# 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,

- $a\sin(kx+\omega t)$
- $-a\cos(kx-\omega t)$
- $-a\cos(kx+\omega t)$
- $-a\sin(kx-\omega t)$

**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 $y^\prime=-a\cos(kx+\omega t)$.

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