A longitudinal standing wave ` y = a cos kx cos omega t` is maintained in a homogeneious medium of density `rho`. Here `omega` is the angular speed and `k` , the wave number and `a` is the amplitude of the standing wave . This standing wave exists all over a given region of space.
The space density of the potential energy `PE = E_(p)(x , t)` at a point `(x , t)` in this space is

Options

A. `E_(p) = ( rho a^(2) omega^(2))/(2)`
B. `E_(p) = ( rho a^(2) omega^(2))/(2) cos^(2) kx sin^(2) omega t`
C. `E_(p) = ( rho a^(2) omega^(2))/(2) sin^(2) kx cos^(2) omega t`
D. `E_(p) = ( rho a^(2) omega^(2))/(2) sin^(2) kx sin^(2) omega t`

Answer

C

Explanation

The given longitudinal standing wave is
`y = a cos kx cos omega t` `(i)`
The nodes of this wave are located where `cos kx = 0 ` (i.e., ) at the values
`x = (lambda)/( 4) , ( 3 lambda)/(4) ,…`
and the antinodes are located where `cos kx = +- (i.e.,)` at the values
`x = 0 , (lambda)/(2) ,...`
At the nodes , the space density of kinetic energy ( kinetic energy per unit vanishes for the nodes , i.e.,
`x = (lambda)/(4) , ( 3 lambda)/(4) `etc.
Also , `y` is maximum at `t = 0` , as we see from Eq.(i). Hence potential energy must be maximum at `t = 0` . Hence the time factor in potential energy density must enter as ` cos^(2) omega t`. Also , the sum of kinetic and potential energy densities must always be constant for a given `x` as it represents total energy at that point.
Hence the potential energy density is
`E_(p) = ( rho a^(2) omega^(2))/(2) sin^(2) kx cos^(2) omega t` `(ii)`
and the kinetic energy density is
`E_(k) = ( rho a^(2) omega^(2))/(2) cos^(2) kx sin^(2) omega t` `(iii)`