Question

Derive expression for average rate at which energy is transported by a progressive
wave propagating in a medium.

Answer 1

Answer on Question #61868-Physics-Mechanics-Relativity

Derive expression for average rate at which energy is transported by a progressive wave propagating in a medium.

Solution

The total average energy of the segment of the medium under consideration at any instant of time is

$$E = \frac{1}{2} dm \omega^{2} a^{2},$$

where $\omega$ is the angular frequency and $a$ is an amplitude.

This equation gives the total energy carried by a progressive wave and transported per cycle through a thin layer of mass $dm$ of the medium.

$$dm = \rho A \Delta x,$$

where $\rho$ is the density, $A$ is the cross-sectional area, $\Delta x$ is the thickness of the layer.

Now, we can write the expression for the power average rate at which energy is transported by a progressive wave propagating in a medium:

$$P = \frac{E}{\Delta t} = \frac{\frac{1}{2} \rho A \Delta x (2\pi f)^{2} a^{2}}{\frac{\Delta x}{v}},$$

where we used the expression $\omega = 2\pi f$ for the frequency and $\Delta t = \frac{\Delta x}{v}$ for the time taken by the wave to cross the layer of thickness $\Delta x$ by the wave propagating with velocity $v$. Then,

$$P = 2\pi^{2} \rho A a^{2} f^{2} v.$$

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