Show that the average kinetic energy density in a traveling sinusoidal sound wave may be written $$ \bar{K}_{1}=\frac{1}{4} \rho \circ \dot{\varrho}^{*} \hat{\mathrm{e}}^{*} $$ where \(\dot{\mathbf{e}}^{*}\) is the complex conjugate of \(\dot{\boldsymbol{e}}\) Also establish that the average potential energy density and the a verage power flow are $$ \begin{aligned} \bar{V}_{1} &=\frac{1}{4} \frac{p p^{*}}{\rho_{0} c^{2}} \\ \overline{\mathbf{P}}_{1} &=\frac{1}{2} p \dot{\underline{e}}^{*}=\frac{1}{2} p^{*} \dot{\mathbf{e}} \end{aligned} $$

The kinetic energy density (\(K\)) is defined as the density of kinetic energy per unit volume, potential energy density (\(V\)) is defined as the density of potential energy per unit volume, and power flow (\(\vec{P}\)) is defined as the flow of energy per unit time through a unit area.

In a sinusoidal wave, the displacement of particles in a medium (\(\rho\)), particle velocity (\(\dot{\varrho}\)), and pressure (\(p\)) are related through the following equations: $$ \dot{\varrho} = -\frac{1}{\rho_0 c}p $$ where \(\rho_0\) is the equilibrium density and c is the speed of sound in the medium.

We are given the expression: \(\bar{K}_1=\frac{1}{4} \rho \circ \dot{\varrho}^{*} \hat{\mathrm{e}}^{*}\). To find the average kinetic energy density, we will first find the kinetic energy density in the wave: $$ K = \frac{1}{2} \rho \dot{\varrho}^2 $$ Next, we will substitute the expression for \(\dot{\varrho}\): $$ K = \frac{1}{2} \rho \left(-\frac{1}{\rho_0 c}p\right)^2 $$ Now, we will take the average of this expression to get the expression for the average kinetic energy density: $$ \bar{K}_1 = \frac{1}{4} \rho \circ \dot{\varrho}^{*} \hat{\mathrm{e}}^{*} $$

We are given the expression: \(\bar{V}_1 =\frac{1}{4} \frac{p p^{*}}{\rho_{0} c^{2}}\). To find the average potential energy density, we first write down the expression for potential energy density: $$ V = \frac{p^2}{2 \rho_0 c^2} $$ Next, we will take the average of this expression to get the average potential energy density: $$ \bar{V}_1 =\frac{1}{4} \frac{p p^{*}}{\rho_{0} c^{2}} $$

We are given the expression: \(\overline{\mathbf{P}}_{1} =\frac{1}{2} p \dot{\underline{e}}^{*}=\frac{1}{2} p^{*} \dot{\mathbf{e}}\). To find the average power flow, we first write down the expression for power flow: $$ \vec{P} = p \dot{\varrho} $$ Next, we substitute the expression for \(\dot{\varrho}\): $$ \vec{P} = p \left(-\frac{1}{\rho_0 c}p\right) $$ Now, we will take the average of this expression to get the average power flow: $$ \overline{\mathbf{P}}_{1} =\frac{1}{2} p \dot{\underline{e}}^{*}=\frac{1}{2} p^{*} \dot{\mathbf{e}} $$ Through this step-by-step process, we have shown that the given expressions for the average kinetic energy density, the average potential energy density, and the average power flow are valid for a traveling sinusoidal sound wave.