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Energy of a Wave

Energy of a Wave :- All waves carry energy. The energy of a wave refers to the amount of energy carried by the oscillations or disturbances propagating through a medium or space. It is the measure of the ability of the wave to do work or cause a change in its surroundings as it propagates through a medium or space and is quantified by factors such as amplitude, frequency, and medium properties.

In this article, we shall be examining the quantitative expression of energy of a wave. To obtain the mathematical expression for the energy of a wave, consider a sinusoidal wave on a string, as shown in the figure below :

Expression for Energy of a Wave

The energy associated with a traveling wave in a stretched string is conveniently expressed as the “energy per wavelength”. The energy of a small segment of the string can be expressed as the sum of the kinetic energy and elastic potential energy of the segment.

The differential form of the elastic potential energy of an element of length dx (mass dm) of string is given by :-

$\displaystyle dU=\frac{1}{2}(force\text{constant}){{y}^{2}}=\frac{1}{2}(dm){{\omega }^{2}}{{y}^{2}}=\frac{1}{2}(\mu dx){{\omega }^{2}}{{y}^{2}}$ …..(1)

Here μ is mass per unit length of the string.

For a plane progressive wave

$\displaystyle y=A\sin (kx-\omega t)$

From equation (1)

$\displaystyle dU=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}{{\sin }^{2}}(kx-\omega t)dx$ …..(2)

The potential energy for a full wavelength can be found by integrating this expression at a given time, for convenience, let us put t=0 and integrate equation (2) :

$\displaystyle {{U}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\int\limits_{0}^{\lambda }{{{\sin }^{2}}(kx)dx}$

$\displaystyle {{U}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\int\limits_{0}^{\lambda }{\left( \frac{1-\cos (2kx)}{2} \right)dx}$

$\displaystyle {{U}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ x-\frac{\sin (2kx)}{2k} \right]_{0}^{\lambda }$

$\displaystyle {{U}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ \left( \lambda -\frac{\sin (2k\lambda )}{2k} \right)-\left( 0-\frac{\sin (2k\times 0)}{2k} \right) \right]$

$\displaystyle {{U}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ \left( \lambda -\frac{\sin (2\times \frac{2\pi }{\lambda }\lambda )}{2k} \right) \right]$

$\displaystyle {{U}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\lambda$ …..(3)

The differential form of the kinetic energy of the element :

$\displaystyle dK=\frac{1}{2}(dm)v_{y}^{2}=\frac{1}{2}(\mu dx)v_{y}^{2}$ …..(4)

The velocity expression for a plane progressive wave :

$\displaystyle {{v}_{y}}=A\omega \cos (kx-\omega t)$

Using this in equation (4) :

$\displaystyle dK=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}{{\cos }^{2}}(kx-\omega t)dx$ …..(5)

The kinetic energy for a full wavelength can be found by integrating above expression at a given time. Again for convenience, let us put t=0 and integrate equation (5) :

$\displaystyle {{K}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\int_{0}^{\lambda }{{{\cos }^{2}}(kx)dx}$

$\displaystyle {{K}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\int_{0}^{\lambda }{\left( \frac{1+\cos (2kx)}{2} \right)}dx$

$\displaystyle {{K}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ x+\frac{\sin 2kx}{2k} \right]_{0}^{\lambda }$

$\displaystyle {{K}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ \left( \lambda +\frac{\sin 2k\lambda }{2k} \right)-\left( 0+\frac{\sin (2k\times 0)}{2k} \right) \right]$

$\displaystyle {{K}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\left[ \left( \lambda +\frac{\sin (2\times \frac{2\pi }{\lambda }\lambda )}{2k} \right) \right]$

$\displaystyle {{K}_{\lambda }}=\frac{1}{4}\mu {{\omega }^{2}}{{A}^{2}}\lambda$ …..(6)

The total energy associated with one wavelength is given by adding equations (3) and (6) :-

$\displaystyle {{E}_{\lambda }}={{U}_{\lambda }}+{{K}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\lambda$ …..(7)

From above result of equation (7), we observe that the energy of a wave depends on following factors:

Amplitude: In many types of waves, including electromagnetic and mechanical waves, the energy carried by the wave is proportional to the square of its amplitude. So, doubling the amplitude results in four times the energy.
Frequency: Higher frequencies generally correspond to waves with higher energy. For example, in electromagnetic waves, higher frequencies correspond to photons with higher energy.
Medium Properties: The properties of the medium through which the wave travels can affect its energy. For example, in mechanical waves like sound waves, the density and elasticity of the medium influence how energy propagates through it.
Wave Speed: The speed at which the wave propagates through the medium also plays a role. However, unlike amplitude and frequency, wave speed does not directly affect the energy of the wave but rather influences how quickly the energy is transferred from one point to another.

Energy of a Wave

Expression for Energy of a Wave

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