Chapter 17

Sound Waves

Introduction to Sound Waves

Waves can move through three-dimensional bulk media.

Sound waves are longitudinal waves.

They travel through any material medium.

Commonly experienced as the mechanical waves traveling through air that result in the human perception of hearing

As the sound wave travels through the air, elements of air are disturbed from their equilibrium positions.

Accompanying these movements are changes in density and pressure of the air.

The mathematical description of sinusoidal sound waves is very similar to sinusoidal waves on a string.

Introduction

Categories of Sound Waves

The categories cover different frequency ranges.

Audible waves are within the sensitivity of the human ear.

Infrasonic waves have frequencies below the audible range.

Ultrasonic waves have frequencies above the audible range.

Introduction

Speed of Sound Waves

The diagram shows the motion of a one-dimensional longitudinal sound pulse moving through a long tube containing a compressible gas.

The piston on the left end can be quickly moved to the right to compress the gas and create the pulse.

Before the piston is moved, the gas has uniform density.

When the piston is suddenly moved to the right, the gas just in front of it is compressed.

Darker region in b

The pressure and density in this region are higher than before the piston was pushed.

Section 17.1

Speed of Sound Waves, cont

When the piston comes to rest, the compression region of the gas continues to move.

This corresponds to a longitudinal pulse traveling through the tube with speed v.

Section 17.1

Producing a Periodic Sound Wave

A one-dimensional periodic sound wave can be produced by causing the piston to move in simple harmonic motion.

The darker parts of the areas in the figures represent areas where the gas is compressed and the density and pressure are above their equilibrium values.

The compressed region is called a compression.

Section 17.1

Producing a Periodic Sound Wave, cont.

When the piston is pulled back, the gas in front of it expands and the pressure and density in this region ball below their equilibrium values.

The low-pressure regions are called rarefactions.

They also propagate along the tube, following the compressions.

Both regions move at the speed of sound in the medium.

The distance between two successive compressions (or rarefactions) is the wavelength.

Section 17.1

Periodic Sound Waves, Displacement

As the regions travel through the tube, any small element of the medium moves with simple harmonic motion parallel to the direction of the wave.

The harmonic position function is

s (x, t) = smax cos (kx – t)

smax is the maximum position of the element relative to equilibrium. This is also called the displacement amplitude of the wave.

k is the wave number.

ω is the angular frequency of the wave.

Note the displacement of the element is along x, in the direction of the sound wave.

Section 17.1

Periodic Sound Waves, Pressure

The variation in gas pressure, P, is also periodic.

P = Pmax sin (kx – t)

Pmax is the pressure amplitude. It is the maximum change in pressure from the equilibrium value.

k is the wave number.

is the angular frequency.

The pressure can be related to the displacement.

This relationship is given by Pmax = B smax k. B is the bulk modulus of the material.

Section 17.1

Periodic Sound Waves, final

A sound wave may be considered either a displacement wave or a pressure wave.

The pressure wave is 90o out of phase with the displacement wave.

The pressure is a maximum when the displacement is zero, etc.

Section 17.1

Speed of Sound in a Gas

Consider an element of the gas between the piston and the dashed line.

Initially, this element is in equilibrium under the influence of forces of equal magnitude.

There is a force from the piston on left.

There is another force from the rest of the gas.

These forces have equal magnitudes of PA. P is the pressure of the gas. A is the cross-sectional area of the tube.

Section 17.2

Speed of Sound in a Gas, cont.

After a time period, Δt, the piston has moved to the right at a constant speed vx.

The force has increased from PA to (P+ΔP)A.

The gas to the right of the element is undisturbed since the sound wave has not reached it yet.

Section 17.2

Impulse and Momentum

The element of gas is modeled as a non-isolated system in terms of momentum.

The force from the piston has provided an impulse to the element, which produces a change in momentum.

The impulse is provided by the constant force due to the increased pressure:

The change in pressure can be related to the volume change and the bulk modulus:

Therefore, the impulse is

( )t A P t ˆ= Δ = Δ Δ∑I F ir r

xvVP B B

V v

=− =

xvAB t

vˆ⎛ ⎞= ⎜ ⎟

⎝ ⎠I ir

Section 17.2

Impulse and Momentum, cont.

The change in momentum of the element of gas of mass m is

Setting the impulse side of the equation equal to the momentum side and simplifying, the speed of sound in a gas becomes.

The bulk modulus of the material is B.

