Standing Waves and Resonance

Pluck a guitar string, blow across the top of a bottle, or push a child on a swing at just the right moment — in each case a wave is doing something quite different from carrying energy from A to B. It is trapped, reflecting back and forth, building up a fixed pattern that seems to sit still while it oscillates. This workbook looks at how that happens, what determines the possible patterns on strings and in pipes, and why matching a driving frequency to a system's own natural frequency can make its amplitude grow dramatically — for better (musical instruments, MRI scanners) or worse (collapsing bridges).

By the end of this workbook you should be able to:

Explain the formation of a standing wave as the superposition of two identical waves travelling in opposite directions.
Identify nodes and antinodes on a standing wave, and describe the relative amplitude and phase of points along it.
Describe and calculate the possible harmonics of standing waves on a string fixed at both ends.
Describe and calculate the possible harmonics of standing waves in open and closed pipes.
Explain natural frequency and resonance, and describe how the amplitude of a driven oscillation depends on the driving frequency.
Describe the effects of light, critical and heavy damping on an oscillating system, and on its resonant response.

Name

1. The formation of standing waves

Every wave you met in the earlier workbooks was a travelling wave: it carried energy steadily away from its source, and every point it passed through repeated the same motion, just slightly delayed. Standing waves are different. A standing wave is a pattern that stays in the same place — it doesn't travel anywhere — even though every point within it is still oscillating.

Standing waves form when two waves of the same amplitude, wavelength and frequency, travelling in opposite directions, overlap and superpose. This happens very naturally: send a wave along a string with one end fixed, and it reflects off that fixed end, creating a second wave travelling back the way it came. The original (incident) wave and the reflected wave then overlap continuously, and if the length of the string is just right for the wavelength involved, the superposition of the two settles into a stable, unmoving pattern.

Standing wave. The pattern produced by the superposition of two waves of equal amplitude, wavelength and frequency, travelling in opposite directions through the same region. Unlike a travelling wave, a standing wave does not transfer energy from one place to another.

Fig. 1.1 Two waves of equal amplitude, wavelength and frequency, travelling in opposite directions, overlap continuously to produce a standing wave. The live simulation below shows their sum.

Live simulation: forming a standing wave

Superposition of two travelling waves

The teal curve moves right; the maroon curve moves left. The bold white curve is their sum.

Interactive simulation — open the online version to watch two counter-travelling waves combine into a standing wave.

Watch the points where the bold resultant curve always crosses zero, however long the simulation runs. However the two component waves shift, their sum is always exactly zero at those fixed points — they never move. Everywhere else, the resultant oscillates up and down, but always about a fixed position, with an amplitude that depends on where along the pattern that point sits.

Check your understanding

1Explain why a standing wave does not transfer energy from one end of it to the other, even though the two travelling waves that form it each individually carry energy.

The two component waves carry equal amounts of energy in opposite directions, so the net energy flow past any point, averaged over a cycle, is zero. Energy is still transported by each individual wave, but the same amount flows each way, cancelling out overall — so the standing wave pattern itself stores energy (it oscillates between kinetic and potential/elastic forms) without transferring it onward.

2Using the simulation above, describe what is special about the points where the resultant (bold) curve always crosses zero, compared with every other point on the pattern.

At those points, the two component waves are always exactly out of phase with each other (crest meeting trough), whatever instant you look at, so their displacements always cancel completely and the resultant displacement is permanently zero. Every other point is only out of phase by varying amounts as time passes, so its resultant displacement rises and falls.

3State the three conditions that two travelling waves must satisfy in order to produce a stable standing wave pattern when they overlap.

The two waves must have the same amplitude, the same wavelength (and therefore the same frequency), and must be travelling in opposite directions through the same region.

2. Nodes and antinodes

A standing wave pattern remains fixed in space, but the amount of oscillation is not the same everywhere along it.

Node. A point on a standing wave where the displacement is always zero.
Antinode. A point on a standing wave where the amplitude of oscillation is greatest.

Between one node and the next, every point oscillates in phase with every other point in that same stretch — they all reach maximum displacement at the same instant, and all pass through zero at the same instant — but with different amplitudes, rising smoothly from zero at the nodes to a maximum at the antinode in the middle. Cross over a node into the next stretch of the pattern, and the oscillation there is exactly out of phase with the one before it: while one side is at its positive maximum, the other is at its negative maximum.

