C.4 Standing waves and resonance

Core idea

A standing wave forms when two waves with the same frequency, wavelength and amplitude travel in opposite directions and superpose. In practice, one wave is often the incident wave and the other is its reflection. The pattern oscillates in place rather than travelling along the medium.

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.

Check the boundary

Do not describe a standing wave as a single wave travelling forward. First identify the two counter-propagating waves and then use superposition to explain the fixed pattern. The model here is limited to two identical opposite-travelling waves; the syllabus does not require superposition of more than two waves.

Identify the positions

A node is a fixed position where the displacement is always zero. An antinode is a fixed position where the amplitude is greatest. Adjacent nodes or adjacent antinodes are separated by $\lambda/2$; a node and its nearest antinode are separated by $\lambda/4$.

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.

Common trap

Do not infer phase only from distance. First locate the nodes: crossing one node changes the phase by $\pi$; staying within the same node-to-node segment leaves the phase difference zero.

Start with the boundary conditions

A fixed end of a string is a displacement node; a free end is a displacement antinode. For air displacement in a pipe, a closed end is a displacement node and an open end is a displacement antinode. These end conditions determine which standing-wave patterns are allowed.

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.

Common trap

Do not use the closed-pipe formula for an open pipe, and do not count pressure nodes or pressure antinodes here: the syllabus asks for air-displacement nodes and antinodes. Also use “first harmonic” for the lowest-frequency mode; the syllabus does not require the terms fundamental or overtone.

Separate the frequencies

The natural frequency is the frequency at which a system oscillates after a disturbance when it is left alone. The driving frequency is imposed by an external periodic force. Resonance occurs when the driving frequency is equal or very close to the system’s natural frequency, producing a large amplitude response.

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.

Common trap

Do not call the driving frequency the natural frequency. A large amplitude alone is not enough to establish resonance: connect it to the driving frequency being close to the natural frequency and to efficient energy transfer.

Read the response peak

Damping removes mechanical energy from an oscillator. On an amplitude-versus-driving-frequency graph, increasing damping lowers the maximum amplitude and makes the peak less sharp. The resonant frequency also shifts slightly to a lower value. These are qualitative changes; the syllabus does not require a detailed damped-oscillator derivation.

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.

Common trap

Do not draw a damped response with a taller peak. More damping lowers the peak and shifts it left on a frequency axis whose driving frequency increases to the right.

Classify the response

Light damping lets the system oscillate about equilibrium while its amplitude decreases gradually. Critical damping returns the system to equilibrium in the shortest time without oscillating. Heavy damping also avoids oscillation, but returns to equilibrium more slowly than critical damping.

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.

Common trap

Critical and heavy damping are both non-oscillatory, but they are not equally fast. Critical damping is the fastest return without overshoot; heavy damping is slower.