chapter 16 outline sound and hearing sound waves pressure fluctuations speed of sound general fluid...
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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
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