Fig. 2.1 The dashed purple curves show the envelope — the maximum extent the pattern ever reaches. The solid teal curve is a snapshot at one instant: it touches zero at exactly the same points (nodes, N) as the envelope, and its amplitude at any position is a fixed fraction of the envelope's height there. The amber arrows show how far each point actually swings — zero at the nodes, greatest at the antinodes (A), and smoothly in between.

	Standing wave	Travelling wave
Energy transfer	No energy is transferred	Energy is transferred through the medium
Amplitude	Varies with position: zero at nodes, maximum at antinodes	The same at every point
Phase	All points between adjacent nodes are in phase; points either side of a node are out of phase	Points one wavelength apart are in phase; other separations are generally out of phase
Wavelength	Twice the distance between adjacent nodes (or adjacent antinodes)	The shortest distance between two points in phase

Check your understanding

4State the phase relationship between two points that lie within the same loop of a standing wave (i.e. between the same pair of adjacent nodes, with no node between them).

They are in phase with each other — they reach maximum displacement at the same instant and pass through zero at the same instant, even though their amplitudes are different.

5State the phase relationship between two points on either side of a node, in adjacent loops of a standing wave.

They are exactly out of phase (in antiphase) with each other — when one side is at its positive maximum, the other is at its negative maximum.

6The distance between two adjacent nodes of a standing wave on a string is measured to be 0.42 m. Determine the wavelength of the standing wave.

The distance between adjacent nodes is always λ/2, so λ = 2 × 0.42 m.

7Explain, using the idea of superposition, why a node is a point of permanently zero displacement.

A node is a point where the two component travelling waves are always exactly out of phase with each other, whatever instant you look at — so by the principle of superposition their displacements always add to zero there, however the individual waves move.

3. Standing wave patterns on strings

If a string fixed at both ends is plucked or vibrated, it can only settle into a standing wave that has a node at each fixed end — the string obviously cannot move at a point that is clamped in place. These fixed-end requirements are called the boundary conditions of the system, and they determine exactly which wavelengths (and so which frequencies) are actually possible.

Boundary conditions. The physical conditions imposed at the ends of a standing wave system, which determine whether there is a node or an antinode there. A fixed end must be a node; a free end must be an antinode.

The simplest possible pattern — a single loop, with a node at each end and one antinode in the middle — is called the first harmonic. It is usually the dominant mode of vibration, but a whole series of other harmonics (modes of vibration) can also occur, each fitting an extra half-wavelength loop into the same string length.

Important IB terminology: the lowest-frequency standing wave a system can support is always called the first harmonic in this course — not the "fundamental". Higher modes are the second harmonic, third harmonic, and so on — the terms "overtone" and "fundamental" are not used.

Fig. 3.1 The first four harmonics of a string of length l, fixed at both ends. Each successive harmonic fits one more half-wavelength loop into the same length.

If a string of length l is fixed at both ends, the wavelength of the first harmonic, λ₀, is 2l — the longest wave that can fit a node at each end. Since v = fλ, the frequency of the first harmonic is:

f₀ = v / λ₀ = v / 2l

where v is the speed of the wave along the string (which depends on its tension and mass per unit length, as covered in Topic C.1). Every other harmonic is a whole-number multiple of this: the wavelengths are 2l, 2l/2, 2l/3, 2l/4 … and the frequencies are f₀, 2f₀, 3f₀, 4f₀ …

The wavelength of the first harmonic is fixed by the length of the system and its boundary conditions. The equation v = fλ then connects that wavelength to the frequency, once the wave speed is known.

Live simulation: standing waves on a string

String fixed at both ends

Harmonic, n = 1

String length L = 0.60 m, first-harmonic frequency f₀ = 200 Hz (fixed for this demo).
λₙ = 2L/n = 1.20 m
fₙ = n × f₀ = 200 Hz

Interactive simulation — open the online version to explore harmonics 1–6 on a fixed string.

Worked example

Worked example 3.1

A cello string has a length of 0.70 m. Waves travel along it at 246 m s⁻¹.

Answer:
λ₀ = 2l = 2 × 0.70 m = 1.40 m
f₀ = v/λ₀ = 246 / 1.40

Check your understanding

8A guitar string of length 0.65 m vibrates in its first harmonic. Waves travel along the string at 191 m s⁻¹. Determine the frequency of the first harmonic.

λ₀ = 2l = 2 × 0.65 m = 1.30 m
f₀ = v/λ₀ = 191/1.30

9A string of length 0.84 m, fixed at both ends, supports a first harmonic of frequency 96 Hz. Determine the wavelength and frequency of the third harmonic on this string.

