Superposition and interference

Two pulses on one rope

Two children hold the ends of a long skipping rope. Each one flicks the rope once. Two pulses run towards each other. What happens when they meet? Do they bounce off each other? Do they crash and stop?

Neither. Look at the picture.

On the left, both pulses are on the same side. When they overlap, the rope rises twice as high as one pulse. On the right, one pulse is up and one is down. When they overlap exactly, the rope is flat for an instant. In both cases the pulses then move on, with the same shape as before. Each pulse passes through the other as if the other was not there.

The principle of superposition

This is a rule for waves on a rope, on water, in air and in many other places. When two or more waves are at one point at the same time, the displacement there is the sum of the displacements of each wave:

We call this the principle of superposition. Displacements on opposite sides have opposite signs, so they can cancel.

In the flat moment on the right, where did the energy go? It did not vanish. The rope is flat, but it is moving. The energy is all kinetic for that instant. A moment later the pulses appear again.

Two waves with a phase difference

Now take two long waves on the same string, with the same amplitude $a$, the same wavelength and the same frequency. The second is shifted by a phase difference $\phi$:

Add them. A rule of trigonometry, $\sin A + \sin B = 2\sin\frac{A + B}{2}\cos\frac{A - B}{2}$, gives

The sum is again a wave of the same wavelength and frequency. Its amplitude is

(Take the size of this; an amplitude is never negative.) Let us look at three cases.

• $\phi = 0$: the waves are in phase. $A = 2a$. They help each other fully. This is constructive interference.
• $\phi = \pi$: the waves are out of phase. $A = 0$. They cancel. This is destructive interference.
• Between these, $A$ lies between $0$ and $2a$. For $\phi = 60^\circ$ and $a = 3\ \text{mm}$, $A = 2 \times 3\ \text{mm} \times \cos 30^\circ = 5.2\ \text{mm}$.

When the two amplitudes are different, $a_1$ and $a_2$, the sum is still a wave of the same frequency, with

For $a_1 = a_2 = a$ this gives the same answer as before.

Intensity

The energy a wave carries each second through each square metre is its intensity. It grows as the square of the amplitude: $I \propto A^2$. So two equal waves in phase, with twice the amplitude, give four times the intensity of one wave. Two waves out of phase give none. The energy is not lost. It goes to the places where the waves are in phase.

If the amplitudes are $a_1$ and $a_2$, the largest and the smallest intensities are in the ratio

Path difference and phase difference

Two sources in step send out waves of wavelength $\lambda$. At a point, one wave has come a distance $x_1$ and the other $x_2$. The difference $\Delta x = x_2 - x_1$ is the path difference. One wavelength of path is a whole cycle, $2\pi$, of phase. So

A path difference of $0$, $\lambda$, $2\lambda$ and so on gives constructive interference. A path difference of $\lambda/2$, $3\lambda/2$ and so on gives destructive interference.

Practise with new numbers

Try it in Python

The program adds two waves point by point along a string. Then it compares the largest value of the sum with the formula $2a\cos(\phi/2)$.

Common mistakes

• Adding amplitudes in every case. $a_1 + a_2$ is right only when the waves are in phase.
• Thinking that waves which cancel destroy energy. The energy moves to other places, or into motion of the medium.
• Forgetting the half angle. For equal waves, $A = 2a\cos(\phi/2)$, not $2a\cos\phi$.
• Adding intensities when the waves are in step. Add the displacements first, then square.

Remember these

• Superposition: the displacements of waves at a point add, $y = y_1 + y_2$.
• Pulses pass through each other unchanged.
• Equal waves with a phase difference $\phi$: $A = 2a\cos(\phi/2)$.
• In phase: $A = 2a$. Out of phase: $A = 0$.
• Intensity $I \propto A^2$. Phase difference $\phi = \dfrac{2\pi}{\lambda}\Delta x$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A pulse $2\ \text{cm}$ high meets a pulse $5\ \text{cm}$ deep on a rope, on the other side. What is the displacement of the rope where they overlap exactly?

2. Two equal waves of amplitude $a$ meet with a phase difference of $90^\circ$. What is the amplitude of the sum?

3. Two waves arrive at a point with a path difference of $1.5\lambda$. Do they help each other or cancel?

4. Two equal waves in phase give an intensity $I$. What intensity does one of them give alone?