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Resonance

Key idea: Explain resonance using forced oscillations, interpret amplitude–frequency response curves, and discuss useful vs dangerous resonance (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Distinguish free and forced oscillations, natural frequency and driving frequency.
  • Interpret resonance response curves, damping effects and practical applications.

1. Definitions (Must Know)

A. Forced oscillation

A forced oscillation occurs when a system is driven by an external periodic force.

B. Natural frequency

The natural frequency is the frequency at which the system oscillates when it is left to vibrate freely.

C. Resonance

Resonance is the maximum steady-state amplitude produced when the driving frequency is close to, or at, the natural frequency. Damping can move the response maximum slightly below the undamped natural frequency.

D. Frequency response curve

A frequency response curve is a graph of oscillation amplitude against driving frequency.

2. Key Ideas (What Earns Marks)

  • Resonance corresponds to a peak in the amplitude–frequency response.
  • Damping reduces the peak amplitude and makes the peak less sharp (broader).
  • Resonance can be useful (amplify a desired response) or dangerous (excessive vibration/stress).
Link: damping and natural frequency

See: Damping and Natural Frequency.

3. Detailed Explanations

A. Response curves (qualitative shapes)

This plot shows the typical effect of damping on the resonance peak (shape only).

Resonance response curves (effect of damping)

Amplitude response versus driving frequency ratio for light, medium, and heavy damping; increasing damping lowers and broadens the peak.

Scroll across the graph to read all labels.

Amplitude response versus driving frequency ratio for light, medium, and heavy damping; increasing damping lowers and broadens the peak.Amplitude response versus driving frequency ratio for light, medium, and heavy damping; increasing damping lowers and broadens the peak.
Increasing damping lowers the maximum amplitude and broadens the resonance peak.
Open full-size graph
View figure data
Values for Resonance response curves (effect of damping)
Driving frequency / natural frequency (unitless)LightModerateHeavier
0.21.04076361780453321.03606084259456031.0197712705600053
0.300000000000000041.09652023190326391.08426967318775751.043650309657746
0.41.18511365784994331.15810263653019871.0748339509808695
0.51.32163720091017961.26491106406735181.1094003924504583
0.60000000000000011.5357377920848781.4147795874357321.139901881468883
0.71.89083558415121231.61669284832118291.154623566040508
0.82.5383654128340481.8582353656179161.139901881468883
0.90000000000000013.8208035995043532.0472212475808571.0871492046706062
1521
1.13.28797974610714451.69857888396249560.8929639290487886
1.21.99521721116905541.3440083264261760.7823969299055072
1.31.35618936220899561.05491336088618430.6794550781149664
1.40000000000000010.99999999999999980.84167451064205960.5890920370328413
1.50.7779098415844140.68599434057003530.5121475197315839
1.60.62795044912916020.57039580949534410.44750008726252544
1.70.52074152231077750.48254591130050010.39338080327355524
1.80.440772487177097660.414242956474593360.34799506017558246

B. Why amplitude peaks at resonance (energy transfer)

The driving force supplies energy each cycle.

Near resonance, the energy transfer adds up efficiently, so amplitude grows until the average power input equals the average power dissipated by damping.

C. Useful vs dangerous resonance (examples)

Useful:

  • sound production in musical instruments (air columns/strings)
  • tuning circuits (selecting a frequency)

Potentially dangerous:

  • large vibrations in bridges/buildings if driven near a natural frequency
  • damage in machines if rotating parts drive vibration near resonance

4. Common Mistakes

  • Mixing up natural frequency with driving frequency (forced oscillation follows the driver).
  • Saying “resonance always bad” (it depends on context; it can be useful or harmful).
  • Forgetting damping limits the amplitude.

5. Exam Tips

  • If asked about the effect of damping on the response curve: “lower peak and broader curve”.
  • If asked how to reduce resonance effects: increase damping or change the natural frequency (e.g. change mass/stiffness).

6. Worked Examples

Modelled example 1

Identify resonance from a graph

Core

Problem

An amplitude–driving-frequency curve has its maximum at 8.0 Hz. What can be inferred about the natural frequency, and what precision limit should be stated?
Study the worked solution

  1. Locate resonance

    Method

    The observed resonance frequency is about 8.0 Hz.

    Reason

    Resonance is identified by the maximum steady amplitude.

    Working

    Response peak at f_d = 8.0 Hz.

  2. Infer the natural frequency

    Method

    The natural frequency is close to 8.0 Hz.

