1The idea in one paragraph

A wave sent along a string reflects from the far end, and the incoming and reflected waves, identical but travelling in opposite directions, superpose into a wave that goes nowhere: a standing wave. Some points, the nodes, never move; the antinodes between them swing hardest. The ends force a node or an antinode to sit there, so only certain wavelengths fit, each a harmonic with its own frequency. Those are the system's natural frequencies. Drive a system at one of them and its amplitude grows large, which is resonance; damping removes energy and lowers the peak.

2How a standing wave forms

Figure 1 draws two identical waves on one line, teal moving right and amber moving left, and their sum in black, found by adding displacements point by point.

The rows are an eighth of a period apart. At t = 0 the two waves coincide and add to twice the amplitude. A quarter-period later each has moved λ/4 in opposite directions and they cancel everywhere: the string is momentarily flat. At half a period they reinforce again, the other way up. Through all of it the black curve never slides sideways, and the dotted points stay at zero at every instant, because there the two waves are always equal and opposite. Those are the nodes.

A standing wave is the superposition of two waves of the same frequency, wavelength and amplitude travelling in opposite directions.

In practice the second wave is a reflection: a wave meets a boundary and comes back over itself, on a string between two supports, in the air of a bottle, between the metal walls of a microwave oven. If the two amplitudes differ, the cancellation is incomplete and there are no true nodes, so exam answers name all three matching properties.

3Nodes, antinodes, amplitude and phase

Figure 2 labels the parts of a standing wave. The dashed grey curves are the envelope: the furthest each point ever gets from equilibrium. The teal curve is the string at one instant.

A node is a point where the displacement is always zero. An antinode is a point where the amplitude is a maximum. The spacings come straight from Figure 1:

• adjacent nodes are λ/2 apart, and so are adjacent antinodes;
• a node and the next antinode are λ/4 apart;
• one full wavelength spans two loops.

Here λ is the wavelength of the two travelling waves that make the pattern, so in a picture you count loops and multiply by λ/2.

Amplitude depends on position. In a travelling wave every point has the same amplitude. In a standing wave the amplitude runs from zero at a node to a maximum at an antinode. In Figure 2, P and Q are in the same loop but P is nearer the node, so P has the smaller amplitude.

Phase is simpler than you expect. Every point between two neighbouring nodes rises, peaks and passes through zero together, so P and Q are in phase even though their amplitudes differ. Cross a node and the motion flips: R, in the next loop, is in antiphase with P and Q, a phase difference of π rad (180°). In a standing wave those are the only two possibilities.

4Boundary conditions: what the ends do

The ends of a string or an air column are its boundary conditions, and they decide which standing waves can exist.

On a string:

• A fixed end cannot move, so it is always a node.
• A free end (a string tied to a light ring sliding on a smooth rod) moves most, so it is always an antinode.

In a pipe, the wave is sound, which is longitudinal, and we describe it by the displacement of the air along the pipe.

• At a closed end the air is up against a wall and cannot move along the pipe, so it is always a displacement node.
• At an open end the air is free to move in and out, so it is always a displacement antinode.

Pressure nodes and antinodes are not required, and the end correction (the antinode sitting slightly beyond an open end) is ignored: put the antinode exactly at the open end. After that, finding harmonics is a drawing problem. Put the right thing at each end, fit the fewest loops that obey both ends, then add loops one at a time.

5Standing waves on strings

Figure 3 shows the first three possible patterns for each of the three kinds of string. In each sketch the two teal curves are the extreme positions a quarter-cycle apart; the dashed line is equilibrium.

The lowest-frequency standing wave is the first harmonic (the guide bans fundamental and overtone), and the nth harmonic has frequency fₙ = n f₁.

Fixed at both ends (panel a), and free at both ends (panel c). Both ends are the same kind, nodes or antinodes, so the length must hold a whole number of half-wavelengths.

One end fixed, one free (panel b). One end is a node and the other an antinode, so the length holds an odd number of quarter-wavelengths.

An even number of quarter-wavelengths would put the same kind of point at both ends, so even harmonics are missing and the second pattern is the third harmonic (3f₁). In every case v, set by the string's tension and mass per unit length, is the same for all harmonics.

Worked example: a guitar string. A string of vibrating length 0.650 m, fixed at both ends, has a first harmonic of 196 Hz. Determine the wave speed on the string and the wavelength and frequency of its third harmonic.

