2.2 Energy transmission by waves  (Page 3/3)

Average power transmitted

The average power transmitted by wave is equal to time rate of transmission of mechanical energy over integral wavelengths. It is equal to :

$$P_{\text{avg}}=\frac{dE}{dt}{|}_{\text{avg}}=2X\frac{1}{4}\mu v{\omega }^2{A}^2=\frac{1}{2}\mu v{\omega }^2{A}^2$$

If mass of the string is given in terms of mass per unit volume, “ρ”, then we make appropriate change in the derivation. We exchange “μ” by “ρs” where “s” is the cross section of the string :

$$P_{\text{avg}}=\frac{1}{2}\rho sv{\omega }^2{A}^2$$

Energy density

Since there is no loss of energy involved, it is expected that energy per unit length is uniform throughout the string. As much energy enters that much energy goes out for a given length of string. This average value along unit length of the string length is equal to the average rate at which energy is being transferred.

The average mechanical energy per unit length is equal to integration of expression over integral wavelength :

$$E_L{|}_{\text{avg}}=2X\frac{1}{4}\mu v{\omega }^2{A}^2=\frac{1}{2}\mu v{\omega }^2{A}^2$$

We have derived this expression for harmonic wave along a string. The concept, however, can be extended to two or three dimensional transverse waves. In the case of three dimensional transverse waves, we consider small volumetric element. We, then, use density,ρ, in place of mass per unit length, μ. The corresponding average energy per unit volume is referred as energy density (u):

$$u=\frac{1}{2}\rho v{\omega }^2{A}^2$$

Intensity

Intensity of wave (I) is defined as power transmitted per unit cross section area of the medium :

$$I=\rho sv{\omega }^2\frac{A^2}{2}s=\frac{1}{2}\rho v{\omega }^2{A}^2$$

Intensity of wave (I) is a very useful concept for three dimensional waves radiating in all direction from the source. This quantity is usually referred in the context of light waves, which is transverse harmonic wave in three dimensions. Intensity is defined as the power transmitted per unit cross sectional area. Since light spreads uniformly all around, intensity is equal to power transmitted, divided by spherical surface drawn at that point with source at its center.

Mechanics of energy transmission

Derivation of expression of energy transmission considering classical concept of power gives an insight into the manner energy is transmitted along the string. We have pointed out that every element pulls element ahead. In this fashion, a string element is worked out by the adjacent element. In this fashion, energy is transmitted from one element preceding to the element following it.

We consider a small string element as shown in the figure. Here, our objective is to evaluate the basic power equation as given :

$$P=\text{Force}X\text{velocity}$$

Now, the string element is pulled by tension “F” as shown in the figure. The y-component of force in the direction of oscillation is :

$$F_y=-F\sin \theta$$

Since angle “θ” is small, sinθ = tanθ. But tanθ is slope of the waveform at x=x. Therefore,

$$Fy=-F\frac{dy}{dx}$$

$$Fy=-F\frac{d}{dx}A\sin \left(kx-\omega t\right)=-FkA\cos \left(kx-\omega t\right)$$

On the other hand, velocity of the small element is :

$$v_p=\frac{\partial y}{\partial t}=-\omega A\cos \left(kx-\omega t\right)$$

Putting these expressions, the power is given as :

$$P=\text{Force}X\text{velocity}$$

$$P=-F\left(\frac{dy}{dx}\right)X\left(\frac{dy}{dt}\right)$$

$$P=-FkA\cos \left(kx-\omega t\right)X-\omega A\cos \left(kx-\omega t\right)$$

$$P=F\omega k{A}^2{\cos }^2\left(kx-\omega t\right)$$

Using $v=\sqrt{\left(\frac{F}{\mu }\right)}$ and $v=\frac{\omega }{k}$ ,

$$P=F\omega k{A}^2{\cos }^2\left(kx-\omega t\right)=\mu v^2\omega \left(\frac{\omega }{v}\right)A^2{\cos }^2\left(kx-\omega t\right)$$

$$P=\mu v{\omega }^2{A}^2\cos {}^2\left(kx-\omega t\right)$$

The average transmission of power is obtained by integrating the expression over integral wavelength,

$$P_{\text{avg}}=\frac{1}{2}\mu v{\omega }^2{A}^2$$

This expression is same as the expression obtained earlier by adding kinetic and elastic potential energy.