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Theme C — Wave Behaviour

C.4.5 The Nature of Resonance — IB Physics HL Revision Notes & Topical Questions

5 min read 9 parts Lesson 31 of 37 1 quiz

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Curriculum Equation Key قوانين الرقم الهيدروجيني والاتزان الأيوني - pH & Acid-Base Equilibrium FormulasFast Exam Solution
pH = -log[H⁺] | pOH = -log[OH⁻] | pH + pOH = 14 | [H⁺][OH⁻] = 10⁻¹⁴
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Lesson summary

C.4.5 The Nature of Resonance — IB PHYSICS SL notes: Explain free and forced oscillations and resonance as the large-amplitude response of a system driven at its natural frequency.Free oscillation — an oscillation with only internal forces acting and no energy inputA system is di…

  • Natural frequency (f0)
  • Driving frequency
  • Swing example
  • Far below f0
  • Far above f0
  • Radio tuning

Part 1 of 7Free oscillations

Free oscillation
  • Free oscillation an oscillation with only internal forces acting and no energy input
  • A system is displaced and then left to oscillate.
  • Strictly it happens only in a vacuum, but vibration in air with no external force counts as free.
Frequency
  • Frequency a free vibration occurs at the natural frequency of the system
Natural frequency (f0)
  • Natural frequency (f0) the frequency of oscillation when the system is allowed to oscillate freely
  • A pendulum, a mass on a spring, a string and an air column each have one or more natural frequencies.
Examples
  • Examples a struck tuning fork is a free vibration

Part 2 of 7Forced oscillations

Forced oscillation
  • Forced oscillation an oscillation produced by a periodic external force
Purpose
  • Purpose a periodic force must be applied to replace the energy lost to damping
  • It does work against resistive forces such as air resistance.
Driving frequency
  • Driving frequency the system is made to oscillate at the frequency of the driver, whatever its own natural frequency, once any transient has died away
Examples
  • Examples glass shattered by a high-pitched sound, a car interior vibrating at speed and a clarinet air column driven by the reed

Part 3 of 7Resonance

1Resonance
  • Resonance when the driving frequency equals the natural frequency the amplitude is at its maximum
2Approach
  • Approach as the driving frequency nears f0, the system gains more energy from the driver
3Cause
  • Cause each push arrives in step with the velocity, so the driver does positive work on the system in every cycle
  • The stored energy builds up until it is balanced by the energy lost per cycle.
  • Energy is transferred most efficiently at resonance.
4Phase
  • Phase at resonance the driver is a quarter of a cycle ahead of the displacement, so the force always acts in the direction of the velocity
5Swing example
  • Swing example pushes timed once per swing give large oscillations
  • Pushes slightly above or below f0 still raise the amplitude, but less.
  • Pushes at the wrong rate hardly help or even hinder.

Part 4 of 7Amplitude–frequency graph

Peak
  • Peak the amplitude of response against driving frequency peaks at or very near f0
Far below f0
  • Far below f0 the amplitude is small and nearly constant and the system follows the driver in phase
Far above f0
  • Far above f0 the amplitude is small and the displacement is nearly opposite in phase to the force
Damping
  • Damping less damping makes the peak taller and narrower, and more damping lowers and broadens it

Part 5 of 7Useful resonance

1Radio tuning
  • Radio tuning an LC circuit is adjusted so its natural frequency matches the station
  • The resonant frequency of an LC circuit is f = 1/(2π√(LC)).
2Musical instruments
  • Musical instruments strings and air columns resonate and the body amplifies the sound
3Microwave ovens
  • Microwave ovens the frequency is absorbed strongly by water molecules
4Magnetic resonance imaging
  • Magnetic resonance imaging it uses the resonant response of hydrogen nuclei in a magnetic field
5The ear
  • The ear different regions of the basilar membrane have different natural frequencies, so each pitch makes a different place vibrate most
6Standing waves
  • Standing waves the harmonics of a string or pipe are its natural frequencies

Part 6 of 7Harmful resonance

Harmful resonance
Structures
  • Structures buildings and bridges can be shaken by earthquakes or wind at a natural frequency
  • A marching crowd can excite a bridge.
Glass
  • Glass a wine glass shatters when sound matches its natural frequency
Vehicles
  • Vehicles a panel can rattle at a particular engine speed
Prevention
  • Prevention change the stiffness or mass, or add dampers

Part 7 of 7More from the specification

More from the specification
Resonant amplitude scale
  • Resonant amplitude scale driving mechanical or acoustic systems at natural frequencies generates dramatic oscillations visible on a large observational scale.
Standing Wave Formation
  • Created when two progressive travelling waves of identical frequency and amplitude travelling in opposite directions superpose after reflection at a boundary.
Nodes and Antinodes in Pipes and Strings
  • Standing waves feature fixed displacement nodes (zero amplitude) and antinodes; adjacent nodes are separated by a distance equal to half the wavelength (λ/2).
    • (maximum amplitude)
Harmonics in Strings and Acoustic Pipes
  • A string fixed at both ends or a pipe open at both ends supports resonant harmonics whose wavelengths satisfy λₙ = 2L/n,
  • where n is an integer.
Resonance Curves and Damped Systems
  • On a frequency-response graph,
  • increasing damping broadens the resonance peak, lowers maximum amplitude,
  • and slightly shifts peak frequency away from natural frequency.
Heavy damping
  • Damping greater than critical, giving a slow return without oscillation
Antinode
  • Point of maximum amplitude on a stationary wave.
Boundary
  • Surface between two media at which waves are partially reflected
Wavelength
  • Distance between a point on one wave and the same point on the next wave
Equilibrium
  • State with zero net force.
Length
  • Resistance is directly proportional to the length of a wire.
Boundary condition
  • Constraint at an edge fixing whether there is a node or antinode.
Equilibrium position
  • Position where the net force is zero.
Progressive wave
  • Wave that transfers energy without transferring matter.

Key terms

TermMeaning
Natural frequencyFrequency of free oscillation of a system.
Driving frequencyFrequency of an applied periodic force.
ResonanceLarge-amplitude response when the driving frequency equals the natural frequency.
Free oscillationOscillation with only internal forces and no energy input.
Forced oscillationOscillation under a periodic external force.

Avoid these mistakes

1
Saying the system oscillates at its natural frequency under forced driving — it oscillates at the driving frequency.
2
Claiming that a large force alone causes resonance — the frequency match is the key.
3
Thinking resonance happens only in mechanical systems.
4
Forgetting that real systems are damped — infinite amplitude occurs only in theory.
5
Assuming the peak is always exactly at f_{0} — with heavy damping it shifts slightly lower.

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