λ = 2l/3 = (2 × 0.84)/3
f = 3 × f₀ = 3 × 96

10Explain why the ends of a string that is fixed at both ends must always be nodes, whichever harmonic is present.

A fixed end is clamped in place and physically cannot move, so its displacement must always be zero — which is exactly the definition of a node. This boundary condition must be satisfied for any standing wave that forms on the string, whatever harmonic it is.

11Using the simulation above, describe how the spacing between adjacent nodes changes as the harmonic number n is increased, for the same fixed string length L.

As n increases, more nodes appear along the same length of string, so adjacent nodes get closer together — the node spacing is L/n, which decreases as n increases.

4. Standing wave patterns in pipes

Standing waves also form in columns of air, such as inside a pipe, tube or bottle — this time as longitudinal sound waves rather than transverse waves on a string. As with strings, the possible patterns depend entirely on the boundary conditions at the two ends of the pipe.

A pipe open at both ends must have an antinode at each end. A pipe closed at one end and open at the other must have a node at the closed end and an antinode at the open end.

Pipes open at both ends

An open end is free to move, so it must be an antinode — the air there is free to oscillate with maximum amplitude. This gives exactly the same set of possible patterns as a string fixed at both ends, just with the node/antinode roles swapped: every harmonic (1st, 2nd, 3rd, 4th …) is possible, and the equations are identical: λ₀ = 2l, f₀ = v/2l, with the same whole-number multiples for higher harmonics.

Fig. 4.1 The first three harmonics in a pipe open at both ends. The two grey rails represent the physical walls of the pipe; the curve is the maximum sideways displacement of the vibrating air at each point, which always reaches the rails (an antinode) at both open ends and touches the centreline (a node) in between — not a transverse wave, since sound in a pipe is longitudinal.

Pipes closed at one end

A pipe closed at one end and open at the other needs a node at the closed end (the air there cannot move) and an antinode at the open end. The simplest pattern that satisfies both conditions at once is a quarter wavelength — not a half wavelength — so the first harmonic has a longer wavelength, and therefore a lower frequency, than an open pipe of the same length:

λ₀ = 4l, f₀ = v/4l

Because the pattern must still end on a node at the closed end and an antinode at the open end, only the odd-numbered harmonics (1st, 3rd, 5th, 7th …) are possible for a pipe closed at one end — the even harmonics simply don't fit the boundary conditions.

Fig. 4.2 The first, third and fifth harmonics in a pipe closed at one end (left) and open at the other (right). The solid navy bar marks the closed end (a node — the pipe wall itself stops the air moving there); the right end is left open, where the wave always reaches the rail (an antinode). The second, fourth … harmonics cannot occur, because they would require an antinode at the closed end.

Live simulation: standing waves in a pipe

Open vs. closed pipe

Boundary conditions

Harmonic

Pipe length L = 0.50 m, speed of sound v = 340 m/s (fixed for this demo).

Interactive simulation — open the online version to compare open- and closed-pipe harmonics.

Worked example

Worked example 4.1

A bicycle pump, closed at one end by a fingertip, behaves as a pipe closed at one end with an effective length of 0.62 m. Take the speed of sound as 343 m s⁻¹.

Answer:
λ₀ = 4l = 4 × 0.62 m = 2.48 m
f₀ = v/λ₀ = 343/2.48

Check your understanding

12An organ pipe of length 1.15 m is open at both ends. Take the speed of sound as 340 m s⁻¹. Determine the frequency of its first harmonic.

λ₀ = 2l = 2 × 1.15 m = 2.30 m
f₀ = v/λ₀ = 340/2.30

13A different pipe, also of length 1.15 m, is closed at one end. Take the speed of sound as 340 m s⁻¹. Determine the frequency of its first harmonic, and explain why only odd-numbered harmonics are possible for this pipe.

λ₀ = 4l = 4 × 1.15 m = 4.60 m
f₀ = v/λ₀ = 340/4.60
Only odd harmonics fit the boundary conditions (a node at the closed end and an antinode at the open end at the same time) — the even-numbered patterns would require an antinode at the closed end, which is not possible.

14Explain why the first-harmonic frequency of a pipe closed at one end is lower than that of an open pipe of the same length.

The closed pipe's first harmonic has a wavelength of 4l, twice as long as the open pipe's first-harmonic wavelength of 2l, because only a quarter wavelength (not a half) fits between a node and an antinode. Since f = v/λ and v is the same in both cases, the longer wavelength gives a lower frequency.

15Using the simulation above, describe how the set of available harmonics changes when you switch the boundary conditions from "open at both ends" to "closed at one end".

