Chapter 16 OutlineSound and Hearing

• Sound waves• Pressure fluctuations

• Speed of sound• General fluid

• Ideal gas

• Sound intensity

• Standing waves• Normal modes

• Instruments

• Interference and beats

Sound

• Sound is a longitudinal wave.

• We normally think of sound in air, but it can travel through any medium.

• The displacement of the medium is along the direction of propagation, so we can also describe the wave in terms of pressure fluctuations.

• The maxima in displacement magnitude correspond to the minima in the gauge pressure magnitude.

Sound Wave Pressure Equation

• We can describe the displacement of the medium using the same equation for waves on a string.

• Note that is now parallel to .

• In terms of pressure,

• is the bulk modulus of the medium.

• The pressure amplitude is

Speed of Sound in a Fluid

• Recall that the speed of a wave on a string is given by

• It depends on the tension (related to the restoring force) and the linear density.

• It seems quite intuitive that the speed of sound in a fluid would take a similar form.

• For a fluid, the bulk modulus, describes the restoring force and the relevant density is the volume density, .

Speed of Sound in an Ideal Gas

• The bulk moduli of gases are much smaller than solids or liquids, but are not constant. In general,

• is the adiabatic constant ( for air), and is the equilibrium pressure of the gas.

• In an ideal gas, .• Combining these with the previous equation for the speed of

sound in a fluid,

• Where is the gas constant, is the temperature in , and is the molar mass of the gas.

Speed of Sound in Air

• At (), what is the speed of sound in air?

• It is very important that the temperature is expressed in ! • For air, , and since the atmosphere is mostly nitrogen, .

• Since humans can hear frequencies from about , this corresponds to wavelengths from about to .

Sound Intensity

• Power can be expressed as force times velocity, so intensity (average power per area) can be expressed as the average of the pressure times velocity.

• The average of is just , and using and , so

Decibel Scale

• Perceived loudness is not directly proportional to sound intensity, but follows a roughly logarithmic relationship.

• If the intensity of a sound is increased by a factor of ten, it will sound about twice as loud.

• The loudness also depends on frequency.

• The sound intensity level, , is given by the decibel scale.

• Where the reference intensity is approximately the threshold of human hearing at .

Sound Intensity Level Example

Standing Sound Waves

• Just as we had standing waves on strings, we can set up standing waves in air columns.

• This is the basis of wind instruments.

• As we showed earlier, we can describe a sound wave in terms of the displacement of the medium or the pressure.

• A pressure node is always a displacement node, and vice versa.

Standing Sound Waves

• Consider a wave traveling down a pipe. When it reaches an end, it will be reflected.

• If the end is closed, the displacement at the end must be the zero.

• This is a displacement node (pressure antinode).

• If the end is open, the pressure must be the same as the atmospheric (constant) pressure.

• This is a pressure node (displacement antinode).

• In the following diagrams, the waves drawn will represent the displacement.

Pipe Open at Both Ends

• If a pipe is open at both ends, there must be a displacement antinode at each end.

• Recall that the distance between adjacent antinodes is .

• The wavelength and frequencies can be found in the same manner we used for strings.

Where

Pipe Closed at One End (Stopped Pipe)

• If a pipe is open at one end and closed at the other, there must be a displacement antinode at one end and displacement node at the other.

• This asymmetry eliminates half of the expected harmonics.

• Also, the fundamental frequency is cut in half.

Where

Standing Waves Example

Interference and Beats

• Standing waves are an example of interference of waves with the same frequency. What if we have two waves with slightly different frequencies?

• The superposition of the waves will look sinusoidal but with a varying amplitude.

Interference and Beats

• The frequency of these beats can be calculated.• If the waves start in phase, they will again be in phase later.

• During this time, the higher frequency wave will have gone through cycles and the lower frequency wave cycles.

and

• Eliminating the and solving for , the beat frequency is simply the difference between the two frequencies.

• This can be quite useful in tuning an instrument.

Beats Example

Chapter 16 SummarySound and Hearing

• Sound waves• Pressure fluctuations:

• Speed of sound• Fluid:

• Ideal gas:

• Sound intensity: • Decibel scale:

Chapter 16 SummarySound and Hearing

• Standing waves• Open-open pipe: , where

• Open-closed pipe: , where

• Interference and beats