Periodic Sound Waves
This topic will help you better comprehend the nature of sound waves. You will learn that pressure variations control what we hear—an important fact for understanding how our ears work.
One can produce a one-dimensional periodic sound wave in a long, narrow tube containing a gas by means of an oscillating piston at one end, as shown in Figure 17.2. The darker parts of the colored areas in this figure represent regions where the gas is compressed and thus the density and pressure are above their equilibrium values. A compressed region is formed whenever the piston is pushed into the tube. This compressed region, called a condensation, moves through the tube as a pulse, continuously compressing the region just in front of itself. When the piston is pulled back, the gas in front of it expands, and the pressure and density in this region fall below their equilibrium values (represented by the lighter parts of the colored areas in Fig. 17.2). These low-pressure regions, called rarefactions, also propagate along the tube, following the condensations. Both regions move with a speed equal to the speed of sound in the medium.
As the piston oscillates sinusoidally, regions of condensation and rarefaction are continuously set up. The distance between two successive condensations (or two successive rarefactions) equals the wavelength λ. As these regions travel through the tube, any small volume of the medium moves with simple harmonic motion parallel to the direction of the wave. If s(x, t) is the displacement of a small volume element from its equilibrium position, we can express this harmonic displacement function as
Where smax is the maximum displacement of the medium from equilibrium (in other words, the displacement amplitude of the wave), k is the angular wave number, and ω is the angular frequency of the piston. Note that the displacement of the medium is along x, in the direction of motion of the sound wave, which means we are describing a longitudinal wave.
As we shall demonstrate shortly, the variation in the gas pressure ΔP, measured from the equilibrium value, is also periodic and for the displacement function in Equation 17.2 is given by
where the pressure amplitude ΔPmax — which is the maximum change in pressure from the equilibrium value — is given by
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 is zero, and the displacement is a maximum when the pressure variation is zero.
Derivation of Equation 17.3
From the definition of bulk modulus (see Eq. 12.8), the pressure variation in the gas is
The volume of gas that has a thickness Δx in the horizontal direction and a crosssectional area A is Vi = AΔx The change in volume ΔV accompanying the pressure change is equal to A Δs, where Δs is the difference between the value of s at x + Δx and the value of s at x. Hence, we can express ΔP as
As Δx approaches zero, the ratio Δs/Δx becomes ∂s/∂x (The partial derivative indicates that we are interested in the variation of s with position at a fixed time.) Therefore,
If the displacement is the simple sinusoidal function given by Equation 17.2, we find that
Because the bulk modulus is given by B = ρν2 (see Eq. 17.1), the pressure variation reduces to
From Equation 16.13, we can write k = ω/ν; hence, ΔP can be expressed as
Because the sine function has a maximum value of 1, we see that the maximum value of the pressure variation is ΔPmax = ρνωsmax (see Eq. 17.4), and we arrive at Equation 17.3:
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