5.1 Principle of superposition

Understanding the Principle of Superposition in Sound Waves

When two or more sound waves occupy the same space at the same time, they combine. The principle of superposition tells you exactly how: the resulting displacement at any point is simply the sum of the individual wave displacements. This one idea underpins interference patterns, standing waves, beats, and most of what you'll study in this unit.

Principle of Superposition in Sound

The principle of superposition applies to linear systems, which includes sound waves traveling through air under normal conditions. "Linear" here means the medium responds proportionally to the wave, so you can just add waves together without worrying about them distorting each other.

In formal terms: when two or more waves overlap, the resulting displacement at any point equals the algebraic sum of the displacements from each individual wave. After the waves pass through each other, they continue on unchanged. The waves don't permanently affect one another.

This principle is what allows you to explain:

• Interference patterns (regions of louder and quieter sound)
• Standing waves (fixed patterns that form in enclosed spaces or on strings)
• Beats (the pulsing effect you hear when two close frequencies overlap)

Interaction of Multiple Sound Waves

Wave interference is what happens whenever waves overlap, and it comes in two main flavors:

• Constructive interference occurs when waves arrive in phase (their peaks and troughs line up). The displacements add together, producing a larger amplitude and therefore louder sound.
• Destructive interference occurs when waves arrive out of phase (one wave's peak lines up with the other's trough). The displacements oppose each other, reducing the amplitude. If two waves have equal amplitude and are exactly half a cycle apart, they cancel completely.

A key detail: when you superpose sinusoidal waves of the same frequency, the frequency stays the same. Only the amplitude and phase of the combined wave change, depending on how the individual waves line up.

For waves of different frequencies, the superposition still holds, but the resulting waveform is more complex. You're no longer getting a simple sinusoid; instead, you get a new wave pattern that is the algebraic sum of the originals.

Problem-Solving for Wave Superposition

The core equation is straightforward:

$y_{total} = y_1 + y_2 + \cdots + y_n$

where each $y$ represents the displacement of an individual wave at a given point and time.

Representing sinusoidal waves. Each wave can be written as:

$y = A \sin(\omega t + \phi)$

where $A$ is amplitude, $\omega$ is angular frequency, and $\phi$ is the phase.

Amplitude for two special cases:

1. Waves perfectly in phase ($\Delta \phi = 0$): $A_{total} = A_1 + A_2$
2. Waves perfectly out of phase ($\Delta \phi = \pi$): $A_{total} = |A_1 - A_2|$

General case with a phase difference. When two waves of equal frequency but different phases overlap, follow these steps:

1. Find the phase difference: $\Delta \phi = \phi_2 - \phi_1$

2. Use the combined amplitude formula: $A_{total} = \sqrt{A_1^2 + A_2^2 + 2A_1 A_2 \cos(\Delta \phi)}$

3. Determine the resulting phase using trigonometric identities or the phasor method.

Phasor addition method. This is especially useful when combining several waves with different phases. You represent each wave as a rotating vector (phasor) with length equal to its amplitude and angle equal to its phase. Add the vectors tip-to-tail, and the resultant vector gives you the combined amplitude and phase. It turns a trig problem into a geometry problem, which is often easier to visualize and compute.

Energy considerations. The energy carried by a wave is proportional to the square of its amplitude ($E \propto A^2$). Because of this, doubling the amplitude means quadrupling the energy. In regions of constructive interference, energy concentrates; in regions of destructive interference, energy is reduced. The total energy across the entire sound field is conserved, though it redistributes spatially.