Chapter 16, Problem 60CP

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 ΔP = −(B ∂s/∂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_max cos (kx − ωt). Show that this function satisfies the wave equation, provided $\omega /k=v=\sqrt{B/\rho }$.

To determine

The force diagram for this element showing the force exerted on the left and the right surface.

Answer to Problem 60CP

The force diagram for this element showing the force exerted on the left and the right surface is

Explanation of Solution

Force diagram contains all the forces acting on the body. It contains the direction of the each force acting on the body represents at its top and bottom end or left and right sides.

The force diagram for this element showing the force exerted on the left and the right surface is shown below.

Figure (1)

The force diagram of the element of gas in Figure (1) indicates the force exerted on the right and left surfaces due the pressure of the gas on the either side of the gas.

To determine

The expression, $-\frac{\partial \left(\Delta P\right)}{\partial x}A\Delta x=\rho A\Delta x\frac{{\partial }^{2}s}{\partial t^2}$ by applying the Newton’s second law to the element.

Answer to Problem 60CP

The expression $-\frac{\partial \left(\Delta P\right)}{\partial x}A\Delta x=\rho A\Delta x\frac{{\partial }^{2}s}{\partial t^2}$ is proved by applying the Newton’s second law to the element.

Explanation of Solution

Let $P\left(x\right)$ represents absolute pressure as a function of $x$.

The net force to the right on the chunk of air in Figure (1) is,

    $+P\left(x\right)A-P\left(x+\Delta x\right)A$

The force due to atmosphere is,

    $F=-\Delta P\left(x+\Delta x\right)A+\Delta P\left(x\right)A$        (1)

Here, $A$ is the area of the surface, $\Delta P(x)$ is the atmospheric pressure on the surface and $\Delta x$ is the smallest distance.

Differentiate the equation (1) with respect to $x$.

    $\begin{array}{c}\frac{\partial F}{\partial x}=\left[-\Delta P\left(x+\Delta x\right)+\Delta P\left(x\right)\right]A\\ =\left[-\frac{\partial \Delta P}{\partial x}x-\frac{\partial \Delta P}{\partial x}\Delta x+\frac{\partial \Delta P}{\partial x}x\right]A\\ =-\frac{\partial \Delta P}{\partial x}\Delta xA\end{array}$

Formula to calculate the mass of the air is,

    $\Delta m=\rho \Delta V$

Here, $\rho$ is the density of the air, $\Delta m$ is the mass of the air and $\Delta V$ is the volume of the air.

Formula to calculate the acceleration is,

    $a=\frac{{\partial }^{2}s}{\partial t^2}$

Here, $s$ is the distance and $t$ is the time.

From Newton’s second law, formula to calculate the Force is,

    $\frac{\partial F}{\partial x}=\Delta ma$        (2)

Substitute $\frac{{\partial }^{2}s}{\partial t^2}$ for $a$, $\rho \Delta V$ for $\Delta m$ and $-\frac{\partial \Delta P}{\partial x}\Delta xA$ for $\frac{\partial F}{\partial x}$ in equation (2).

    $-\frac{\partial \Delta P}{\partial x}\Delta xA=\rho \Delta V\frac{{\partial }^{2}s}{\partial t^2}$

Conclusion:

Therefore the expression, $-\frac{\partial \left(\Delta P\right)}{\partial x}A\Delta x=\rho A\Delta x\frac{{\partial }^{2}s}{\partial t^2}$ by applying the Newton’s second law to the element.

To determine

The wave equation for sound is $\frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}$.

Answer to Problem 60CP

The following wave equation for sound is $\frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}$.

Explanation of Solution

 The value of the $\Delta P$ is $B\frac{\partial s}{\partial x}$.

From part (b), the given expression is,

    $-\frac{\partial \left(\Delta P\right)}{\partial x}A\Delta x=\rho A\Delta x\frac{{\partial }^{2}s}{\partial t^2}$

Substitute $B\frac{\partial s}{\partial x}$ for $\Delta P$.

    $\begin{array}{c}-\frac{\partial }{\partial x}\left(B\frac{\partial s}{\partial x}\right)A\Delta x=\rho A\Delta x\frac{{\partial }^{2}s}{\partial t^2}\\ \frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}\end{array}$

Thus, the wave equation for sound is $\frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}$.

Conclusion:

Therefore, the wave equation for sound is $\frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}$.

To determine

The function $s\left(x,t\right)=s_{\max }\cos \left(kx-\omega t\right)$ satisfies the wave equation $\frac{\omega }{k}=v=\sqrt{\frac{B}{\rho }}$.

Answer to Problem 60CP

The function $s\left(x,t\right)=s_{\max }\cos \left(kx-\omega t\right)$ satisfies the wave equation $\frac{\omega }{k}=v=\sqrt{\frac{B}{\rho }}$.

Explanation of Solution

The given wave equation is,

    $s\left(x,t\right)=s_{\max }\cos \left(kx-\omega t\right)$        (3)

Apply the trial solution in the above equation.

Double differentiate the equation (1) with respect to $x$.

    $\begin{array}{c}\frac{\partial s}{\partial x}=-k{s}_{\max }\sin \left(kx-\omega t\right)\\ \frac{{\partial }^{2}s}{\partial x^2}=-k^2{s}_{\max }\cos \left(kx-\omega t\right)\end{array}$

Double differentiate the equation (1) with respect to $t$.

    $\begin{array}{c}\frac{\partial s}{\partial t}=+\omega s_{\max }\sin \left(kx-\omega t\right)\\ \frac{{\partial }^{2}s}{\partial t^2}=-\omega s_{\max }\cos \left(kx-\omega t\right)\end{array}$

The wave equation for sound in part (c) is,

    $\frac{B}{\rho }\frac{{\partial }^{2}s}{\partial x^2}=\frac{{\partial }^{2}s}{\partial t^2}$        (4)

Substitute $-\omega s_{\max }\cos \left(kx-\omega t\right)$ for $\frac{{\partial }^{2}s}{\partial t^2}$ and $-k^2{s}_{\max }\cos \left(kx-\omega t\right)$ for $\frac{{\partial }^{2}s}{\partial x^2}$ in equation (2).

    $\begin{array}{c}\frac{B}{\rho }\left(-k^2{s}_{\max }\cos \left(kx-\omega t\right)\right)=-{\omega }^2{s}_{\max }\cos \left(kx-\omega t\right)\\ \frac{B}{\rho }{k}^2={\omega }^2\\ \frac{\omega }{k}=\sqrt{\frac{B}{\rho }}\\ \frac{\omega }{k}=\sqrt{\frac{B}{\rho }}\end{array}$

Thus, the function $s\left(x,t\right)=s_{\max }\cos \left(kx-\omega t\right)$ satisfies the wave equation $\frac{\omega }{k}=v=\sqrt{\frac{B}{\rho }}$.

Conclusion:

Therefore, the function $s\left(x,t\right)=s_{\max }\cos \left(kx-\omega t\right)$ satisfies the wave equation $\frac{\omega }{k}=v=\sqrt{\frac{B}{\rho }}$.