Applications and Problems of Resonance

Resonance is not just a theoretical concept — it has profound practical consequences, both useful and dangerous. In this lesson, we examine the most important applications of resonance that you may encounter in Edexcel A-Level Physics, as well as the engineering challenges that arise when resonance is undesirable.

Desirable Resonance

Musical Instruments

Musical instruments rely on resonance to produce their characteristic sounds. When a guitar string is plucked, it vibrates at its natural frequencies (fundamental and harmonics). The body of the guitar acts as a resonator — the air cavity inside the body has natural frequencies that are driven by the vibrating string. The body resonates, amplifying the sound and giving the guitar its characteristic tone.

Without resonance, the string alone would produce a very quiet sound. The resonating body increases the amplitude of the sound waves dramatically. Different instruments have differently shaped resonating cavities, which emphasise different harmonics and produce their unique timbres.

Organ Pipes and Wind Instruments

In organ pipes, a column of air resonates at frequencies determined by the pipe's length. The musician excites the air column (by blowing or using a bellows), and resonance amplifies the standing wave at the natural frequency. Changing the effective length of the air column (by opening valves or covering holes) changes the natural frequency and hence the pitch.

Radio Tuning

A radio receiver contains an electrical circuit with a variable capacitor and inductor — an LC circuit. This circuit has a natural frequency of electrical oscillation. When you tune the radio, you adjust the capacitor to change the circuit's natural frequency until it matches the frequency of the desired radio station. The circuit resonates with that station's signal, amplifying it while signals at other frequencies produce negligible response because they are far from resonance.

The quality of tuning — how well the radio separates adjacent stations — depends on the sharpness of the resonance peak. A lightly damped circuit gives a sharp peak (selective tuning), while a heavily damped circuit gives a broad peak (poor selectivity).

MRI Scanners

Magnetic resonance imaging (MRI) is one of the most important medical applications of resonance. Hydrogen nuclei (protons) in the body precess around the direction of the applied magnetic field at a specific frequency called the Larmor frequency. A radio-frequency pulse at exactly this frequency causes the protons to resonate, absorbing energy and tipping their magnetic axes.

When the pulse ends, the protons relax back to their original alignment, emitting radio-frequency signals. The rate of relaxation differs between tissue types (fat, muscle, bone, tumour), allowing detailed images to be constructed.

Microwave Cooking

The oscillating electric field in a microwave oven is set to approximately 2.45 GHz, close to a rotational absorption frequency of water molecules. Water molecules absorb energy resonantly, increasing their rotational kinetic energy and therefore the temperature of the food.

Undesirable Resonance

Bridges

The Tacoma Narrows Bridge collapse in 1940 remains the most dramatic example of destructive resonance. Modern bridge design accounts for resonance in several ways:

Aerodynamic shaping of the deck to prevent vortex shedding
Dampers installed at key points to absorb vibrational energy
Structural analysis to ensure natural frequencies do not coincide with likely driving frequencies (wind, traffic, pedestrians)
Forcing soldiers to break step when crossing bridges, preventing periodic driving at the natural frequency

Buildings in Earthquakes

Earthquakes produce ground vibrations containing a range of frequencies. If any of these frequencies match a building's natural frequency of lateral oscillation, resonance can cause the building to sway violently. Taller buildings have lower natural frequencies and are more vulnerable to the low-frequency components of earthquake vibrations.

Earthquake-resistant design strategies:

Tuned mass dampers: A large mass (sometimes hundreds of tonnes) suspended on springs or pendulums near the top of the building. Designed to oscillate out of phase with the building, absorbing and dissipating energy.
Base isolation: Flexible bearings between the building and its foundations that absorb ground motion.
Cross-bracing and shear walls: Structural elements that increase stiffness and change the natural frequency.
Active damping systems: Computer-controlled actuators that apply corrective forces in real time.

Taipei 101, a 508-metre skyscraper in Taiwan, has a 730-tonne tuned mass damper suspended between the 87th and 92nd floors. During typhoons and earthquakes, the damper swings opposite to the building, reducing sway amplitude by up to 40%.

Machinery Vibration

Rotating machinery (engines, turbines, washing machines) can experience resonance when the rotational frequency matches a natural frequency of the structure. This can cause excessive vibration, noise, fatigue failure, and bearing damage.

Solutions include:

Designing the system so that operating frequencies are well away from natural frequencies
Adding damping (rubber mounts, viscous dampers)
Balancing rotating components carefully
Using vibration absorbers (additional masses tuned to the problem frequency)

Aircraft Structures

Aircraft wings and control surfaces have natural frequencies of vibration. Aerodynamic forces during flight can drive these at certain speeds, potentially causing flutter — a self-reinforcing oscillation that can destroy the wing. Aircraft are carefully tested to ensure that flutter speeds are well above the maximum operating speed, and mass balances are added to control surfaces to shift natural frequencies away from dangerous values.

