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Vibrations and Waves

15 min read

Simple harmonic motion

A body moves with simple harmonic motion (SHM) if its acceleration is proportional to its displacement from a fixed point and always directed towards that point. Examples are a mass on a spring and a simple pendulum swinging through a small angle.

A is the amplitude, ω the angular frequency. The speed is greatest at the centre and zero at the ends.
Graphs of simple harmonic motionThree graphs against time for a mass on a spring. The displacement is a cosine curve. The velocity is a negative sine curve a quarter of a cycle ahead of the displacement, and the acceleration is the negative of the displacement curve. The velocity is greatest when the displacement is zero, and the acceleration is greatest at the ends of the motion.xvatime
The period of a spring depends on the mass and the spring constant; the period of a pendulum on its length and g, not on the mass or (for small angles) the amplitude.
Try itPeriod of a mass on a spring and of a pendulum

Choose the mass and spring constant, and the length of a pendulum. Find each period.

Open the simple pendulum lab
Measure the period for different lengths.
Energy changes between kinetic and elastic potential energy.
The total energy is constant and proportional to the square of the amplitude.

Damping and resonance

In damped oscillations friction removes energy, so the amplitude falls. Light damping lets the oscillation continue for many cycles; critical damping returns the system to rest in the shortest time (car shock absorbers, door closers); heavy damping returns slowly. In forced oscillations, a periodic force drives the system. When the driving frequency equals the natural frequency there is resonance: the amplitude is very large. Resonance is useful in radio tuning and musical instruments but dangerous for bridges and buildings.

Superposition, interference and standing waves

The principle of superposition: when two waves meet, the resultant displacement is the sum of the individual displacements. Waves with the same frequency and constant phase difference are coherent. Constructive interference happens when the path difference is a whole number of wavelengths (nλ); destructive interference when it is an odd number of half wavelengths ((n + ½)λ).

Standing waves on a stringA string fixed at both ends can vibrate in patterns called harmonics. The fundamental has one loop and the wavelength is twice the string length. The second harmonic has two loops and the third has three. Points that never move are nodes, and points of largest movement are antinodes.fundamental: L = λ/22nd harmonic: L = λ3rd harmonic: L = 3λ/2nodeantinode (largest movement)

A standing (stationary) wave is made by two waves of the same frequency travelling in opposite directions. It has nodes (zero amplitude) and antinodes (maximum amplitude), half a wavelength apart. A string fixed at both ends has harmonic frequencies f = nv ÷ 2L.

Young’s double-slit experimentLight of one colour passes through two narrow slits very close together. The waves from the slits overlap and interfere. Where crests meet crests the waves reinforce and there is a bright fringe, and where crests meet troughs they cancel and there is a dark fringe. The fringe spacing is wavelength times the screen distance divided by the slit separation.screentwo slits (separation s)Dbrightdarkfringe spacing = λD ÷ s
Young’s double slits: x = fringe spacing, D = distance to the screen, s = slit separation. For a diffraction grating with spacing d, the maximum of order n appears at angle θ.
Try itYoung's fringes

Choose the wavelength of the light, the slit separation and the distance to the screen. Find the fringe spacing.

Diffraction and the Doppler effect

Diffraction is the spreading of waves at an edge or gap. A single slit of width a gives a central maximum of angular half-width θ ≈ λ ÷ a. The Doppler effect is the change in the observed frequency when the source and observer move relative to each other. For a source moving towards a stationary observer, f′ = f v ÷ (v − v_s): the pitch of an approaching ambulance siren is higher and drops as it passes. The Doppler effect of light shows that distant galaxies are moving away.

Check yourself In SHM the acceleration is greatest…
Check yourself The period of a pendulum is 2 s. If the length is made four times longer, the period becomes…
Check yourself Two waves meet with a path difference of λ/2. The interference is…

Key terms

Simple harmonic motion
Motion in which acceleration is proportional to displacement and directed towards the equilibrium point.
Resonance
A large amplitude oscillation when the driving frequency equals the natural frequency.
Coherent waves
Waves with a constant phase difference.
Doppler effect
The change in observed frequency due to relative motion.

Summary

  • SHM: a = −ω²x; T = 2π√(m/k) for a spring and T = 2π√(L/g) for a pendulum.
  • Damping removes energy; resonance gives large amplitudes.
  • Superposition gives interference and standing waves; Young’s fringes x = λD/s.
  • Diffraction and the Doppler effect are wave behaviours.
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