Energy Transfer of a Wave

ENERGY TRANSFER OF A WAVE

The mechanical erergy is transferred through the vibration of the string.

As a wave propagates in a medium (for example along a string), it transports energy. It means a vibrating string has more energy than a string that is not vibrating.

Let us consider a string under uniform tension T and m the mass / unit length.

If a small element of the string of length dx is considered, then its mass is m dx.

As the string is vibrating, then the kinetic energy of this small element is (Fig.3.18)

where dy is the vertical displacement of the portion of a string and

x is the direction of propagation of the wave in a string.

Now the string is under tension T.

When the string is displaced from equilibrium, a segment associated with interval dx has a length dl (Fig. 3.18).

Thus under tension, a small segment of the string has expanded by an amount

So work is done for this expansion and it is stored as potential energy.

If the string is vibrating with displacement.

Therefore, the potential energy is

Substituting (4) in (5) gives

If we substitute Eq.(4) in (1), then

Comparing eqn (4) and eqn. (6)

dU = dK          …. (8)

⸫ The total energy is

(or)

The quantity is called the linear energy density.

At any point, for example at x = 0, the average value of sin2 (ω t) over one period is 1/2, so,

This relation is called as average energy density

As the average power transmitted by the wave is

therefore Eq.(11) becomes

Eq.(13) states that wave power is directly proportional to the speed or velocity v at which energy moves along the wave.