The principle of superposition

By the end of this lesson

By the end you can state the principle of superposition, use it to work out the resultant displacement where two waves meet, and explain what happens when identical waves arrive in phase or in antiphase.

Key to the symbols

The main symbols used in this lesson:

Symbol	Meaning
$A_1$	amplitude of the first wave
$A_2$	amplitude of the second wave

Two people, one rope

Imagine two friends shaking opposite ends of a long rope at the same time. Two pulses race towards each other across the rope.

What happens when they meet? Nothing dramatic — the rope simply does both motions at once. Its actual displacement is the sum of the two separate displacements.

That one idea is the principle of superposition, and it explains interference, stationary waves, noise-cancelling headphones and the colours in a soap bubble.

Definition

Principle of superposition

When two or more waves meet at a point, the resultant displacement at that point is the vector sum of the displacements of the individual waves.

Displacement is a signed number

Displacement tells you how far a point on the wave sits above or below its rest position. Up is positive, down is negative.

So "add the displacements" is ordinary arithmetic with signs:

wave 1	wave 2	resultant
$+3\ \text{mm}$	$+3\ \text{mm}$	$+6\ \text{mm}$
$+3\ \text{mm}$	$-3\ \text{mm}$	$0$
$+4\ \text{mm}$	$-1\ \text{mm}$	$+3\ \text{mm}$

The waves do not bounce off each other or push each other aside. They pass through each other and carry on with their original shapes.

In phase and in antiphase

When two waves at a point are in phase — crest meets crest — their displacements have the same sign, so the resultant amplitude is

$A = A_1 + A_2$

This is constructive superposition, and it gives the largest possible resultant.

When they are in antiphase — crest meets trough — the displacements have opposite signs, so the resultant amplitude is

$A = |A_1 - A_2|$

This is destructive superposition. If the amplitudes are equal, the resultant is zero and that point does not move at all.

Two identical waves are in phase when the path difference is $0, \lambda, 2\lambda, \ldots$ and in antiphase when it is $\dfrac{\lambda}{2}, \dfrac{3\lambda}{2}, \ldots$

Worked example: Adding two displacements at a point

At point P, a pulse on a spring gives a displacement of $+6.0\ \text{mm}$ . A second pulse arrives at P and gives $-2.5\ \text{mm}$ .

(a) Find the resultant displacement. (b) Repeat if the second pulse had instead given $+2.5\ \text{mm}$ .

Have a go first?

Common mistakes

When two waves cancel, their energy disappears.

The displacement is momentarily zero, but energy is not destroyed. It is stored in the medium or shifted to neighbouring points, and it reappears as the waves move on.

The larger wave always wins, so the resultant can never be zero.

Displacements add as signed numbers. Equal and opposite displacements give zero, however large each wave is.

Superposition only works if the two waves are identical.

It works for all waves, whatever their amplitudes, frequencies or directions. Identical waves just give the neat cases of doubling or complete cancellation.

In the exam

"State the principle of superposition" is usually worth two marks: the displacements of the individual waves add, at a point, as vectors.

For a diagram of two pulses, show the addition with signs — for example $+5 - 3 = +2\ \text{mm}$ — and state the direction.

If the question says the waves are in phase or antiphase, quote the amplitude rule $A_1 + A_2$ or $|A_1 - A_2|$ and say which one applies.

Check yourself

Fill in the blanksQuestion 1 · Recall

CalculateQuestion 2 · Recall

Answer the 2 questions above to continue.