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Superposition

Level 1 - Physics topic page in Mechanical Waves.

Principle

In a linear medium, overlapping waves add displacement point by point.

Superposition is the reason interference patterns appear: the medium responds to the sum of disturbances, not to one wave at a time.

Notation

\(y_1(x,t)\)
first wave displacement
\(\mathrm{m}\)
\(y_2(x,t)\)
second wave displacement
\(\mathrm{m}\)
\(y(x,t)\)
resultant displacement
\(\mathrm{m}\)
\(A\)
individual wave amplitude
\(\mathrm{m}\)
\(\Delta\phi\)
phase difference
\(\mathrm{rad}\)
\(A_{\mathrm{res}}\)
resultant amplitude
\(\mathrm{m}\)

Method

Derivation 1: Add the displacements directly

For a linear wave equation, the sum of two solutions is another solution. That means the medium displacement is the algebraic sum of the individual displacements.

Superposition rule
\[y(x,t)=y_1(x,t)+y_2(x,t)\]

The graph shows two sinusoidal waves at the same instant and the displacement produced when they overlap.

0123456-2.2-1.101.12.2xyy1y2y

Derivation 2: Add two equal sinusoids with a phase difference

Now take two waves with the same amplitude, frequency, and wavenumber:

\[ y_1=A\\cos(kx-\\omega t) \] \[ y_2=A\\cos(kx-\\omega t+\\Delta\\phi) \]

Use the cosine-sum identity to rewrite the total as a single sinusoid.

Add the waves
\[y=y_1+y_2\]
Trig identity
\[y=2A\cos\left(\frac{\Delta\phi}{2}\right)\cos\left(kx-\omega t+\frac{\Delta\phi}{2}\right)\]
Resultant amplitude
\[A_{\mathrm{res}}=2A\cos\left(\frac{\Delta\phi}{2}\right)\]

Derivation 3: Interpret interference

The phase difference controls whether the waves reinforce or cancel.

Constructive case
\[\Delta\phi=0\Rightarrow A_{\mathrm{res}}=2A\]
Destructive case
\[\Delta\phi=\pi\Rightarrow A_{\mathrm{res}}=0\]

Rules

These are the compact results from the method above.

Superposition
\[y=y_1+y_2\]
Equal-wave resultant
\[A_{\mathrm{res}}=2A\cos\left(\frac{\Delta\phi}{2}\right)\]

Examples

Question
Two equal sinusoidal waves each have amplitude
\[3.0\,\mathrm{mm}\]
and phase difference
\[120^\circ\]
Find the resultant amplitude.
Answer
Convert the phase difference to radians or use the known cosine value:
\[A_{\mathrm{res}}=2A\cos\left(\frac{120^\circ}{2}\right)=2(3.0\,\mathrm{mm})\cos60^\circ=3.0\,\mathrm{mm}\]

Checks

  • Superposition adds displacements, not energies.
  • Complete cancellation requires equal amplitudes and a phase difference of \(\\pi\).
  • Constructive interference doubles amplitude and therefore quadruples intensity or power in many wave models.
  • Interference can vary from point to point because phase difference can depend on position.

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Boundary Conditions