For the open pipe, every harmonic (1st, 2nd, 3rd, 4th …) is available. Switching to a closed pipe removes all of the even-numbered harmonics, leaving only the odd-numbered ones (1st, 3rd, 5th …), and the first-harmonic frequency drops because its wavelength is now 4l instead of 2l.

5. Natural frequency and resonance

Pluck a ruler clamped to a desk, tap a wine glass, or strike a tuning fork, and each will vibrate at its own characteristic frequency (or set of frequencies) once disturbed and left alone.

Natural frequency. The frequency at which a system oscillates when it is disturbed and then left to vibrate freely, without any external influence.

The natural frequency depends only on the physical properties of the system — its dimensions, mass, stiffness or tension — not on how hard it was disturbed. A simple mass on a spring, for example, has a natural frequency set by its mass m and the spring constant k (Topic C.1).

16Given the time period T of an oscillating system, what is its natural frequency f₀? Then state the equation for the natural frequency of a mass m on a spring of spring constant k (Topic C.1), in terms of m and k.

f₀ = 1/T, for any oscillating system.
For a mass–spring system: f₀ = (1/2π) × √(k/m)

Forced vibrations and driving frequency

If instead a system is continuously pushed by an external periodic force — such as repeatedly pushing a child on a swing — it undergoes a forced vibration at the frequency of that external force, called the driving frequency, which need not be the same as the system's natural frequency.

Forced vibration. An oscillation that occurs because an external periodic (driving) force acts on a system, potentially at a frequency different from the system's own natural frequency.

Resonance

Something special happens when the driving frequency matches the natural frequency: the amplitude of the oscillation grows, often dramatically. This is resonance.

Resonance. The large increase in amplitude (and energy) of an oscillation that occurs when a periodic driving force acts on a system at the same frequency as the system's natural frequency, and in phase with its motion.

A graph of amplitude against driving frequency — a frequency–response graph — rises to a sharp maximum at the resonant frequency, which is at, or very close to, the natural frequency of the system.

Live simulation: resonance

Amplitude vs. driving frequency

Damping level

Driving frequency = 1.00 × natural frequency

Relative amplitude at this driving frequency: —

Interactive simulation — open the online version to explore how amplitude depends on driving frequency and damping.

The resonant frequency is at, or very close to, the natural frequency — but the height and sharpness of the peak depend strongly on the amount of damping present, as you can see by switching damping level in the simulation above. Greater damping produces a lower, broader peak, and shifts the resonant frequency very slightly below the natural frequency.

Worked example

Worked example 5.1

A trolley of mass 0.40 kg is attached to a spring of spring constant 65 N m⁻¹ and set oscillating horizontally.

Answer:
f₀ = (1/2π) × √(k/m) = (1/2π) × √(65/0.40)

Check your understanding

17A 0.25 kg mass hangs from a spring of spring constant 18 N m⁻¹ and is set oscillating vertically. Determine its natural frequency.

f₀ = (1/2π) × √(k/m) = (1/2π) × √(18/0.25)

18State what is meant by resonance, in terms of driving frequency and natural frequency.

Resonance occurs when a system is driven by a periodic external force whose frequency equals (and is in phase with) the system's own natural frequency, causing the amplitude of oscillation to increase to a maximum.

19Using the simulation above, describe how the amplitude changes as the driving frequency is moved further away from the natural frequency, in either direction.

The amplitude falls away from its maximum on both sides of the natural frequency — it does not matter whether the driving frequency is higher or lower than the natural frequency, moving away from it in either direction reduces the amplitude.

20Soldiers marching in step are traditionally told to "break step" (stop marching in time with each other) when crossing a bridge. Explain why, in terms of resonance.

Marching in step applies a strong periodic driving force to the bridge at the marching frequency. If that frequency happened to match the bridge's natural frequency, resonance could build the bridge's oscillation amplitude to a dangerously large value. Breaking step removes the regular, synchronised driving force, so this risk is avoided.

6. Damping

No real oscillator vibrates forever. Resistive forces — friction, air resistance, internal forces in the material — act against the motion and remove energy from the system, transferring it to the surroundings as heat (and sometimes sound).

Damping. The dissipation of energy from an oscillator due to resistive forces, causing the amplitude of oscillation to decrease over time.

The amount of damping present can vary enormously between systems, and is usually described using three broad categories.

Light damping — the system continues to oscillate for a long time, with its amplitude decreasing gradually.
Critical damping — the system returns to equilibrium as quickly as possible, without oscillating at all.
Heavy damping — the resistive forces are so large that the system takes a long time to return to equilibrium, again without oscillating.

