Explain standing waves on strings and in pipes and relate harmonics to resonance

What this dot point is asking

WACE wants you to explain standing waves as the superposition of two travelling waves, locate nodes and antinodes, and find the allowed frequencies for strings and pipes. This is where superposition produces something that does not appear to move along at all.

How a standing wave forms

When a wave reflects off a boundary, the reflected wave travels back through the incoming wave. Where the two are always in phase, constructive interference gives an antinode of maximum oscillation; where they are always out of phase, destructive interference gives a node of zero displacement. The pattern of nodes and antinodes stays fixed in space, so the wave appears to stand still while points between nodes oscillate.

Strings fixed at both ends

A string fixed at both ends must have a node at each end. The longest wave that fits has a single antinode in the middle, so its wavelength is twice the string length: $\lambda_1=2L$. Successive harmonics fit more half-wavelengths, giving

The fundamental (first harmonic) is $f_1=v/2L$, and higher harmonics are whole-number multiples of it.

Pipes

An open pipe (open at both ends) has an antinode at each end and behaves like the string case, with $f_n=nv/2L$ for all integers $n$. A pipe closed at one end has a node at the closed end and an antinode at the open end, so only odd harmonics exist:

This is why a closed pipe of a given length sounds an octave lower than an open pipe of the same length.

Resonance

Every system has natural frequencies equal to its harmonic series. When an external driver oscillates at one of these frequencies, energy is fed in efficiently each cycle and the standing wave grows to a large amplitude. This is resonance, responsible for the loud note of a tuned instrument and, destructively, for structures shaking apart when driven at a natural frequency.

Counting half-wavelengths

To find a harmonic, sketch the pattern, count nodes and antinodes against the boundary conditions, and read off how many half-wavelengths fit the length. Mislabelling an open pipe as closed (or vice versa) changes which harmonics exist, so identify the ends first.