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Chapter 17

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.

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Chapter 17

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  1. Chapter 17 Sound Waves

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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

  7. 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

  8. 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 – wt) • 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

  9. Periodic Sound Waves, Pressure • The variation in gas pressure, DP, is also periodic. • DP = DPmax sin (kx – wt) • DPmax is the pressure amplitude. • It is the maximum change in pressure from the equilibrium value. • k is the wave number. • w is the angular frequency. • The pressure can be related to the displacement. • This relationship is given by DPmax = B smax k. • B is the bulk modulus of the material. Section 17.1

  10. 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

  11. 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

  12. 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

  13. 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 Section 17.2

  14. 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 r Section 17.2

  15. 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. Section 17.2

  16. 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. Section 17.2

  17. 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

  18. Speed of Sound in Gases, Example Values Section 17.2

  19. Energy of Periodic Sound Waves • Consider an element of air with mass Dm and length Dx. • 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

  20. 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. Section 17.3

  21. 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 rv(wsmax)2 • Therefore, the intensity of a periodic sound wave is proportional to the • Square of the displacement amplitude • Square of the angular frequency Section 17.3

  22. Intensity, cont. • In terms of the pressure amplitude, Section 17.3

  23. 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

  24. 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. Section 17.3

  25. 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, b Section 17.3

  26. 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. • b is in decibels (dB) • Threshold of pain: I = 1.00 W/m2; b = 120 dB • Threshold of hearing: I0 = 1.00 x 10-12 W/ m2 corresponds to b = 0 dB Section 17.3

  27. Sound Level, Example • What is the sound level that corresponds to an intensity of 2.0 x 10-7 W/m2 ? b = 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

  28. Sound Levels Section 17.3

  29. 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

  30. Loudness and Frequency, cont. Section 17.3

  31. 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

  32. 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

  33. Doppler Effect, Observer Moving, cont • The speed of the sound is v, the frequency is ƒ, and the wavelength is l • 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. Section 17.4

  34. 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

  35. 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. Section 17.4

  36. 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. Section 17.4

  37. 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

  38. 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

  39. 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

  40. Doppler Effect, Submarine Example cont. • Approaching each other: • Receding from each other: Section 17.4

  41. 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 q = 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

  42. 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

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