Principle of Superposition and Stationary Waves

Principle of Superposition

If two or more waves are travelling in a medium at the same time, then the resultant displacement of waves is equal to the vector sum of individual displacement of the wave.

Let \(\vec y_1, \vec y_2, \vec y_3 \dots \dots \vec y_n \) be the displacements of different waves travelling in medium at same time. Then, according to the principle of superposition

Resultant displacement,

$$ \vec y = \vec y_1+\vec y_2+ \vec y_3+\dots \dots +\vec y_n $$

If two waves each of same amplitude ‘a’ superposition the resultant displacement is

$$\begin{align*} y &= a + a = 2a \: (\text {constructive interference}) \\ \text {and} \\ y &= a - a = 0 \: (\text {destructive interference}) \\ \end{align*}$$

The principle of superposition can be used to explain many wave phenomena. Some of them are as follows:

1. When two waves of same frequency moving in the same direction superpose, constructive interference of waves is produced.
2. When two waves of same frequency moving in the opposite direction superpose, stationary waves are produced.
3. When two waves of slightly different frequency moving in the same direction superpose, beats are produced.

Stationary Wave

If two waves having same amplitude and frequency are travelling in opposite direction on the medium at the same time then the resultant displacement of the wave becomes zero and the waves come in rest or stationary state. This type of wave is called stationary wave or standing wave. There are different point appear in a stationary wave.

Nodes

Two different waves meet at a point are called nodes. In node particles of the medium are in permanent rest.

Antinodes

The maximum displacement of the wave in either side from equilibrium position is called antinodes. The antinode particles are not in permanent rest.

Expression of Stationary Wave:

Consider two waves having displacement y₁ and y₂ travelling in opposite direction with same amplitude and frequency on the medium at same time. Then, the general progressive wave equation for first wave is given by

$$\begin{align*} \vec y_1 &= a\sin (\omega t – kx) \dots (i) \end{align*} and general progressive wave equation for second wave is given by,\begin{align*}\vec y_2 &= a\sin (\omega t + kx) \dots (ii) \\ \text {where,} \: K &= \frac {2\pi }{\lambda }\\ &= \text {wave number or propagation constant,} \\ \text {According to principle of superposition. } \\ \text {The resultant displacement of two wave} \\ \end{align*}$$

$$\begin{align*}\vec y &= \vec y_1 + \vec y_2 \\ \vec y &= a\sin (\omega t – kx) + a\sin (\omega t + kx) \\ &= a[\sin (\omega t – kx) + \sin (\omega t + kx)] \\ &= a[2\sin \frac {(\omega t – kx + \omega t + kx)}{2}] \cos (\frac {\omega t – kx - \omega t – kx}{2}) \\ &= a[2\sin \omega t . \cos (-kx)] \\ &= 2a\sin \omega t. \cos kx \\ \vec y &= 2a\cos kx . \sin \omega t \dots (iii) \\ \end{align*}$$

Equation (iii) gives the expression for stationary wave where, \( 2a\cos kx \)gives the amplitude of the stationary wave and \( \sin \omega t \)gives the nature of oscillation$$\begin{align*} \text {If} \: 2a\cos kx = 0 \\ kx &= \left (x + \frac 12 \right ) \pi \\ \frac {2\pi x}{\lambda } &= \left (x + \frac 12 \right )\pi \\ \text {or,} \: x &= \left (x + \frac 12 \right ) \frac {\pi }{2} \\ \end{align*}$$

Equation (iv) represents the position of node where displacement of wave is 0 and n is interger.$$\begin{align*} \text {If} \: \cos kx &= \pm 1 \\ kx &= n\pi \\ \frac {2\pi }{\lambda } . x &= x\pi \\ \text {or,} \: x &= \frac {n .\lambda }{2} \dots (v) \\ \end{align*}$$

Equation} (v) represents the position of antinodes of stationary wave where displacement is maximum. If \(\sin \omega t = \pm 1\) Then the wave oscillates in either side of mean position $$\begin{align*} \text {If} \: \sin \omega t = 0 \\ \text {The wave just crosses the nodes} \\ \end{align*}$$

Properties of Stationary Wave

1. When two progressive waves of same amplitude and frequency travel in a medium in opposite direction to each other, a stationary wave is produced.
2. Nodes and antinodes are formed alternatively in the wave. A node is a position of zero displacement and maximum strain whereas an antinode is the position of maximum displacement and zero strain. However, if these waves travel in the same direction, they produce an interference pattern.
3. All the particles except at the nodes vibrate simple harmonically with the period equal to that of the wave.
4. The amplitude of vibration gradually increases from zero at node to maximum at antinode.
5. The medium splits into segments and all the particles of segment vibrate in phase. The particles in one segment have a phase difference of p with the particles in the neighboring segment.
6. The disturbance does not travel forward; there is no transfer of energy.
7. The disturbance between two adjacent nodes is λ/2 and that between two adjacent antinodes is also λ/2. The distance between a node and adjacent node is λ/4.
8. The velocity and acceleration of all the particles separated by a distance λ are the same at the given instant.

Difference between a Progressive and a Stationary Wave