Waves – Chapter 14

Transverse Wave Pulse

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.

http://paws.kettering.edu/~drussell/demos.html

A wave is a disturbance that propagates from one place to another.

Very cool explanations and animations (Dan Russel, Kettering):

Transverse Harmonic WaveTime = 0

Time = T

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 :

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:

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:

Together, the wave equation:

a wiggle, in space

Each point in SHM

Together: a wiggle moving with time

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:

Out to Sea

t

t + Δt

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 ?

Out to Sea

t

t + Δt

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 v = f λλ = = λλ / T, / T, hence T = T = λλ / v / v. If λλ = = 20 m20 mand v = v = 5 m/s5 m/s, then T = T = 4 4 secssecs.

The time to go from a crest to a trough is only T/T/22 (half ahalf a

periodperiod), so it takes 2 secs2 secs !!

Types of Waves

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

Sound Waves

Sound waves are longitudinal waves, similar to the waves on a Slinky:

Here, the wave is a series of compressions and stretches.

Speed of Wave on a String

For a string, the wave speed is determined by:

1. the tension in the string, and2. the mass per unit length of the string.

The speed of a wave is determined by the properties of the material through which is

propagates

A larger mass per unit length results in a slower wave speed.

As the tension in the string increases, the speed of waves on the string

increases.

Speed of Wave on a String

The speed increases when the tension increases, and when the mass per length decreases.

speed of a wave on a string:

Wave ReflectionWhen 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.

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.

Sound Waves

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

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.

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.

Sound Bite I

a) the frequency f

b) the wavelength λ

c) the speed of the wave

d) both f and λ

e) both vwave and λ

When a sound wave

passes from air into

water, what

properties of the

wave will change?

Wave speed must change (different medium).Wave speed must change (different medium).

Frequency does not change (determined by the source).

Now, v = fv = fλλ and because vv has changed and ff is

constant thenλλ must also changemust also change.

Sound Bite I

a) the frequency f

b) the wavelength λ

c) the speed of the wave

d) both f and λ

e) both vwave and λ

When a sound wave

passes from air into

water, what

properties of the

wave will change?

Follow-upFollow-up: Does the wave speed increase or decrease in water?: Does the wave speed increase or decrease in water?

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

a) about the same distance

b) about 3 miles

c) about 10 miles

d) about 30 miles

e) about 100 miles

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?

Sound Intensity II

a) about the same distance

b) about 3 miles

c) about 10 miles

d) about 30 miles

e) about 100 miles

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?

Sound Intensity II

Remember that intensity

drops with the inverse inverse

square of the distancesquare of the distance, so if intensity drops by a factor of 10, the other person must be ~10 farther away, which is about a factor of 3.

The Doppler EffectThe 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.

The Doppler Effect, moving Observer

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

The distance between peaks is the wavelengthThe time between peaks is:

(stationary) (moving)

Since the distance between peaks is the same:

The Doppler Effect, moving SourceThe 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?

The Doppler Effect, moving Source

Given the speed of sound, this new wavelength corresponds to a specific frequency:

minus for source moving toward observerplus for source moving away from observer

The Doppler Effect

These results can be combined for the case where both observer and source are moving:

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?

Under the right conditions, the shock wavefront as a jet goes supersonic will

condense water vapor and become visible

Doppler radar showing the “hook echo” characteristic

of tornado formation.

Doppler effect can measure velocity.

Doppler blood flow meter

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

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

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 sourceapproach of the source toward the stationary observer, the frequency is shifted higherfrequency is shifted higher.

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

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 you are now a moving observer approaching the sound waveapproaching the sound wave of frequency f1 so you will hear

an even higher frequencyeven higher frequency.

Doppler Effect

Superposition and InterferenceWaves 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

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

Two-dimensional waves exhibit interference as well. This is an example of an interference pattern.

A: Constructive

B: Destructive

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.

Constructive: Δl = nλλλλ

Destructive: Δl = (n+1/2)λλλλ

L

A

B

a) intensity increases

b) intensity stays the same

c) intensity goes to zero

d) impossible to tell

Interference

Speakers A and B emit sound waves of λ = 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?

L

A

B

If λλ = 1 m = 1 m, then a shift of 2.5 m2.5 m corresponds to 2.52.5λ, which puts the two waves out of phaseout of phase, leading to

destructive interferencedestructive interference. The sound intensity will therefore go to zero.

Speakers A and B emit sound waves of λ = 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

Interference

Follow-upFollow-up: What if you : What if you move speaker move speaker AA back back

by 4 m?by 4 m?

Standing WavesA standing wave is fixed in location,

but oscillates with time.

The fundamental, or lowest, frequency on a fixed string has a wavelength twice the length of the string.

These waves are found on strings with both ends fixed, or vibrating columns of air, such as in a musical

instrument.

Higher frequencies are called harmonics.

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 Harmoni

c

Second Harmoni

c

Third Harmoni

c

First Harmoni

c

Second Harmoni

c

Third Harmoni

c

Musical Strings

In a piano, the strings vary in both length and density. This gives the sound box of a grand piano its characteristic shape.

A guitar has strings that are all the same length, but the density varies.

Musical instruments are usually designed so that the variation in tension between the different strings is small; this helps prevent warping and other damage.

Standing Waves IA 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

b

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.

Standing Waves IA 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

Every point in in SHM, with the amplitude fixed for each position

Standing Waves in Air TubesStanding 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).

Standing WavesWith one end closed and one open:

the fundamental wavelength is four times the length of the pipe, and only odd-numbered harmonics appear.

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.

Musical Tones

Frequency doubles for octave steps of the same note

Human Perception: equal steps in pitch are not additive steps, but rather equal multiplicative factors

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.

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

Beats

Beats

1 1

2 2

1 2 1 2

1 2 1 2

cos 2

cos 2

cos 2 cos 2

2 cos 2 cos 22 2

Slow Fast

y A f t

y A f t

y y A f t A f t

f f f fA

Beats

If two sounds are very close in frequency, their sum also has a periodic time dependence: f beat = |f1 - f2|, NOT

Beats are an interference pattern in time, rather than in space.

1 2

2

f f