c) about 10 miles
d) about 30 miles
e) about 100 miles
Remember that intensity drops with the inverse square of the distance, so if intensity drops by a factor of 10, the other person must be ~ farther away, which is about a factor of 3.

23. The Doppler Effect
The Doppler effect is the change in pitch of a sound when the source and observer are moving with respect to each other.
When an observer moves toward a source, the wave speed appears to be higher. Since the wavelength is fixed, the frequency appears to be higher as well.

24. The Doppler Effect, moving Observer
The distance between peaks is the wavelength
The time between peaks is:
(stationary)
(moving)
Since the distance between peaks is the same:
The new observed frequency f’ is:
If the observer were moving away from the source, only the sign of the observer’s speed would change

25. The Doppler Effect, moving Source
The Doppler effect from a moving source can be analyzed similarly. Now it is the wavelength that appears to change:
In one period, how far does the wave move?
So how far apart are the peaks?

26. The Doppler Effect, moving Source
Given the speed of sound, this new wavelength corresponds to a specific frequency:
minus for source moving toward observer
plus for source moving away from observer

27. The Doppler Effect
These results can be combined for the case where both observer and source are moving:

28. The Doppler Effect
The Doppler shift for a moving source compared to that for for a moving observer.
The two are similar for low speeds but then diverge.
If the source moves faster then the speed of sound, a sonic boom is created.
What if the observer is moving away at the speed of sound?
What if the source is moving away at the speed of sound?

29. Under the right conditions, the shock wavefront as a jet goes supersonic will condense water vapor and become visible

30. Doppler effect can measure velocity.
Doppler radar showing the “hook echo” characteristic of tornado formation.
ECHO – moving observer becomes moving source
Doppler blood flow meter

31. Here a Doppler ultrasound measurement is used to verify sufficient umbilical blood flow in early pregnancy

32. Doppler Effect
You are heading toward an island in a speedboat and you see your friend standing on the shore, at the base of a cliff. You sound the boat’s horn to alert your friend of your arrival. If the horn has a rest frequency of f0, what frequency does your friend hear ?
a) lower than f0
b) equal to f0
c) higher than f0
Answer: c

33. Doppler Effect
You are heading toward an island in a speedboat and you see your friend standing on the shore, at the base of a cliff. You sound the boat’s horn to alert your friend of your arrival. If the horn has a rest frequency of f0, what frequency does your friend hear ?
a) lower than f0
b) equal to f0
c) higher than f0
Due to the approach of the source toward the stationary observer, the frequency is shifted higher.

34. Doppler Effect
In the previous question, the horn had a rest frequency of f0, and we found that your friend heard a higher frequency f1 due to the Doppler shift. The sound from the boat hits the cliff behind your friend and returns to you as an echo. What is the frequency of the echo that you hear?
a) lower than f0
b) equal to f0
c) higher than f0 but lower than f1
d) equal to f1
e) higher than f1
Answer: e

35. Doppler Effect
In the previous question, the horn had a rest frequency of f0, and we found that your friend heard a higher frequency f1 due to the Doppler shift. The sound from the boat hits the cliff behind your friend and returns to you as an echo. What is the frequency of the echo that you hear?
a) lower than f0
b) equal to f0
c) higher than f0 but lower than f1
d) equal to f1
e) higher than f1
The sound wave bouncing off the cliff has the same frequency f1 as the one hitting the cliff (what your friend hears). For the echo, you are now a moving observer approaching the sound wave of frequency f1 so you will hear an even higher frequency.

36. Superposition and Interference
Waves of small amplitude traveling through the same medium combine, or superpose, by simple addition.
If two pulses combine to give a larger pulse, this is constructive interference (left). If they combine to give a smaller pulse, this is destructive interference (right).
constructive
destructive

37. constructive destructive
Two waves with distance to the source different by whole integer wavelengths Nλ
Two waves with distance to the source different by half-integer wavelengths Nλ

38. Two-dimensional waves exhibit interference as well
Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.
A: Constructive
B: Destructive

39. Superposition and Interference
If the sources are in phase, points where the distance to the sources differs by an equal number of wavelengths will interfere constructively; in between the interference will be destructive.

40. Constructive: l = n : , 2 , 3 ....
Destructive: l = (n+1/2)  :  /2, 3  /2, 5  /2...

41. Interference
Speakers A and B emit sound waves of  = 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A is moved back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell L A B

42. Follow-up: What if you move speaker A back by 4 m?
Interference
Speakers A and B emit sound waves of  = 1 m, which interfere constructively at a donkey located far away (say, 200 m). What happens to the sound intensity if speaker A is moved back 2.5 m?
a) intensity increases
b) intensity stays the same
c) intensity goes to zero
d) impossible to tell
If  = 1 m, then a shift of 2.5 m corresponds to 2.5 , which puts the two waves out of phase, leading to destructive interference. The sound intensity will therefore go to zero. A
Follow-up: What if you move speaker A back by 4 m? L B

43. A standing wave is fixed in location, but oscillates with time.
Standing Waves
A standing wave is fixed in location, but oscillates with time.
These waves are found on strings with both ends fixed, or vibrating columns of air, such as in a musical instrument.
The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string.
Higher frequencies are called harmonics.

44. Standing Waves on a String
Points on the string which never move are called nodes; those which have the maximum movement are called antinodes.
There must be an integral number of half-wavelengths on the string (must have nodes at the fixed ends).
This means that only certain frequencies (for fixed tension, mass density, and length) are possible.
First Harmonic
Second Harmonic
Third Harmonic

45. First Harmonic
Second Harmonic
Third Harmonic

46. Musical Strings
Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.
A guitar has strings that are all the same length, but the density varies.
In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape.

47. Standing Waves I
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string a b
Answer: a

48. Standing Waves I
A string is clamped at both ends and plucked so it vibrates in a standing mode between two extreme positions a and b. Let upward motion correspond to positive velocities. When the string is in position b, the instantaneous velocity of points on the string:
a) is zero everywhere
b) is positive everywhere
c) is negative everywhere
d) depends on the position along the string
Observe two points:
Just before b
Just after b
Both points change direction before and after b, so at b all points must have zero velocity.
Every point in in SHM, with the amplitude fixed for each position

49. Standing Waves in Air Tubes
Standing waves can also be excited in columns of air, such as soda bottles, woodwind instruments, or organ pipes.
A sealed end must be at a NODE (N),
an open end must be an ANTINODE (A).

51. Standing Waves With one end closed and one open:
the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

52. Standing Waves If the tube is open at both ends:
both ends are antinodes, and the sequence of harmonics is the same as that on a string.

53. Musical Tones
Human Perception: equal steps in pitch are not additive steps, but rather equal multiplicative factors
Frequency doubles for octave steps of the same note
The frets on a guitar are used to shorten the string.
Each fret must shorten the string (relative to the previous fret) by the same fraction, to make equal spaced notes.

54. Beats
Two waves with close (but not precisely the same) frequencies will create a time-dependent interference

55. Beats

56. Beats Beats are an interference pattern in time, rather than in space.
If two sounds are very close in frequency, their sum also has a periodic time dependence: f beat = |f1 - f2|, NOT