Resonance Decision Tree for Exam Questions

flowchart TD
    A["Is resonance occurring?"] -->|"f_d = f₀ and\namplitude is large"| B["Yes — Resonance"]
    A -->|"f_d ≠ f₀ or\namplitude is small"| C["No — Off-resonance\nforced oscillation"]
    B --> D["Is it desirable?"]
    D -->|"Yes"| E["Musical instruments\nRadio tuning\nMRI scanning\nMicrowave cooking"]
    D -->|"No"| F["How to reduce it?"]
    F --> G["Add damping\n(reduces peak amplitude)"]
    F --> H["Change natural frequency\n(shift f₀ away from f_d)"]
    F --> I["Remove periodic driver\n(break step on bridge)"]
    F --> J["Add a tuned mass damper\n(absorbs energy at f₀)"]

Damping as the Universal Solution

The common thread in managing unwanted resonance is damping. By adding damping to a system:

The maximum amplitude at resonance is reduced
The resonance peak is broadened, making the system less sensitive to a specific driving frequency
Energy is dissipated before it can accumulate to dangerous levels

The trade-off is that damping also reduces the system's response when resonance is desired (e.g., in musical instruments or radio circuits). Engineering design always involves balancing the benefits and drawbacks of damping for the specific application.

Worked Example 1: Washing Machine Resonance

A washing machine drum has a mass of 15 kg and is mounted on springs with a combined spring constant of 6000 N m⁻¹. Calculate the spin speed (in rpm) at which resonance would occur.

Solution:

f₀ = (1/2π)√(k/m) = (1/2π)√(6000/15) = (1/2π)√400 = (1/2π) × 20 = 20/(2π) ≈ 3.18 Hz

In rpm: 3.18 × 60 ≈ 191 rpm

The washing machine should be designed so that the operating spin speed (typically 1000–1400 rpm) is well above this resonant frequency. The machine passes through resonance briefly during spin-up, which is why washing machines often vibrate noticeably at low spin speeds.

Worked Example 2: Building Natural Frequency

A 50-metre tall building has a natural frequency of lateral oscillation of 0.5 Hz. An earthquake produces ground vibrations with significant energy between 0.1 Hz and 5 Hz. Is resonance likely?

Solution:

The building's natural frequency (0.5 Hz) falls within the earthquake's frequency range (0.1–5 Hz), so resonance is possible and the building is at risk. Taller buildings have lower natural frequencies, which overlap more with the low-frequency content of earthquake vibrations.

If a tuned mass damper were added, it should be designed with a natural frequency close to 0.5 Hz to absorb energy at the building's resonant frequency.

Summary Table

Application	Resonance Type	How It Works	Key Physics
Musical instruments	Desirable	Sound box resonates with vibrating string/air	f₀ of cavity matches harmonic frequencies
Radio tuning	Desirable	LC circuit tuned to station frequency	Variable capacitor adjusts f₀
MRI	Desirable	RF pulse at Larmor frequency of protons	Precise frequency matching for resonance
Microwave oven	Desirable	EM waves at water molecule rotation frequency	2.45 GHz matches molecular resonance
Bridge destruction	Undesirable	Wind vortices at bridge natural frequency	Periodic driving force at f₀
Earthquake damage	Undesirable	Ground vibrations at building natural frequency	Low-frequency seismic waves match tall buildings
Machine vibration	Undesirable	Rotation frequency matches structural natural frequency	RPM ÷ 60 = f_d, compare with f₀
Aircraft flutter	Undesirable	Aerodynamic forces drive wing at natural frequency	Speed-dependent driving frequency

Real-World Application: The Millennium Bridge, London

When the Millennium Bridge opened in June 2000, it began swaying alarmingly as pedestrians crossed it. The problem was a form of resonance: pedestrians naturally walk at about 2 Hz (vertical) and 1 Hz (lateral). The bridge's lateral natural frequency was close to 1 Hz. As the bridge swayed slightly, pedestrians unconsciously synchronised their steps to maintain balance, which increased the driving force and amplified the oscillation — a positive feedback loop.

The solution was to add 37 fluid viscous dampers and 52 tuned mass dampers, increasing the damping to reduce the resonance peak. The bridge reopened in February 2002 and has been stable since. This case demonstrates that even modern engineering can be caught out by resonance in unexpected ways.

A-Level Deep Dive: Applications and Problems of Resonance

Spec mapping

Edexcel 9PH0 specification Topic 13 — Oscillations treats forced oscillations and resonance as the climax of the topic, requiring candidates to recognise the characteristic resonance curve, describe how damping affects the height and width of the peak, and apply these ideas to engineered and natural systems (refer to the official specification document for exact wording). Resonance is examined in Paper 2 (Practical Skills in Physics II) and contributes to the synoptic content tested in Paper 3. Although the topic begins with simple harmonic motion, the resonance sub-strand demands the highest analytical depth: candidates must distinguish between driven and natural frequency, evaluate the role of damping, and connect the mathematical condition $f_d \approx f_0$ to design decisions made by engineers, musicians and clinicians. The Edexcel data and formulae booklet provides $T = 2\pi\sqrt{m/k}$ and $T = 2\pi\sqrt{l/g}$ but does not give a closed-form expression for the resonance amplitude — this must be reasoned about qualitatively from the resonance curve.