The density of the material is

( )xm pvv A t ˆΔ = Δ = Δp v ir r

Bv

=

Section 17.2

Speed of Sound Waves, General

The speed of sound waves in a medium depends on the compressibility and the density of the medium.

The compressibility can sometimes be expressed in terms of the elastic modulus of the material.

The speed of all mechanical waves follows a general form:

For a solid rod, the speed of sound depends on Young’s modulus and the density of the material.

elastic property

inertial propertyv =

Section 17.2

Speed of Sound in Air

The speed of sound also depends on the temperature of the medium.

This is particularly important with gases.

For air, the relationship between the speed and temperature is

The 331 m/s is the speed at 0o C.

TC is the air temperature in Celsius.

Tv C(331 m/s) 1

273= +

Section 17.2

Relationship Between Pressure and Displacement

The pressure amplitude and the displacement amplitude are related by

ΔPmax = B smax k

The bulk modulus is generally not as readily available as the density of the gas.

By using the equation for the speed of sound, the relationship between the pressure amplitude and the displacement amplitude for a sound wave can be found.

ΔPmax = ρ v ω smax

Section 17.2

Speed of Sound in Gases, Example Values

Section 17.2

Energy of Periodic Sound Waves

Consider an element of air with mass m and length x.

Model the element as a particle on which the piston is doing work.

The piston transmits energy to the element of air in the tube.

This energy is propagated away from the piston by the sound wave.

Section 17.3

Power of a Periodic Sound Wave

The rate of energy transfer is the power of the wave.

The average power is over one period of the oscillation.

( )x

avg

Power

Power Av s2 2max

1

2ρ ω

=

=

F vr rg

Section 17.3

Intensity of a Periodic Sound Wave

The intensity, I, of a wave is defined as the power per unit area.

This is the rate at which the energy being transported by the wave transfers through a unit area, A, perpendicular to the direction of the wave.

In the case of our example wave in air,

I = ½ v(smax)2

Therefore, the intensity of a periodic sound wave is proportional to the

Square of the displacement amplitude

Square of the angular frequency

( )avg

PowerI

A=

Section 17.3

Intensity, cont.

In terms of the pressure amplitude,

( )2

max

2

PI

vρ

Δ=

Section 17.3

A Point Source

A point source will emit sound waves equally in all directions.

This can result in a spherical wave.

This can be represented as a series of circular arcs concentric with the source.

Each surface of constant phase is a wave front.

The radial distance between adjacent wave fronts that have the same phase is the wavelength λ of the wave.

Radial lines pointing outward from the source, representing the direction of propagation, are called rays.

Section 17.3

Intensity of a Point Source

The power will be distributed equally through the area of the sphere.

The wave intensity at a distance r from the source is

This is an inverse-square law.

The intensity decreases in proportion to the square of the distance from the source.

( ) ( )avg avg

Power PowerI

A r 24π= =

Section 17.3

Sound Level

The range of intensities detectible by the human ear is very large.

It is convenient to use a logarithmic scale to determine the intensity level,

10logo

I

I

⎛ ⎞= ⎜ ⎟

⎝ ⎠

Section 17.3

Sound Level, cont

I0 is called the reference intensity.

It is taken to be the threshold of hearing.

I0 = 1.00 x 10-12 W/ m2

I is the intensity of the sound whose level is to be determined.

is in decibels (dB)

Threshold of pain: I = 1.00 W/m2; = 120 dB

Threshold of hearing: I0 = 1.00 x 10-12 W/ m2 corresponds to = 0 dB

Section 17.3

Sound Level, Example

What is the sound level that corresponds to an intensity of 2.0 x 10-7 W/m2 ?

= 10 log (2.0 x 10-7 W/m2 / 1.0 x 10-12 W/m2)

= 10 log 2.0 x 105

= 53 dB

Rule of thumb: A doubling in the loudness is approximately equivalent to an increase of 10 dB.

Section 17.3

Sound Levels

Section 17.3

Loudness and Frequency

Sound level in decibels relates to a physical measurement of the strength of a sound.

We can also describe a psychological “measurement” of the strength of a sound.

Our bodies “calibrate” a sound by comparing it to a reference sound.

This would be the threshold of hearing.

Actually, the threshold of hearing is 10-12 W/m2 only for 1000 Hz.

There is a complex relationship between loudness and frequency.

Fig. 17.7 shows this relationship:

The white area shows average human response to sound.

The lower curve of the white area shows the threshold of hearing.

The upper curve shows the threshold of pain.

Section 17.3

Loudness and Frequency, cont.

Section 17.3

The Doppler Effect

The Doppler effect is the apparent change in frequency (or wavelength) that occurs because of motion of the source or observer of a wave.

When the relative speed of the source and observer is higher than the speed of the wave, the frequency appears to increase.

When the relative speed of the source and observer is lower than the speed of the wave, the frequency appears to decrease.

Section 17.4

Doppler Effect, Observer Moving

The observer moves with a speed of vo.

Assume a point source that remains stationary relative to the air.

It is convenient to represent the waves as wave fronts.

These surfaces are called wave fronts.

The distance between adjacent wave fronts is the wavelength.

Section 17.4

Doppler Effect, Observer Moving, cont

The speed of the sound is v, the frequency is ƒ, and the wavelength is

When the observer moves toward the source, the speed of the waves relative to the observer is v ’ = v + vo.

The wavelength is unchanged.

The frequency heard by the observer, ƒ ’, appears higher when the observer approaches the source.

The frequency heard by the observer, ƒ ’, appears lower when the observer moves away from the source.

ƒ ' ƒov v

v

+⎛ ⎞=⎜ ⎟⎝ ⎠

ƒ ' ƒov v

v

−⎛ ⎞=⎜ ⎟⎝ ⎠

Section 17.4

Doppler Effect, Source Moving

Consider the source being in motion while the observer is at rest.

As the source moves toward the observer, the wavelength appears shorter.

As the source moves away, the wavelength appears longer.

Section 17.4

Doppler Effect, Source Moving, cont

When the source is moving toward the observer, the apparent frequency is higher.

When the source is moving away from the observer, the apparent frequency is lower.

ƒ ' ƒs

v

v v

⎛ ⎞=⎜ ⎟−⎝ ⎠

ƒ ' ƒs

v

v v

⎛ ⎞=⎜ ⎟+⎝ ⎠

Section 17.4

Doppler Effect, General

Combining the motions of the observer and the source

The signs depend on the direction of the velocity.

A positive value is used for motion of the observer or the source toward the other.

A negative sign is used for motion of one away from the other.

ƒ ' ƒo

s

v v

v v

⎛ ⎞+=⎜ ⎟−⎝ ⎠

Section 17.4

Doppler Effect, final

Convenient rule for signs.

The word “toward” is associated with an increase in the observed frequency.

The words “away from” are associated with a decrease in the observed frequency.

The Doppler effect is common to all waves.

The Doppler effect does not depend on distance.

Section 17.4

Doppler Effect, Water Example

A point source is moving to the right .

The wave fronts are closer on the right.

The wave fronts are farther apart on the left.

Section 17.4

Doppler Effect, Submarine Example

Sub A (source) travels at 8.00 m/s emitting at a frequency of 1400 Hz.

The speed of sound in the water is 1533 m/s.

Sub B (observer) travels at 9.00 m/s.

What is the apparent frequency heard by the observer as the subs approach each other? Then as they recede from each other?

Section 17.4

Doppler Effect, Submarine Example cont.

Approaching each other:

Receding from each other:

( )( )

1533 m s 9.00 m sƒ ' ƒ (1400 )

1533 m s 8.00 m s

1416

o

s

v vHz

v v

Hz

⎛ ⎞+ +⎛ ⎞+= = ⎜ ⎟⎜ ⎟ ⎜ ⎟− − +⎝ ⎠ ⎝ ⎠=

( )( )

1533 m s 9.00 m sƒ ' ƒ (1400 )

1533 m s 8.00 m s

1385

o

s

v vHz

v v

Hz

⎛ ⎞+ −⎛ ⎞+= = ⎜ ⎟⎜ ⎟ ⎜ ⎟− − −⎝ ⎠ ⎝ ⎠=

Section 17.4

Shock Waves and Mach Number

The speed of the source can exceed the speed of the wave.

The envelope of these wave fronts is a cone whose apex half-angle is given by sin =v/vS.

This is called the Mach angle.

The ratio vs / v is referred to as the Mach number .

The relationship between the Mach angle and the Mach number is

Section 17.4

sins s

vt v

v t v = =

Shock Wave, final

The conical wave front produced when vs > v is known as a shock wave.

This is a supersonic speed.

The shock wave carries a great deal of energy concentrated on the surface of the cone.

There are correspondingly great pressure variations.

Section 17.4