Lecture 18

Waves and Sound

Reading and Review

Summary of Simple Harmonic MotionThe position as a function of time:

The angular frequency:

From this comparison with circular motion, we can see:

The velocity as a function of time:

The acceleration as a function of time:

Common Simple Harmonic Oscillators• Period of a mass on a spring:

• Total energy in simple harmonic motion:

• A simple pendulum with small amplitude exhibits simple harmonic motion

• Period of a simple pendulum:

• Period of a physical pendulum:

Damping and Resonance• Oscillations where there is a nonconservative force are called damped.

• Underdamped: the amplitude decreases exponentially with time:

• Critically damped: no oscillations; system relaxes back to equilibrium in minimum time

• Overdamped: also no oscillations, but slower than critical damping• An oscillating system may be driven by an external force

• Resonance occurs when the driving frequency is equal to the natural frequency of the system

A mass on a spring oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

a) total energy will increase

b) total energy will not change

c) total energy will decrease

Reading Quiz Reading Quiz Energy in SHM IEnergy in SHM I

A mass on a spring oscillates in simple harmonic motion with amplitude A. If the mass is doubled, but the amplitude is not changed, what will happen to the total energy of the system?

a) total energy will increase

b) total energy will not change

c) total energy will decrease

The total energy is equal to the initial value of the elastic potential energy, which is PEs = kA2. This

does not depend on mass, so a change in mass will not affect the energy of the system.

Reading Quiz Reading Quiz Energy in SHM IEnergy in SHM I

Follow-upFollow-up: What happens if you double the amplitude?: What happens if you double the amplitude?

A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change?

a) period will increase

b) period will not change

c) period will decrease

Reading Quiz Reading Quiz Spring on the MoonSpring on the Moon

A mass oscillates on a vertical spring with period T. If the whole setup is taken to the Moon, how does the period change?

a) period will increase

b) period will not change

c) period will decrease

The period of simple harmonic motion depends only on the mass and the spring constant and does not depend on the acceleration due to gravity. By going to the Moon, the value of g has been reduced, but that does not affect the period of the oscillating mass–spring system.

Reading Quiz Reading Quiz Spring on the MoonSpring on the Moon

Follow-upFollow-up: How would a pendulum change between the earth and the : How would a pendulum change between the earth and the moon?moon?

A pendulum is suspended from the ceiling of an elevator. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward?

a) period will increase

b) period will not change

c) period will decrease

Pendulum in ElevatorPendulum in Elevator

A pendulum is suspended from the ceiling of an elevator. When the elevator is at rest, the period is T. What happens to the period when the elevator is accelerating upward?

a) period will increase

b) period will not change

c) period will decrease

Pendulum in ElevatorPendulum in Elevator

Follow-upFollow-up: What if the elevator moves up with constant velocity?: What if the elevator moves up with constant velocity?

When the elevator accelerates upward, the string must apply a net vertical force of (mg + maelevator). This is equivalent to changing the effective value of g due to the acceleration of the elevator. Because the period depends inversely on g, and the effective value of g increased, then the period of the pendulum will decrease (i.e., its frequency will increase and it will swing faster).

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:

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:

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.

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.

Wave Motion IWave Motion I

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled A ?

A

a)

b)

c)

d)

e) velocity is zero

Wave Motion IWave Motion I

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled A ?

The velocity of an

oscillating particle

is (momentarilymomentarily) zerozero

at its maximum

displacement.

A

Follow-upFollow-up: What is the acceleration of the particle at point A?: What is the acceleration of the particle at point A?

a)

b)

c)

d)

e) velocity is zero

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled B ?

a)

b)

c)

d)

e) zero

Wave Motion IIWave Motion II

B

Consider a wave on a string moving to the right, as shown below.

What is the direction of the velocity of a particle at the point labeled B ?

The wave is moving

to the rightright, so the

particle at B has to

be moving upwardmoving upward

in the next instant of

time.

B

Wave Motion IIWave Motion II

Follow-upFollow-up: What is the acceleration of the particle at point : What is the acceleration of the particle at point BB??

a)

b)

c)

d)

e) zero

Out to SeaOut 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 SeaOut 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 !!

Lunch TimeLunch Timea) 0.3 mm

b) 3 cm

c) 30 cm

d) 300 m

e) 3 km

Microwaves travel with the speed of

light, c = 3 108 m/s. At a frequency

of 10 GHz these waves cause the water

molecules in your burrito to vibrate.

What is the microwave wavelength?

1 GHz = 1 Gigahertz = 109 cycles/sec

H

H

O

Lunch TimeLunch Time

We know vwave = = f λ

so λ = =

λ = 3 = 3 10 10−−22 mm = 3 = 3 cmcm

3 108 m s

10 109 Hz

a) 0.3 mm

b) 3 cm

c) 30 cm

d) 300 m

e) 3 km

Microwaves travel with the speed of

light, c = 3 108 m/s. At a frequency of

10 GHz these waves cause the water

molecules in your burrito to vibrate.

