Stationary wave equation, energy and power transfer in a stationary wave

STATIONARY WAVE EQUATION, ENERGY AND POWER TRANSFER IN A STATIONARY WAVE

In general, there is no transmission of energy across any plane in either direction of a stationary wave, because, if two progressive waves of equal energy travel in opposite direction, then the resultant flow of energy will be equal to zero.

Let us now find the energy wave equation of the stationary and prove that the transfer of energy is zero.

Proof

(i) Stationary Wave equation

Let us consider a progressive wave moving with amplitude 'A' wavelength 'λ' and velocity 'v' travelling from left to right along the X‒axis.

Then, the displacement of the transmitted wave at any point shall be written as

 y₁ = A sin(2π/ λ)  (vt‒x) [Transmitted wave]            ………..(1)

If the wave is reflected, then the wave will move from right to left, i.e., in the opposite direction, thus the displacement of the reflected wave shall be written as

y₂ = A sin(2π/ λ)  (vt+x) [Reflected wave]            ………..(2)

The resultant displacement y = y₁ + y₂

Substituting eqn. (1) and eqn. (2) in eqn. (3) we get

Since sin C + sin D = 2 sin [(C+D)/2] cos [(C‒D)/2], we can write the above equation as

Equation (4) represents the simple harmonic wave equation of a stationary wave.

(ii) Energy transfer in a stationary wave

We know the angular frequency ω = 2πν / λ

The stationary wave equation (4) shall be written as

 y = 2A cos(2πx/λ) sin ωt                    ……………(5)

∴ Particle velocity dy/dt shall be obtained by differentiating equation (5) with respect to 't'.

When there is a stationary wave, then, there will be excess pressure in the medium.

∴ The excess pressure due to stationary wave in the medium shall be written as

 P = ‒ E dy/dx                      …………..(7)

Differentating eqn (5) with respect to 'x' we get

Substituting eqn (8) in eqn (7) we get

∴ The workdone against the excess pressure (or) the energy transferred per unit area in a small internal of time dt shall be written as

(or) Energy transfer in time 'dt' = P (dy/dt) dt           ……….(10)

∴ The total energy transferred in a periodic time of T seconds across the unit area shall be obtained by integrating eqn (10) from 0 to "T"

i.e., Total Energy transfer = ₀∫^r P (dy/dt) dt               ………….(11)

We know the rate of energy transfer (or) power transfer = Total Energy Transfer / Total Time taken                …………..(12)

Substituting eqn (11) in eqn (12), we get

Rate of energy (or) power transfer =

......... (13)

Substituting eqn (6) and eqn (9) in eqn (13) we get

Rate of energy (or) power transfer = 0

Thus we can see that for a stationary wave, the rate of energy transfer in zero, which implies that no energy is Transferred in a stationary wave.