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.
If a graph `E ( = E_(p) + E_(k))` versus `t` , i.e., total space energy density verus time were drawn at the instants of time `t = 0` and `t = T//4`, between two successive nodes separated by distance `lambda//2` which of the following graphs correctly shows the total energy `(E)` distribution at the two instants.

Options

A.
B.
C.
D.

Answer

A

Explanation

The graphs shown are between two successive nodes , say
`x_(1) = lambda//4 at N_(1)` and `x_(2) = 3 lambda//4 at N_(2)`.
Total energy density is
`E = E_(p) + E_(k))`
`= ( rho a^(2) omega^(2))/(2) [ sin^(2) kx cos^(2) omega t + cos^(2) kx sin^(2) omega t]`
Putting ` x = lambda//2` , ( first antinode)
`(E)_(lambda//2) = ( rho a^(2) omega^(2))/(2) [ sin^(2) omega t]`
At ` t = 0 , (E)_(lambda//2) = 0` and at `t = (T)/(4)`
`(E)_(lambda//2) = ( rho a^(2) omega^(2))/(2)`
This is truly reflected in the graph `(a)`.