C.4 Standing waves and resonance

Build the physical model

At each point, add the displacements of the two waves. Where they always cancel, the amplitude is zero: these fixed positions are nodes. Where they reinforce most strongly, the amplitude is greatest: these fixed positions are antinodes. The wave pattern repeats every half-wavelength, so adjacent nodes and adjacent antinodes are separated by $\lambda/2$, while a node and its nearest antinode are $\lambda/4$ apart.

Interpret what is and is not moving

The particles of the medium still oscillate between nodes and antinodes, but the locations of the nodes and antinodes do not move. A standing wave does not transfer energy progressively from one end to the other in the way a travelling wave does; energy is stored and exchanged locally within each segment between adjacent nodes.

Read relative amplitude

Every point between two adjacent nodes oscillates at the same frequency, but its amplitude depends on position: zero at a node, maximum at an antinode, and intermediate elsewhere. The standing-wave envelope therefore describes amplitude, not a travelling displacement profile at one instant.

Read phase

Points in the same segment between adjacent nodes oscillate in phase. Points in neighbouring segments oscillate in antiphase, with phase difference $\pi$ (180°). At a node the phase is not useful to assign because the displacement amplitude is zero.

Use the string patterns

For a string fixed at both ends, or with two free ends, the nth harmonic has $n$ half-wavelengths in length $L$: $\lambda_n=2L/n$ and $f_n=nv/(2L)$. For one fixed and one free end, the allowed patterns contain an odd number of quarter-wavelengths: $\lambda_n=4L/(2n-1)$ and $f_n=(2n-1)v/(4L)$, with $n=1,2,3,\ldots$.

Apply the same geometry to pipes

An open pipe has displacement antinodes at both ends and follows the two-open-end pattern. A closed pipe has a displacement node at the closed end and an antinode at the open end, so only the odd sequence of harmonics is allowed. Use $v=f\lambda$ after finding the wavelength from the boundary pattern. End corrections for open pipes are not required.

Explain the large amplitude

At resonance, the driving force supplies energy efficiently to the oscillator each cycle because its timing is well matched to the motion. The amplitude rises until the energy supplied per cycle is balanced by energy dissipated. Greater energy dissipation means a smaller maximum amplitude.

Read a frequency-response graph

Plot amplitude against driving frequency. The peak identifies the resonant frequency; the peak height is the maximum amplitude. A practical system may have its peak slightly displaced from its undamped natural frequency when damping is significant, but the syllabus requires only a qualitative frequency-response analysis.

Recognize useful and destructive resonance

Resonance is useful when a large, frequency-selective response is wanted, such as tuning a receiver or producing a strong musical sound. It can be destructive when repeated driving builds damaging oscillations in a bridge, building or machine. Designs then change the natural frequency, avoid the matching driving frequency, or add damping.

Connect damping to energy

More damping means more energy is dissipated during each cycle. The driver must supply that lost energy, but the oscillator cannot build up as large an amplitude before input and loss balance. With little damping, energy accumulates more efficiently and the resonance peak is taller and narrower.

Apply the model

If a suspension or bridge is damped, the oscillation amplitude is reduced and the resonant response occurs at a slightly lower driving frequency. This can be useful for controlling vibration, although damping also reduces the sharpness of frequency selection.

Choose the response from the design goal

A system that must settle quickly without repeated oscillation is adjusted close to critical damping. Too little damping allows repeated crossings of equilibrium; too much damping resists the motion so strongly that the return takes longer.