IB Physics HL C.4 Standing waves and resonance Questions

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Practise C.4 by reading standing-wave diagrams, calculating string and pipe harmonics, and interpreting resonance curves and damping responses from experimental data.

125 questions in this syllabus scope, with question parts counted together.

Syllabus
First assessment 2025
Course
Physics HL
Level
HL
Exam points & syllabus

Exam points

  • Explain standing-wave formation as superposition of identical waves travelling in opposite directions, usually an incident wave and its reflection.
  • Identify the reflecting boundary in the actual apparatus: a fixed string end, closed pipe end or water surface terminating the air column.
  • Compare standing and travelling waves using energy propagation, position-dependent amplitude, relative phase or allowed wavelengths.
  • Nodes have zero amplitude; antinodes have maximum amplitude. An instant when every particle is at equilibrium does not make every point a node.
  • Particles in the same standing-wave loop oscillate in phase; each intervening node reverses the phase by π.
  • Adjacent antinodes are λ/2 apart and opposite in phase; alternate antinodes can be λ apart and in phase.
  • When drawing a second point’s time trace, preserve the period and determine its amplitude and phase separately from the standing pattern.
  • Standing-wave particles share a period but can differ in amplitude and speed; a travelling wave’s snapshot heights do not directly give amplitudes.
  • Use the stated sign convention for longitudinal displacement and compare snapshots to determine horizontal particle position and velocity.
  • For the second air-column resonance above water, displacement nodes occur at the water surface and two-thirds up the air column.
  • A string fixed at both ends has λn = 2L/n and fn = nv/(2L); count the actual loops before selecting the harmonic index.
  • An open-open pipe has λ1 = 2L; a closed-open pipe has λn = 4L/n with odd n only.
  • Compare length and boundary changes together: closing one end halves the first-harmonic frequency at equal length, while shortening the pipe raises it.
  • In a closed-open pipe, a three-quarter-wave pattern gives λ = 4L/3; the next mode above the first harmonic has three times its frequency.
  • Successive resonant air-column lengths at fixed frequency differ by λ/2; first and second closed-open lengths are λ/4 and 3λ/4.
  • At fixed wave speed and string length, frequency scales with harmonic index; n loops have n + 1 nodes including the endpoints.
  • For fixed-end strings, even harmonics have a central node; if the second-harmonic frequency is f, these frequencies are nf.
  • Use supplied wave-speed laws as given and show the requested precision or fundamental SI units.
  • For a centre-clamped first-harmonic rod with free ends, use a central displacement node and end antinodes, then retain frequency when sound enters air.
  • A closed-open pipe supports odd multiples of its first frequency; an even multiple fails the boundary conditions.
  • Natural frequency describes free oscillation; resonance occurs when periodic driving is close to that frequency and produces a large response.
  • For repeated bumps or wheel passages, calculate driving frequency from speed divided by spacing, then compare with the response peak.
  • Link resonance to large amplitudes when explaining discomfort, structural damage or possible collapse.
  • For a supplied ion–spring lattice model, matching infrared frequency to the lattice natural frequency can produce resonant absorption.
  • During driven startup, net energy input raises amplitude; at steady amplitude, input balances energy dissipated by damping.
  • Damping involves resistive energy loss; free damped oscillations show decreasing displacement peaks.
  • For small friction, sketch decaying amplitude with approximately unchanged period; do not extend that approximation to arbitrary damping.
  • Increasing damping reduces the maximum forced amplitude and shifts the resonant response towards lower driving frequency.
  • Follow each source’s extreme-frequency behaviour when comparing response curves; lower resonance need not mean a changed limiting static displacement.
  • Compare two distinct oscillators separately from changing damping in one oscillator; read both peak height and peak position.
  • A swing’s height has two maxima per full oscillation; with a 3.0 s period, draw decreasing height peaks every 1.5 s.
  • Critical damping gives the fastest return to equilibrium without overshoot or a complete oscillation.
  • Distinguish critical return from slower heavy damping, decaying light-damped oscillations and constant-amplitude undamped motion.
  • After the driver stops in a critically damped system, continue smoothly from the actual displacement towards equilibrium without crossing it.