    Reason

    The response is largest when driving and natural frequencies are close.

    Working

    f₀ ≈ 8.0 Hz.

  3. Limit the claim

    Method

    Do not claim exact equality from the graph alone.

    Reason

    Damping can shift the response maximum slightly below the undamped natural frequency.

    Working

    Inference: close to, not necessarily exactly, 8.0 Hz.

Guided practice 2

Qualitative damping comparison

About 3 min

Problem

Two systems have the same natural frequency. Curve A has a sharper, higher resonance peak than curve B. Which system is more lightly damped?

Try this before viewing the solution

More lightly damped

Hints

Hint 1: use both visual features
Less dissipation produces a taller and more selective response peak.
View solution step by step

  1. Use peak height

    Method

    System A is more lightly damped.

    Reason

    Less energy is dissipated per cycle, allowing a larger resonant response.

    Working

    Higher peak ⇒ lighter damping.

  2. Use peak sharpness

    Method

    The narrower peak confirms the classification.

    Reason

    Light damping gives stronger frequency selectivity.

    Working

    Sharper peak ⇒ lighter damping.

Common misconception 3

Resonant driving frequency

Find and correct the mistake

Learner claim

A damped system has natural frequency 15 Hz. A learner states that the response maximum must occur at exactly 15.000 Hz in every damping condition. Diagnose the precision of the claim.

Try this before viewing the solution

Supported conclusion

View solution step by step

  1. State the resonance condition

    Method

    The largest response occurs when f_d is close to f₀.

    Reason

    Near this condition, the driver transfers energy efficiently each cycle.

    Working

    f_d ≈ f₀ ≈ 15 Hz

  2. Apply the damping caveat

    Method

    Exact equality is not guaranteed.

    Reason

    Damping can shift the amplitude maximum slightly below the undamped natural frequency.

    Working

    Use “approximately 15 Hz” unless a model specifies otherwise.

Examiner practice 4

Interpreting “broader peak” (band of frequencies)

3 marks

Examination question

System A is lightly damped and system B is more heavily damped. Identify which responds over a wider band of driving frequencies, and compare its peak amplitude and curve width. [3 marks]

Try this before viewing the solution

View solution step by step

  1. Identify the wider response

    1 mark

    Method

    System B responds over a wider frequency band.

    Reason

    Greater damping broadens the resonance response.

    Working

    Wider band: system B.

  2. Compare peak height

    1 mark

    Method

    System B has the lower maximum amplitude.

    Reason

    More input energy is dissipated, limiting the resonant response.

    Working

    Greater damping ⇒ lower peak.

  3. Compare curve width

    1 mark

    Method

    System B’s curve is broader and less sharp.

    Reason

    The response is less frequency-selective.

    Working

    Greater damping ⇒ broader peak.

Challenge 5

Reducing dangerous resonance

Minimal support

Independent transfer

A bridge develops large wind-driven oscillations near a natural frequency. Propose two distinct engineering controls and explain how each reduces the response.

Try this before viewing the solution

Hints

Hint 1: change dissipation or detune
One approach changes the height of the response curve; another changes where the natural frequency lies.
View solution step by step

  1. Increase damping

    Method

    Add suitable dampers or another energy-dissipation mechanism.

    Reason

    Greater damping lowers the resonance peak and limits steady amplitude.

    Working

    Control 1: increase damping.

  2. Shift the natural frequency

    Method

    Change stiffness or mass distribution to move f₀ away from the dominant driving frequency.

    Reason

    Detuning prevents the wind forcing from remaining close to the response maximum.

    Working

    Control 2: change structural f₀.

7. Mind Stretchers

Mind stretcher 1: Why “sharpness” matters (selectivity)Extension

Explain why light damping can be useful for tuning (e.g. selecting one frequency) but risky for structures.

Show Answer

Light damping gives a sharp response peak: the system responds strongly to a narrow range of frequencies (useful for tuning/selectivity). But it also means if the driving frequency matches the natural frequency, the amplitude can become very large, which can be dangerous in structures.

Mind stretcher 2: Where does the energy go at steady state?Extension

At resonance, the amplitude reaches a steady value (it stops increasing). Explain what must be true about power input and power dissipated, and where the dissipated energy goes.

Show Answer

At steady state, the average power input from the driving force equals the average power dissipated by damping.

The dissipated energy is transferred mainly to internal energy (heating) of the oscillator and the surroundings, and sometimes to sound.

Mind stretcher 3: Optional (Enrichment)Extension

A. Videos (optional intuition)

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027