Topic 14.6 Wave Interference and Standing Waves: apply superposition to interference and find the harmonics of standing waves.

What this topic is asking

The College Board (Topic 14.6) wants you to apply the superposition principle to constructive and destructive interference, and to describe how standing waves form, with their nodes, antinodes and harmonics on a string or in a pipe.

The superposition principle

Superposition is the rule that makes interference possible: where waves meet, you simply add their displacements at each instant. Two waves can reinforce or cancel depending on how their crests and troughs line up, but neither wave is permanently altered, they emerge on the other side as if nothing had happened. This adding of displacements is the foundation of all interference, both for sound and (in the later topics) for light.

Constructive and destructive interference

Whether overlapping waves reinforce or cancel depends on their relative phase. In phase, the peaks coincide and add (louder sound, brighter light); out of phase, a peak meets a dip and they cancel (silence, darkness). For two sources, this is set by the path difference: a whole number of wavelengths arrives in phase (constructive), while an extra half wavelength arrives out of phase (destructive). This path-difference rule is exactly what governs the double-slit pattern of light in Topic 14.8.

Standing waves and harmonics

A standing wave forms when two identical waves travel in opposite directions and superpose, typically a wave and its reflection off a boundary. The result is a stationary pattern with nodes (points of zero motion, where the waves always cancel) and antinodes (points of maximum motion, where they always reinforce). On a string fixed at both ends, only wavelengths that put a node at each end fit, giving the harmonics: $\lambda_n = \dfrac{2L}{n}$ and $f_n = n f_1$, where the fundamental ($n = 1$) has $L = \lambda/2$. Driving the string at one of these frequencies produces large-amplitude resonance. The strategic role of this topic is that interference and standing waves are the clearest demonstration of wave behavior, and they reappear directly for light: the superposition and path-difference ideas here become the bright and dark fringes of double-slit interference and diffraction (Topics 14.7 to 14.9), tying the wave nature of sound to the wave nature of light.

Try this

Q1. State what happens at a point where a crest of one wave meets a crest of an equal wave. [1 point]

• Cue. Constructive interference: the displacements add to double the amplitude.

Q2. A string fixed at both ends has a fundamental of $80$ Hz. State the frequency of its second harmonic. [1 point]

• Cue. $160$ Hz ($f_2 = 2 f_1$).