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Energy of a Plane Progressive Wave

In case of a progressive wave, new waves are continuously formed at the head of the wave. This means that there is continuous transfer of energy in the direction of propagation of the wave. Thus energy is supplied from the source. The energy transferred per second also corresponds to the energy possessed by the particles in a length v, where v the velocity of the wave. The energy of the particles in partly kinetic and partly potential. The kinetic energy is due to the velocity of the vibrating particles. For a particle executing simple harmonic motion, the velocity is  maximum at the mean position and it is zero at the extreme positions. Consequently the kinetic energy of the particle at the mean position is maximum and zero at the extreme positions. similarly the particles also possess potential energy due to their displacements from their mean positions. At the extreme positions the potential energy of a particle is maximum and at the mean position the potential energy is minimum.

 

In longitudinal wave motion, compressions and rarefactions are formed. The energy distribution is not uniform over the wave. At points of no velocity there is no compression and the particle do not possess energy at these points. At points of maximum velocity, there is compression and the particle possess maximum energy. However in the case of progressive wave motion, there is no transfer of the medium in the direction of propagation of the wave, but there  is always transfer of energy in the direction of propagation of the wave.

 

Analytical Treatment

 

The equation of a simple harmonic wave is

y=a sin 2π /λ (vt –x)

 

The particle velocity u at any instant be obtained by differentiating equation (1) with respect to time

 

 

 

 

The   acceleration of the particle at that instant

 

F= du/dt : f =d2 y /dt2

 

Differentiating equation (2) with respect to time

 

 

 

The negative sign shows that acceleration is directed towards the mean position

 

Potential Energy

 

To move the particle from its mean position to a distance y, work has to be done against acceleration.

 

Workdone for a displacement dy = F dy

Let p be the density of the medium

Workdone per unit volume for a displacement dy

 

 

 

Total work done for displacement y

 

 

 

 

Potential energy per unit volume = [4π2 pv2 2]

 

 

 

 

 

 

Kinetic1 energy per unit  volume = 1/2 p u2

 

 

 

Total energy per unit volume

E = Kinetic energy + Potential energy

 

 

 

The average kinetic energy per unit volume and the average potential energy per unit volume are equal and each is equal to half the total energy per unit volume

 

Average kinetic energy per unit volume = n2 pn2 a2

Average potential energy per unit volume = n2 pn2 a2

 

Suppose that the area of cross-section of a parallel beam of radiation is 1 unit and the

velocity of wave is v

volume = 1 x v = v

 

Energy transfer per unit area per second= E x v

= 2n2pn2a2v

Energy transfer per second also called current per unit area of cross-section. It is to be remembered that the potential and kinetic energies of every particle will change with time, but the average kinetic energy per unit volume and the average potential energy per unit volume remains constant. The current density per unit area of cross-section and the total energy per unit volume also remains constant.