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Intensity of a wave | Power of a Wave

Intensity of a wave | Power of a Wave :- The power of a wave refers to the rate at which energy is transferred by the wave per unit of time. In other words, it is the amount of energy carried by the wave over a specific period.

Intensity of a wave is a measure of the energy flux or power per unit area carried by the wave as it propagates through a medium or space. It represents the concentration of energy within a given area perpendicular to the direction of wave propagation. It is measured in units of watts per square meter (W/m²).

Power in a string wave

(Intensity of a Wave | Power of a Wave)

In the article Energy of a Wave we derived an expression for energy associated with a traveling wave in a stretched string as given below :

$\displaystyle {{E}_{\lambda }}={{U}_{\lambda }}+{{K}_{\lambda }}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\lambda$

here

μ = mass per unit length of the string (in kilograms per meter, kg/m)

ω = angular frequency of the wave (in radians per second)

$A =$ amplitude of the wave (in meters, m)

λ = wavelength of the string wave (in meters, m)

Now in one time period (T), the string wave moves a distance equal to one wavelength(λ) as shown in figure below :

So from the definition, the power transported by the string wave,

$\displaystyle P=\frac{{{E}_{\lambda }}}{T}=\frac{\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\lambda }{T}=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}\frac{\lambda }{T}$

$\displaystyle P=\frac{1}{2}\mu {{\omega }^{2}}{{A}^{2}}v$ …..(1)

Where:

$μ$ is the linear mass density of the string (in kilograms per meter, kg/m),
$ω$ is the angular frequency of the wave (in radians per second),
$A$ is the amplitude of the wave (in meters, m),
$v$ is the speed of the wave along the string (in meters per second, m/s).

Power and intensity of a wave travelling through a medium

(Intensity of a Wave | Power of a Wave)

When a particle executes harmonic oscillations, there exists oscillation energy which keeps on changing between kinetic and potential forms. In a progressive wave, this energy is transferred through the medium with a velocity v. The amplitude of the particle velocity is given by

$\displaystyle {{v}_{0}}=A\omega$

Energy per unit volume

$\displaystyle U=\frac{1}{2}\rho v_{0}^{2}=\frac{1}{2}\rho {{A}^{2}}{{\omega }^{2}}$

Where ρ is the density of the medium.

If S is the cross sectional area of the medium, the energy associated with a volume SΔx , will be

$\displaystyle \Delta E=U\Delta v=\frac{1}{2}\rho {{A}^{2}}{{\omega }^{2}}S\Delta x$

So power becomes

$\displaystyle P=\frac{\Delta E}{\Delta t}=\frac{1}{2}\rho v{{\omega }^{2}}{{A}^{2}}S$

Hence Intensity

$\displaystyle I=\frac{\Delta E}{S\Delta t}=\frac{P}{S}=\frac{1}{2}\rho v{{\omega }^{2}}{{A}^{2}}$ …..(2)

Further in case of sound waves the displacement amplitude is related to the pressure amplitude through the relation :

$\displaystyle {{p}_{0}}=\rho vA\omega$

[ $\displaystyle [{{p}_{0}}=BAk={{v}^{2}}\rho A\frac{\omega }{v}=\rho vA\omega ]$ and $\displaystyle v=\sqrt{\frac{B}{\rho }},k=\frac{\omega }{v}$ ]

Hence

$\displaystyle I=\frac{1}{2}\rho v{{\omega }^{2}}{{\left( \frac{{{p}_{0}}}{\rho v\omega } \right)}^{2}}$

$\displaystyle \Rightarrow I=\frac{1}{2}\frac{p_{0}^{2}}{\rho v}$ …..(3)

Again using $\displaystyle {{p}_{0}}=BAk=BA\frac{\omega }{v}$

$\displaystyle \Rightarrow \omega =\frac{{{p}_{0}}v}{BA}$

Putting this value in equation (2),

$\displaystyle I=\frac{1}{2}\rho v{{\omega }^{2}}{{A}^{2}}=\frac{1}{2}\rho v{{A}^{2}}{{\left( \frac{{{p}_{0}}v}{BA} \right)}^{2}}$

$\displaystyle I=\frac{1}{2}\left( \rho {{v}^{2}} \right)v{{\left( \frac{{{p}_{0}}}{B} \right)}^{2}}=\frac{1}{2}Bv{{\left( \frac{{{p}_{0}}}{B} \right)}^{2}}$