A classic slip: the sound you hear has the string's frequency, 196 Hz, but not its wavelength. In air at 340 m s⁻¹ the sound's wavelength is 340 ÷ 196 = 1.73 m.

6Standing waves in pipes

Figure 4 does the same for pipes. The curves are a graph of displacement amplitude along the pipe; the air itself moves left and right, not up and down.

Closed ends behave like fixed ends, open ends like free ends.

So a pipe closed at one end has a first harmonic half the frequency of an open pipe of the same length (λ₁ = 4L, not 2L), and only odd harmonics.

Worked example: open or closed? A pipe resonates at 450 Hz and at 750 Hz, and at no frequency in between. The speed of sound is 340 m s⁻¹. Determine whether the pipe is open at both ends or closed at one end, and find its length.

Reading it from data. A Paper 1B resonance tube is closed at one end, with its length changed while a speaker sounds above it. For the first harmonic f₁ = (v/4) × (1/L), so f₁ against 1/L is a straight line through the origin with gradient v/4, as in the invented results of Figure 5.

Choosing axes that make a straight line, then interpreting the gradient, is the Paper 1B skill.

7Damping: light, critical and heavy

A real oscillator left alone loses energy to friction, air resistance or forces inside the material. That is damping: a force opposing the velocity, so it always takes energy out. Figure 6 shows the three degrees the guide names, each for an oscillator pulled aside and released from rest.

Light damping. The system still oscillates, but the amplitude decreases gradually, cycle by cycle, with the period almost unchanged. A plucked guitar string is lightly damped.

Critical damping. The system returns to equilibrium in the shortest possible time without oscillating, with no overshoot. Car suspension is designed close to this, so the body settles after a bump without bouncing; so is a self-closing door.

Heavy damping. No oscillation, but the return to equilibrium is slower than with critical damping: it creeps back, like a pendulum in thick oil. The large resistive force prevents overshoot but also slows the return, so heavy damping is not the fastest.

8Driven oscillations and resonance

Every oscillating system has a natural frequency, f₀, at which it oscillates if displaced and released; a string or pipe has a whole set, its harmonics. Push the system with a periodic force at a driving frequency f_d, and after a short settling time it oscillates at f_d, not at f₀. This is a forced oscillation, and what depends on f_d is the amplitude.

Resonance is the large increase in amplitude when the driving frequency equals, or is very close to, the natural frequency. Figure 7 plots steady amplitude against driving frequency for three levels of damping.

Why the peak happens, in terms of energy. At f₀ the driving force stays in step with the velocity, so it does positive work throughout every cycle. The amplitude grows until the energy lost to damping per cycle equals the energy supplied per cycle. Away from f₀ the force is out of step for part of each cycle and takes energy back, so the steady amplitude is smaller.

What damping does to the curve (qualitative only, as in Figure 7):

1. The maximum amplitude is lower, because losses balance the input at a smaller amplitude.
2. The resonant frequency is slightly lower. The peak moves a little below f₀; with light damping the shift is too small to see.
3. The peak is broader: a moderate response over a wide range of frequencies rather than a violent one at a single frequency.

With no damping the amplitude at resonance would grow without limit; real peaks are high but finite.

Standing waves are resonance. A string or pipe has a natural frequency for each harmonic, and energy fed in at one of them builds a large standing wave. That is why the resonance tube in section 6 goes loud at particular lengths: there its natural frequency matches the fork.

9Resonance: useful and destructive

Useful resonance.

• Musical instruments. An organ pipe's air column or a guitar string resonates at its harmonics, turning a small input into a loud, clear note.
• Tuning a radio. The receiving circuit's natural frequency is adjusted until it matches one station's frequency, which it then picks out from all the others.
• Quartz clocks. A quartz crystal driven at its natural frequency oscillates very steadily, and counting the oscillations keeps time.
• Molecules and infrared. Carbon dioxide and water vapour molecules have natural vibration frequencies in the infrared, so they absorb the Earth's infrared emission at those frequencies: the link to the greenhouse effect in B.2.

Destructive resonance.

• Bridges and buildings. Wind, footsteps or earthquakes can drive a structure near its natural frequency. London's Millennium Bridge swayed sideways under crowds when it opened in 2000 and was closed until dampers were fitted; tall buildings in earthquake zones carry damping systems for the same reason.
• Machinery and vehicles. An engine speed that matches the natural frequency of a panel makes it rattle, and repeated large oscillation can crack metal through fatigue.

The cure is always one of two: move the natural frequency away from the driving frequency, or add damping to cut the peak.