Serway & Jewett — Wave Motion: Problem 60

In Section 16.7, we derived the speed of sound in a gas using the impulse–momentum theorem applied to the cylinder of gas in Figure 16.20. Let us find the speed of sound in a gas using a different approach based on the element of gas in Figure 16.18. Proceed as follows. (a) Draw a force diagram for this element showing the forces exerted on the left and right surfaces due to the pressure of the gas on either side of the element. (b) By applying Newton’s second law to the element, show that \[ -\frac{\partial(\Delta P)}{\partial x} A \Delta x = \rho A \Delta x \frac{\partial^2 s}{\partial t^2} \] (c) By substituting \(\Delta P = -B(\partial s / \partial x)\) (Eq. 16.30), derive the following wave equation for sound: \[ \frac{B}{\rho} \frac{\partial^2 s}{\partial x^2} = \frac{\partial^2 s}{\partial t^2} \] (d) To a mathematical physicist, this equation demonstrates the existence of sound waves and determines their speed. As a physics student, you must take another step or two. Substitute into the wave equation the trial solution \(s(x, t) = s_{\text{max}} \cos(kx - \omega t)\). Show that this function satisfies the wave equation, provided \(\omega/k = v = \sqrt{B/\rho}\).