# Stationary waves

The stationary waves are produced when two similar progressive waves travelling in the opposite directions are superposed. \begin{align} y_1 & =A_1\sin(kx-\omega t),\\ y_2 &=A_2\sin(kx+\omega t) \end{align} The displacement of the superposed wave is given by \begin{align} y &=y_1+y_2 \\ &=(2A\cos kx) \sin \omega t \end{align}

The displacement is zero at the nodes and it is maximum at the antinodes. The location of nodes and antinodes is given by \begin{align} x&=\left\{\begin{array}{ll} \left(n+\tfrac{1}{2}\right)\tfrac{\lambda}{2}, & \text{nodes;} \quad n=0,1,2,\ldots\\ n\tfrac{\lambda}{2},& \text{antinodes.} \quad n=0,1,2,\ldots \end{array}\right. \nonumber \end{align}

Standing waves are usually formed when a progressive wave is reflected from a rigid medium. Two common examples are strings and organ pipes.

## Problems from IIT JEE

Problem (IIT JEE 1988): A wave represented by the equation $y=a\cos(kx-\omega t)$ is superimposed with another wave to form a stationary wave such that point $x=0$ is a node. The equation for the other wave is,

1. $a\sin(kx+\omega t)$
2. $-a\cos(kx-\omega t)$
3. $-a\cos(kx+\omega t)$
4. $-a\sin(kx-\omega t)$

Solution: A stationary wave is formed by the superposition of two identical waves travelling in the opposite directions. The point $x=0$ is a node if displacement of the resultant wave is zero at this point at all times. Thus, the wave superimposed on $y=a\cos(kx-\omega t)$ is $y^\prime=-a\cos(kx+\omega t)$.