Chapter 17

Sound Waves: part one

Introduction to Sound Waves

Sound waves are longitudinal waves They travel through any material medium The speed of the wave depends on the

properties of the medium The mathematical description of sinusoidal

sound waves is very similar to sinusoidal waves on a string

Categories of Sound Waves

The categories cover different frequency ranges Audible waves are within the sensitivity of the

human ear Range is approximately 20 Hz to 20 kHz

Infrasonic waves have frequencies below the audible range

Ultrasonic waves have frequencies above the audible range

Speed of Sound Waves

Use a compressible gas as an example with a setup as shown at right

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 the

diagram

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

The speed of the piston is not the same as the speed of the wave

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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:

elastic property

inertial propertyv

Speed of Sound in Liquid or Gas

The bulk modulus of the material is B The density of the material is The speed of sound in that medium is

Bv

Speed of Sound in a Solid Rod

The Young’s modulus of the material is Y The density of the material is The speed of sound in the rod is

Yv

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

C(331 m/s) 1273 C

Tv

Speed of Sound in Gases, Example Values

Note temperatures, speeds are in m/s

Speed of Sound in Liquids, Example Values

Speeds are in m/s

Periodic Sound Waves

A compression moves through a material as a pulse, continuously compressing the material just in front of it

The areas of compression alternate with areas of lower pressure and density called rarefactions

These two regions move with the speed equal to the speed of sound in the medium

Periodic Sound Waves, Example

A longitudinal wave is propagating through a gas-filled tube

The source of the wave is an oscillating piston

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

Use the active figure to vary the frequency of the piston

Periodic Sound Waves, cont

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 from the equilibrium

position This is also called the displacement amplitude of

the wave

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 also given by Pmax = vsmax

k is the wave number (in both equations) is the angular frequency (in both equations)

s (x, t) = smax cos (kx – t)

17.4 Traveling Sound Waves

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.

Example, Pressure and Displacement Amplitudes

17.5: Interference

17.5: Interference

Phase difference can be related to path length difference L, by noting that a phase difference of 2 rad corresponds to one wavelength.

Therefore,

Fully constructive interference occurs when is zero, 2, or any integer multipleof 2.

Fully destructive interference occurs when is an odd multiple of :

Standing Waves

The resultant wave will be y = (2A sin kx) cos t

This is the wave function of a standing wave There is no kx – t term, and

therefore it is not a traveling wave

In observing a standing wave, there is no sense of motion in the direction of propagation of either of the original waves

Standing Waves in Air Columns

Standing waves can be set up in air columns as the result of interference between longitudinal sound waves traveling in opposite directions

The phase relationship between the incident and reflected waves depends upon whether the end of the pipe is opened or closed

Waves under boundary conditions model can be applied

Standing Waves in Air Columns, Closed End

A closed end of a pipe is a displacement node in the standing wave The rigid barrier at this end will not allow

longitudinal motion in the air The closed end corresponds with a pressure

antinode It is a point of maximum pressure variations The pressure wave is 90o out of phase with the

displacement wave

Standing Waves in Air Columns, Open End

The open end of a pipe is a displacement antinode in the standing wave As the compression region of the wave exits the

open end of the pipe, the constraint of the pipe is removed and the compressed air is free to expand into the atmosphere

The open end corresponds with a pressure node It is a point of no pressure variation

Standing Waves in an Open Tube

Both ends are displacement antinodes The fundamental frequency is v/2L

This corresponds to the first diagram The higher harmonics are ƒn = nƒ1 = n (v/2L) where n = 1, 2,

3, …

Standing Waves in a Tube Closed at One End

The closed end is a displacement node The open end is a displacement antinode The fundamental corresponds to ¼ The frequencies are ƒn = nƒ = n (v/4L) where n = 1, 3, 5,

…

More About Instruments

Musical instruments based on air columns are generally excited by resonance

The air column is presented with a sound wave rich in many frequencies

The sound is provided by: A vibrating reed in woodwinds Vibrations of the player’s lips in brasses Blowing over the edge of the mouthpiece in a flute

Resonance in Air Columns, Example

A tuning fork is placed near the top of the tube

When L corresponds to a resonance frequency of the pipe, the sound is louder

The water acts as a closed end of a tube

The wavelengths can be calculated from the lengths where resonance occurs

Spatial and Temporal Interference

Spatial interference occurs when the amplitude of the oscillation in a medium varies with the position in space of the element This is the type of interference discussed so far

Temporal interference occurs when waves are periodically in and out of phase There is a temporal alternation between

constructive and destructive interference

Beats

Temporal interference will occur when the interfering waves have slightly different frequencies

Beating is the periodic variation in amplitude at a given point due to the superposition of two waves having slightly different frequencies

Beat Frequency

The number of amplitude maxima one hears per second is the beat frequency

It equals the difference between the frequencies of the two sources

The human ear can detect a beat frequency up to about 20 beats/sec

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Beats, Final

The amplitude of the resultant wave varies in time according to

Therefore, the intensity also varies in time

The beat frequency is ƒbeat = |ƒ1 – ƒ2|

1 2resultant

ƒ ƒ2 cos2

2A A t