Chapter 17 Sound Waves (1)

Sound waves are the most common example of longitudinal waves. They travel

through any material medium with a speed that depends on the properties of the

medium.

Sound waves are divided into three categories that cover different frequency

ranges. (1) Audible waves lie within the range of sensitivity of the human ear. They can

be generated in a variety of ways, such as by musical instruments, human voices, or

loudspeakers. (2) Infrasonic waves have frequencies below the audible range. Elephants

can use infrasonic waves to communicate with each other, even when separated by

many kilometers. (3) Ultrasonic waves have frequencies above the audible range. You

may have used a “silent” whistle to retrieve your dog. The ultrasonic sound it emits is

easily heard by dogs, although humans cannot detect it at all. Ultrasonic waves are also

used in medical imaging.

17.1 Speed of Sound Waves

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

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

Speed of Sound in Various Media

 

The speed of sound waves in a medium depends on the compressibility and density of the medium. If the medium is a liquid or a gas and has a bulk modulus B and density r, the speed of sound waves in that medium is

                  (17.1)

It is interesting to compare this expression with Equation 16.18 for the speed of transverse waves on a string, 

 In both cases, the wave speed depends on an elastic property of the 

medium—bulk modulus B or string tension T—and on an inertial property of the medium—r or µ. In fact, the speed of all mechanical waves follows an expression of the general form:

For longitudinal sound waves in a solid rod of material, for example, the speed of sound depends on Young’s modulus Y and the density (r). Table 17.1 provides the speed of sound in several different materials.

The speed of sound also depends on the temperature of the medium. For sound traveling through air, the relationship between wave speed and medium temperature is

where 331 m/s is the speed of sound in air at 0°C, and TC is the air temperature in degrees Celsius. Using this equation, one finds that at 20°C the speed of sound in air is approximately 343 m/s.

Answer: (c). Temperature. Although the speed of a wave is given by the product of its wavelength (a) and frequency (b), it is not affected by changes in either one. The amplitude (d) of a sound wave determines the size of the oscillations of elements of air but does not affect the speed of the wave through the air.

17.2 Periodic Sound Waves

§  pressure variations control what we hear

§  Include the  compression and rarefactions

§  As the piston oscillates sinusoidally, regions of compression and rarefaction are continuously set up

The distance between two successive compressions (or two successive rarefactions) equals the wavelength (l). As these regions travel through the tube, any small element of the medium moves with simple harmonic motion parallel to the direction of the wave. If s(x, t) is the position of a small element relative to its equilibrium position¹ we can express this harmonic position function as

 (17.2)

where s_max is the maximum position of the element relative to equilibrium. This is often called the displacement amplitude of the wave. The parameter k is the wave number and w is the angular frequency of the piston. Note that the displacement of the element is along x, in the direction of propagation of the sound wave, which means we are describing a longitudinal wave.

The variation in the gas pressure (DP) measured from the equilibrium value is also periodic. For the position function in Equation 17.2, DP is given by

            (17.3)

1 We use s(x, t) here instead of y(x, t) because the displacement of elements of the medium is not  perpendicular to the x direction.

where the pressure amplitude DP_max—which is the maximum change in pressure from the equilibrium value—is given by

              (17.4)

Thus, we see that a sound wave may be considered as either a displacement wave or a pressure wave. A comparison of Equations 17.2 and 17.3 shows that the pressure wave is 90° out of phase with the displacement wave. Graphs of these functions are shown in Figure 17.3. Note that the pressure variation is a maximum when the displacement from equilibrium is zero, and the displacement from equilibrium is a maximum when the pressure variation is zero.

Figure 17.3 (a) Displacement

amplitude and (b) pressure

amplitude versus position for a

sinusoidal longitudinal wave.

(c). Because the bottom of the bottle is a rigid barrier, the displacement of elements of air at the bottom is zero. Because the pressure variation is a minimum or a maximum when the displacement is zero, and the pulse is moving downward, the pressure variation at the bottom is a maximum

17.3 Intensity of Periodic Sound Waves

The intensity of a wave, or the power per unit area is the rate at which the energy being transported by the wave transfers through a unit area A perpendicular to the direction of travel of the wave: 

 

Now consider a point source emitting sound waves equally in all directions. From everyday experience, we know that the intensity of sound decreases as we move farther from the source. We identify an imaginary sphere of radius r centered on the source. When a source emits sound equally in all directions, we describe the result as a spherical wave. The average power Pav emitted by the source must be distributed uniformly over this spherical surface of area 4pr². Hence, the wave intensity at a distance r from the source is

This inverse-square law, which is reminiscent of the behavior of gravity in Chapter 13, states that the intensity decreases in proportion to the square of the distance from the source.