Question

A standing wave is formed by the superposition of two waves travelling in opposite directions. The transverse displacement is given by
$y\left(x,t\right)=0.5\sin \left({\displaystyle \frac{5\pi }{4}}x\right)\cos \left(200\pi t\right)$
What is the speed of the travelling wave moving in the positive $x$ direction?
($x$ and $t$ are in meter and second, respectively)
(A) $120m/s$
(B) $180m/s$
(C) $90m/s$
(D) $160m/s$

Answer 1

Hint: First compare the equation given in the question with the general standing wave equation generated due to superposition of two waves travelling in opposite directions. We can get the value of angular frequency and propagation constant from here. Then using these two values, calculate the velocity of the wave.
Formulas used:
$v={\displaystyle \frac{\omega }{k}}$ where $v$ is the velocity of the wave, $w$is its angular frequency and $k$ is its propagation constant.

Complete step by step answer
Standing or stationary waves can be generated in two ways:
1. By moving the medium in a direction opposite to the wave.
2. By the superposition of two waves with equal frequency travelling in opposite directions.
In this question, the standing wave is generated due to the superposition of two waves.
Let us consider two waves travelling in positive $x$ direction and negative $x$ direction represented by the equations
$y_1=a\sin \left(kx-\omega t\right)$ and $y_2=a\sin \left(kx+\omega t\right)$
Where $a$ is the amplitude of the waves, $k$is the propagation constant and $w$ is the angular velocity.
By the principle of superposition the resultant standing wave is given by,
$y=y_1+y_2$
$$\begin{gathered} \Rightarrow y=a\left[\sin \left(kx-\omega t\right)+\sin \left(kx+\omega t\right)\right] \\ \Rightarrow y=a\left[\sin kx\cos \omega t-\cos kx\sin \omega t+\sin kx\cos \omega t+\cos kx\sin \omega t\right] \\ \Rightarrow y=2a\sin kx\cos \omega t \\ \Rightarrow y=A\sin kx\cos \omega t \end{gathered}$$
Comparing this equation with the equation given in the question we have,
$k={\displaystyle \frac{5\pi }{4}}$ and $\omega =200\pi$
Now velocity of a wave is given by the formula, $v={\displaystyle \frac{\omega }{k}}$
So, $v={\displaystyle \frac{200\pi }{{\displaystyle \frac{5\pi }{4}}}}={\displaystyle \frac{200\pi \times 4}{5\pi }}=160m/s$
Therefore, the speed of the travelling wave moving in positive $x$ direction is $160m/s$

So, the correct option is D.

Note: An example of standing wave formation is in the open ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore.