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Superposition, Interference & Standing Waves

Source lecture(s): SC133 Lec 23

Intuition

Two waves in the same medium don't collide — they add, point by point, instant by instant, then pass through each other unchanged. That politeness (linearity) has loud consequences: waves can reinforce into double heights (constructive interference), cancel into silence (destructive), or — when a wave meets its own reflection — freeze into a standing wave that oscillates in place. Every musical instrument is a standing-wave machine.

The superposition principle

\[y_\text{total}(x,t) = y_1(x,t) + y_2(x,t)\]

Two equal sinusoids offset by phase \(\phi\):

\[y = \left[2A\cos\frac{\phi}{2}\right]\sin\left(kx - \omega t + \frac{\phi}{2}\right)\]
  • \(\phi = 0\): amplitude \(2A\) — fully constructive
  • \(\phi = \pi\) (half a wavelength of path difference): amplitude \(0\) — fully destructive

Path-difference rule for two sources in step: constructive where \(\Delta L = m\lambda\), destructive where \(\Delta L = (m + \tfrac12)\lambda\) — the fringe logic that reappears with light (SC134) and in the FDTD double-slit simulation.

Standing waves

Add identical waves traveling in opposite directions (e.g., incident + reflected):

\[y = A\sin(kx - \omega t) + A\sin(kx + \omega t) = \boxed{\,2A\sin kx\,\cos\omega t\,}\]

Space and time have separated: every point oscillates in step, with a position-dependent amplitude. Nodes (never move) sit every \(\lambda/2\); antinodes (maximum swing) between them. No net energy travels — it's trapped, sloshing between the nodes.

Normal modes of a string

A string fixed at both ends demands nodes at \(x = 0, L\), so only certain wavelengths fit:

\[\lambda_n = \frac{2L}{n}, \qquad f_n = \frac{nv}{2L} = n f_1, \qquad n = 1, 2, 3, \dots\]

The fundamental \(f_1 = v/2L\) sets the pitch; the harmonics \(2f_1, 3f_1,\dots\) color the timbre — why a guitar and violin playing the same note sound different. Tuning = changing \(v = \sqrt{F_T/\mu}\) (tension pegs) or \(L\) (frets).

These discrete allowed patterns are physics' first eigenmode problem — the same mathematics that later quantizes PDE solutions (PHY621) and electron orbitals.

Worked example: which harmonics?

A 0.9 m string has wave speed 360 m/s. A tuner drives it at 800 Hz. Does it resonate?

\(f_1 = v/2L = 200\,\text{Hz}\); harmonics at 200, 400, 600, 800 ✓ — yes, the \(n = 4\) mode, with 3 interior nodes.

Try it live

The wave studio superposes two waves live — flip the second wave's direction to build standing waves and watch nodes appear; detune the frequencies slightly to preview beats.

Common mistakes

  • Expecting waves to bounce off each other. They overlap and continue — only the displacement adds while they share space.
  • "Destructive interference destroys energy." Energy is redistributed to the constructive regions; the books always balance.
  • Confusing nodes with antinodes, and node spacing (\(\lambda/2\)) with \(\lambda\).
  • Forgetting boundary conditions choose the modes. One fixed + one free end (organ pipes closed at one end) gives odd harmonics only — different physics, same recipe.

Knowledge graph position

Prerequisites: Traveling waves. Leads to: Beats, Sound, musical acoustics, eigenmode thinking everywhere.

Quiz

Q1 (computational). Two speakers, in phase, face you: one 3.0 m away, the other 3.85 m. For sound at 400 Hz (\(v = 340\) m/s), loud or quiet?

Answer

\(\lambda = 0.85\,\text{m}\); \(\Delta L = 0.85\,\text{m} = 1\lambda\) — constructive: loud.

Q2 (conceptual). Noise-cancelling headphones exploit which phenomenon, and what must they generate?

Answer

Destructive interference: an "anti-noise" wave with equal amplitude and opposite phase (\(\phi = \pi\)) to the incoming sound at the ear — superposition sums to near-silence.

Q3 (multiple choice). On a string fixed at both ends vibrating in its third harmonic, the number of nodes (including the ends) is: (a) 2 (b) 3 (c) 4

Answer

(c). \(n = 3\) means three half-wavelength loops: nodes at both ends + two interior — four total, three antinodes.