Covered syllabus points

C.4.1—Standing wave formationC.4.2—Nodes and antinodesC.4.3—Strings and pipesC.4.4—ResonanceC.4.5—Damping and resonanceC.4.6—Types of damping

Question 1

[Maximum number: 2]

A loudspeaker emits sound of frequency 210 Hz into a pipe with one open and one closed end. The diagram shows a representation of the standing wave established in the pipe.

Figure for Question 1 — IB Physics HL

The length of the pipe is 1.20 m .

Outline how the standing wave is formed in the pipe.

Question 1

[Maximum number: 2]

Question (a)

(a)

The solid line represents the standing wave at time t and the dotted line represents the standing wave at an instant later. The dot is the equilibrium position of a particle P in the pipe. The up arrow indicates displacements to the right and the down arrow displacements to the left.

Figure for Question (a) — IB Physics HL

On the diagram, draw

[ 2 ]

Question (i)

(i)

a dot to indicate the approximate position of P at time t,

[ 1 ]

Question (ii)

(ii)

an arrow to indicate the velocity of P at time t.

[ 1 ]

Question (b)

(b)

A loudspeaker emits sound of frequency 210 Hz into a pipe with one open and one closed end. The diagram shows a representation of the standing wave established in the pipe.

Figure for Question (b) — IB Physics HL

The length of the pipe is 1.20 m .

[ 1 ]

Question (i)

(i)

Determine the wavelength of the wave.

[ 1 ]

Question (c)

(c)

The frequency of sound is reduced to 140 Hz . Explain why a standing wave will not be formed in the pipe.

[ 2 ]

Question 1

[Maximum number: 2]

Question (a)

(a)

The solid line represents the standing wave at time t and the dotted line represents the standing wave at an instant later. The dot is the equilibrium position of a particle P in the pipe. The up arrow indicates displacements to the right and the down arrow displacements to the left.

Figure for Question (a) — IB Physics HL

On the diagram, draw

[ 2 ]

Question (i)

(i)

a dot to indicate the approximate position of P at time t,

[ 1 ]

Question (ii)

(ii)

an arrow to indicate the velocity of P at time t.

[ 1 ]

Question (b)

(b)

A loudspeaker emits sound of frequency 210 Hz into a pipe with one open and one closed end. The diagram shows a representation of the standing wave established in the pipe.

Figure for Question (b) — IB Physics HL

The length of the pipe is 1.20 m .

[ 1 ]

Question (i)

(i)

Determine the wavelength of the wave.

[ 1 ]

Question (c)

(c)

The frequency of sound is reduced to 140 Hz . Explain why a standing wave will not be formed in the pipe.

[ 2 ]

Question 1

[Maximum number: 2]

This question is in two parts. Part 1 is about the oscillation of a mass. Part 2 is about the photoelectric effect.
Part 1 Oscillation of a mass
A mass of 0.80 kg rests on a frictionless surface and is connected to two identical springs both of which are fixed at their other ends. A force of 0.030 N is required to extend or compress each spring by 1.0 mm . When the mass is at rest in the centre of the arrangement, the springs are not extended.

Explain how the model in (b) predicts how the lattice will absorb a certain wavelength of electromagnetic infrared radiation.

Question 1

[Maximum number: 2]

A mass is attached to a vertical spring. The other end of the spring is attached to the driver of an oscillator.

Figure for Question 1 — IB Physics HL

The mass is performing very lightly damped harmonic oscillations. The frequency of the driver is higher than the natural frequency of the system. At one instant the driver is moving downwards.

The oscillator is switched off. The system has a Q factor of 22. The initial amplitude is 10 cm . Determine the amplitude after one complete period of oscillation.

Question 1

[Maximum number: 2]

W in (a) is immersed in a beaker of oil. As a result of this immersion the oscillations of W are critically damped. Describe what is meant by critically damped.