What is the microwave wavelength?

1 GHz = 1 Gigahertz = 109 cycles/sec

H

H

O

/

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.

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.

Ultrasound

Frequencies higher than human perception have common use in medical imaging

Sound Bite ISound 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 then λλ must also changemust also change.

Sound Bite ISound 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?

If you fill your lungs with helium and then try talking, you sound like Mickey Mouse. What conclusion can you reach about the speed of sound in helium, compared to that of air?

a) speed of sound is less in helium

b) speed of sound is the same in helium

c) speed of sound is greater in helium

d) this effect has nothing to do with the speed in helium

Speed of Sound IIISpeed of Sound III

a) speed of sound is less in helium

b) speed of sound is the same in helium

c) speed of sound is greater in helium

d) this effect has nothing to do with the speed in helium

The higher pitch implies a higher frequency. In turn, because v = fλ, this means that the speed of the wave has increased (as long as the wavelength, determined by the length of the vocal chords, remains constant).

Speed of Sound IIISpeed of Sound III

Follow-upFollow-up: Why is the speed of sound greater in helium than in : Why is the speed of sound greater in helium than in air?air?

If you fill your lungs with helium and then try talking, you sound like Mickey Mouse. What conclusion can you reach about the speed of sound in helium, compared to that of air?

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

Sound Intensity

Expressed in terms of power,

Sound intensity from a point source will decrease as the

square of the distance.

R1

R2

Surface area of a sphere = 4π r2

If power output is constant in time, then intensity compared to distance from source:- at R1 must be P / (4πR1

2)- at R2 must be P / (4πR2

2)

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 IISound 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 IISound Intensity II

Remember that intensity drops with the

inverse square of the inverse square of the

distancedistance, so if intensity drops by a factor of 10, the other person must be √10 farther away, which is about a factor of 3.

Intensity Level

When you listen to a variety of sounds, a sound that seems twice as loud as another is actually ten times more intense. Therefore, we use a logarithmic scale to define intensity values.

Here, I0 is the faintest sound that can be heard:

[dB]

Sound Intensity Level

Intensity level is described using the decibel, dB.

The loudness of sound doubles with each increase in intensity level of 10 dB.

a) about the same

b) about 10 times

c) about 100 times

d) about 1000 times

e) about 10,000 times

A quiet radio has an intensity level of about 40 dB. Busy street traffic has a level of about 70 dB. How much greater is the intensity of the street traffic compared to the radio?

Decibel Level IIDecibel Level II

increase by 10 dB →→ increase intensity by factor of 101 (10)

increase by 20 dB →→ increase intensity by factor of 102 (100)

increase by 30 dB increase by 30 dB →→→→ increase intensity by factor of 10 increase intensity by factor of 1033

(1000)(1000)

a) about the same

b) about 10 times

c) about 100 times

d) about 1000 times

e) about 10,000 times

A quiet radio has an intensity level of about 40 dB. Busy street traffic has a level of about 70 dB. How much greater is the intensity of the street traffic compared to the radio?

Decibel Level IIDecibel Level II

Follow-upFollow-up: What decibel level gives an intensity a million times : What decibel level gives an intensity a million times greater?greater?

When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?

a) more than 120 dB

b) 120 dB

c) between 60 dB and 120 dB

d) 60 dB

e) less than 60 dB

Decibel Level IDecibel Level I

When Mary talks, she creates an intensity level of 60 dB at your location. Alice talks with the same volume, also giving 60 dB at your location. If both Mary and Alice talk simultaneously from the same spot, what would be the new intensity level that you hear?

a) more than 120 dB

b) 120 dB

c) between 60 dB and 120 dB

d) 60 dB

e) less than 60 dB

Recall that a difference of 10 dB in intensity level corresponds to a factor of 101 in intensity. Similarly, a difference of 60 dB in corresponds to a factor of 106 in intensity!! In this case, with two voices adding up, the intensity increases by only a factor of 2, meaning that the intensity level is higher by an amount equal to Δ = 10 log(2) = 3 dB. The new intensity level is = 63 dB.

Decibel Level IDecibel Level I

Human perception (loudness) is logarithmic, frequency dependent

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:

If the observer is moving TOWARDS the source:

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:

The Doppler Effect

Here is a comparison of the Doppler shifts for a moving source and 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.

The Doppler Effect

Combining results gives us the case where both observer and source are moving:

The Doppler effect has many practical applications: weather radar, speed radar, medical diagnostics, astronomical measurements.

Doppler effect can measure velocity.

Here a Doppler ultrasound verifies that the umbilical blood flow in early pregnancy is healthy.

At left, a Doppler radar shows the hook echo characteristic of tornado formation. At right, a medical technician is using a Doppler blood flow meter.