Worked example with full mark scheme

Question (8 marks):

A car of mass $1200 \text{ kg}$ is supported on four identical springs. When fully loaded with passengers (total $1500 \text{ kg}$ ), the body sits $3.0 \text{ cm}$ lower than when empty.

(a) Show that the spring constant of each spring is approximately $2.5 \times 10^4 \text{ N m}^{-1}$ . (3)

(b) Estimate the natural frequency of vertical oscillation for the empty car. (2)

(c) Explain, with reference to resonance and damping, why critical (or near-critical) damping is preferred over light damping in a vehicle suspension system. (3)

Solution with mark scheme:

(a) The extra weight of $300 \text{ kg}$ produces an extra force $F = mg = 300 \times 9.81 = 2943 \text{ N}$ shared between four springs, so each spring carries an additional $735.75 \text{ N}$ .

M1 — recognising that the four springs share the load equally and computing the per-spring force.

Using $F = kx$ with $x = 0.030 \text{ m}$ :

$k = \dfrac{735.75}{0.030} \approx 2.45 \times 10^4 \text{ N m}^{-1}$

M1 — correct rearrangement and substitution.

A1 — answer to two significant figures $\approx 2.5 \times 10^4 \text{ N m}^{-1}$ .

(b) The four springs in parallel give an effective stiffness $k_\text{eff} = 4k \approx 1.0 \times 10^5 \text{ N m}^{-1}$ . With the empty mass $m = 1200 \text{ kg}$ :

$f_0 = \dfrac{1}{2\pi}\sqrt{\dfrac{k_\text{eff}}{m}} = \dfrac{1}{2\pi}\sqrt{\dfrac{1.0 \times 10^5}{1200}} \approx 1.5 \text{ Hz}$

M1 — combining the spring constants in parallel.

A1 — correct natural frequency to two significant figures.

(c) B1 — light damping produces a sharp, tall resonance peak: small road inputs near $1.5 \text{ Hz}$ would generate large oscillations, making the ride uncomfortable and the car difficult to control.

B1 — critical damping returns the car to equilibrium in the shortest time without overshoot, so a single bump produces one short response rather than a sustained oscillation.

B1 — over-damping would make the car feel "wooden" because it could not respond quickly to road texture; near-critical damping is the engineering compromise that minimises both the resonance peak and the response time.

Total: 8 marks (M3 A2 B3, split as shown).

The mark scheme rewards candidates who explicitly link the resonance curve shape to a design decision — this is the AO2/AO3 step that distinguishes a complete answer.

Specimen question modelled on the Edexcel 9PH0 Paper 2 format

Question (6 marks): A pedestrian footbridge has a measured lateral natural frequency of $0.95 \text{ Hz}$ . Engineers are concerned about pedestrian-induced lateral resonance.

(a) State the condition under which resonance will occur as pedestrians cross the bridge. (1)

(b) Sketch a graph of bridge oscillation amplitude against pedestrian step frequency for two cases: light damping and heavy damping. Label both curves clearly. (3)

(c) Suggest, with reasoning, two engineering modifications that could reduce the resonance amplitude without altering the bridge's natural frequency. (2)

Mark scheme decomposition by AO:

(a)

B1 (AO1.1) — driving frequency (pedestrian lateral step rate, typically near $1 \text{ Hz}$ ) approximately equals the natural frequency of the bridge.

(b)

B1 (AO1.2) — both curves peaked near $f_0 = 0.95 \text{ Hz}$ , returning toward zero at low and high frequencies.
B1 (AO2.1) — light-damping curve drawn taller and narrower (sharper peak).
B1 (AO2.1) — heavy-damping curve drawn lower and broader, with peak shifted slightly to lower frequency.

(c)

B1 (AO3.1) — fitting tuned mass dampers, which oscillate out of phase with the bridge and absorb energy at the resonant frequency.
B1 (AO3.1) — adding viscous dampers between bridge sections to dissipate kinetic energy as heat, broadening and lowering the peak.

Total: 6 marks split AO1 = 2, AO2 = 2, AO3 = 2. The AO3 marks in (c) reward candidates who name a specific damping mechanism rather than vague "add weight" suggestions.

Synoptic links

Connects to:

Topic 13 — Forced oscillations and SHM: the resonance peak is only meaningful in the driven regime. The same equation of motion $m\ddot{x} + b\dot{x} + kx = F_0\cos(\omega t)$ produces SHM in the limit $b \to 0$ , $F_0 \to 0$ , and produces resonance when $\omega \to \omega_0$ .
Topic 8 — Magnetic resonance imaging (MRI/NMR): protons in a strong magnetic field precess at the Larmor frequency; an applied RF pulse at exactly that frequency tips the spins. This is mechanical resonance in a quantum-mechanical guise — the matching condition $f_\text{RF} = f_\text{Larmor}$ is identical in form to $f_d = f_0$ .
Topic 6 — AC circuits with inductance and capacitance: an LC circuit driven by an AC source has natural angular frequency $\omega_0 = 1/\sqrt{LC}$ . Tuning a radio is electrical resonance: rotating the variable capacitor adjusts $f_0$ to match the broadcast carrier frequency. The resistance $R$ plays the role of damping.