Question

The stationary wave \[y = 2a\left( {\sin kx\cos \omega t} \right)\], in a closed organ pipe, is the result of the superposition of \[y = a\sin \left( {\omega t - kx} \right)\] and:
A. \[y = - a\cos \left( {\omega t + kx} \right)\]
B. \[y = a\cos \left( {\omega t + kx} \right)\]
C. \[y = a\sin \left( {\omega t - kx} \right)\]
D. \[y = - a\sin \left( {\omega t + kx} \right)\]

Answer 1

Hint:
The above problem can be resolved using the fundamentals of the stationary wave. Along with this, the concept of the superposition principle is also required to be applied. In this problem, the mathematical equation for the stationary wave is given, then by applying the mathematical tools, we are supposed to solve the equation. Then by observing the result obtained in the final equation, one can get the desired answer.

Complete step by step solution
The expression for the given stationary wave is,
\[y = 2a\left( {\sin kx\cos \omega t} \right)\]
Solve the above equation by adding and subtracting the term, \[a\cos kx\sin \omega t\].
\[\begin{array}{l}
y = 2a\left( {\sin kx\cos \omega t} \right)\\
\Rightarrow y = a\left( {\sin kx\cos \omega t} \right) + a\left( {\sin kx\cos \omega t} \right) + a\cos kx\sin \omega t - a\cos kx\sin \omega t\\
\Rightarrow y = a\left( {\sin kx\cos \omega t} \right) + a\cos kx\sin \omega t + a\left( {\sin kx\cos \omega t} \right) - a\cos kx\sin \omega t\\
\Rightarrow y = a\sin \left( {\omega t + kx} \right) - a\sin \left( {\omega t - kx} \right)
\end{array}\]
 From the above result it is clear that the additional wave will be \[ - a\sin \left( {\omega t + kx} \right)\].
Therefore, the result of the superposition is \[a\sin \left( {\omega t - kx} \right)\] and \[ - a\sin \left( {\omega t + kx} \right)\] and option (D) is correct.

Note:
To resolve the given problem, one must know the meaning of the stationary wave. The name of the standing wave also denotes the stationary wave. The standing wave's critical concept is that the wave's amplitude is always constant while undergoing the propagation. In addition to this, the fundamental of wave propagation like the superposition principle of the wave. In this principle, the superposition principle accounts for the mathematical formulation of two waves such that each wave contributes to the maximum amplitude.