Figure 17-27 shows the output from a pressure monitor...

Figure $17-27$ shows the output from a pressure monitor mounted at a point along the path taken by a sound wave of a single frequency traveling at 343 $\mathrm{m} / \mathrm{s}$ through air with a uniform density of $1.21 \mathrm{~kg} / \mathrm{m}^{3}$. The vertical axis scale is set by $\Delta p_{s}=5.0 \mathrm{mPa}$. If the displacement function of the wave is $s(x, t)=s_{m} \cos (k x-\omega t)$, what are (a) $s_{m}$, (b) $k$, and (c) $\omega$ ? The air is then cooled so that its density is $1.35 \mathrm{~kg} / \mathrm{m}^{3}$ and the speed of a sound wave through it is $320 \mathrm{~m} / \mathrm{s}$. The sound source again emits the sound wave at the same frequency and same pressure amplitude. What now are (d) $s_{\mathrm{m}}$, (e) $k$, and (f) $\omega$ ?

Key Concepts

Effect of Medium Properties on Sound Waves

The properties of the medium, such as density (?) and the speed of sound (c), play a crucial role in determining the behavior of sound waves. Changes in these properties affect the acoustic impedance (Z = ?c) of the medium, which in turn influences the relationship between pressure amplitude and displacement amplitude. When the medium is altered, even if the sound source’s frequency and pressure amplitude remain constant, parameters like displacement amplitude and wave number can change. Understanding this dependency is fundamental in the study of acoustics and wave propagation.

Wave Number and Angular Frequency

The wave number, k, and the angular frequency, ?, are parameters that characterize the spatial and temporal oscillations of a sound wave, respectively. The wave number is related to the wavelength by k = 2?/?, and the angular frequency is related to the frequency by ? = 2?f. Additionally, the speed of sound in the medium is given by c = ?/k. These relationships are essential for linking the measurable properties of waves (like speed and frequency) to their mathematical representation.

Displacement Wave Function

This concept involves representing sound waves as functions that oscillate in both space and time, typically using the cosine (or sine) function. The displacement function s(x, t) = s? cos(kx - ?t) describes how individual particles in the medium oscillate about their equilibrium positions, where s? is the maximum displacement (amplitude), k is the wave number, and ? is the angular frequency. This mathematical form is fundamental in understanding how sound waves propagate through a medium.

Pressure?Displacement Relationship

In sound wave phenomena, there is an intrinsic relationship between the pressure variation in the medium and the displacement of its particles. The local pressure variation can be derived from the spatial derivative of the displacement function, leading to a relationship between the pressure amplitude (?p?) and the displacement amplitude (s?). This relationship typically involves the medium's density (?) and speed of sound (c) as factors determining how efficiently displacement is converted to pressure changes, and it is key to analyzing acoustic measurements.

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