Live simulation: types of damping

Displacement vs. time

Damping type

The pale dashed curve shows the undamped case for comparison.

Interactive simulation — open the online version to compare light, critical and heavy damping.

Both critical and heavy damping bring the system back to equilibrium without any oscillation at all — the difference between them is only how quickly this happens. Critical damping is the fastest possible return to equilibrium without overshooting; heavier damping actually returns more slowly than critical damping, because the very large resistive force also opposes the motion needed to get back to equilibrium in the first place.

Real-world resonance and damping

Engineers spend a great deal of effort managing resonance and damping in real structures. A car's suspension is designed to be close to critically damped, so that after going over a bump it settles quickly without bouncing repeatedly. Bridges and skyscrapers are designed so that their natural frequencies don't coincide with likely driving frequencies from wind, footsteps or earthquakes — and where that can't be guaranteed, damping is deliberately built in. The Millennium Bridge in London had to close soon after opening in 2000 because pedestrians' footsteps drove it into resonance; the fix was to add extra dampers. Resonance isn't always unwanted, though — magnetic resonance imaging (MRI) scanners deliberately drive hydrogen nuclei in the body at their resonant frequency to produce detailed medical images, and a microwave oven heats food because its radiation frequency is resonant with water molecules.

Check your understanding

21Describe the difference between light damping and heavy damping, in terms of whether the system oscillates and how long it takes to stop moving.

With light damping, the system continues to oscillate for a relatively long time, with its amplitude decaying gradually. With heavy damping, the system does not oscillate at all — the resistive forces are so large that it takes a long time (compared to its natural period) to creep back to equilibrium.

22State which type of damping is used in a car's suspension system, and explain why that choice is preferred over the alternatives.

A car's suspension is designed to be close to critically damped. This is preferred because it returns the car body to its equilibrium position as quickly as possible after a bump, without the discomfort (and loss of control) of repeated bouncing that would occur with lighter damping, and without the sluggish, slow response of heavier damping.

23Using the simulation above, compare the total time taken for the light-damping case and the heavy-damping case to reach approximately zero displacement.

The light-damping case actually takes longer overall to settle to (approximately) zero displacement than the heavily-damped case appears to at first glance, because it keeps oscillating with slowly-decaying amplitude — but neither is as fast as the critically-damped case, which reaches zero soonest of all without overshooting.

24Explain, using the idea of damping, why the resonance peak in Section 5's simulation becomes shorter and broader as the damping level is increased.

Greater damping dissipates more energy from the oscillation at every driving frequency, so the maximum amplitude that can build up at resonance is smaller (a shorter peak). Because damping resists the motion over a wider range of driving frequencies too, not just exactly at resonance, the amplitude falls off more gently either side of the peak, making it broader.

Glossary

Standing (stationary) wave: The stable, unmoving pattern produced by the superposition of two waves of equal amplitude, wavelength and frequency travelling in opposite directions.
Node: A point on a standing wave where the displacement is always zero.
Antinode: A point on a standing wave where the amplitude of oscillation is greatest.
Boundary conditions: The physical conditions at the ends of a standing wave system (fixed or free) that determine whether there is a node or antinode there.
Harmonic: One of the possible standing wave patterns (modes of vibration) a system can support.
First harmonic: The standing wave pattern with the lowest possible frequency (and longest possible wavelength) for a given system. Higher harmonics are whole-number multiples of this frequency. ("Fundamental" and "overtone" are not used in this course.)
Open pipe: A pipe open at both ends, requiring an antinode at each end; supports every harmonic.
Closed pipe: A pipe closed at one end and open at the other, requiring a node at the closed end and an antinode at the open end; supports only odd-numbered harmonics.
Natural frequency: The frequency at which a system oscillates when disturbed and then left to vibrate freely, without external influence.
Forced vibration: An oscillation driven by an external periodic force, potentially at a frequency different from the system's natural frequency.
Driving frequency: The frequency of an external periodic force acting on an oscillating system.
Resonance: The large increase in amplitude of an oscillation that occurs when the driving frequency equals the natural frequency of the system.
Resonant frequency: The driving frequency at which resonance (maximum amplitude) occurs; at, or very close to, the natural frequency.
Damping: The dissipation of energy from an oscillator due to resistive forces, reducing its amplitude over time.
Critical damping: The degree of damping that returns a system to equilibrium in the shortest possible time without oscillating.
Heavy damping: Damping strong enough to prevent oscillation entirely, but which returns the system to equilibrium more slowly than critical damping.