Oscillations, waves and sound

Anything with mass and a restoring force can vibrate: a car on its springs, a building swaying in the wind, a pendulum, a guitar string, the air in a pipe, the quartz crystal in a watch. Vibrations matter to engineers for two opposite reasons. They are useful, in clocks, sensors, loudspeakers, ultrasound and communication; and they can be destructive, shaking machines apart, fatiguing structures and annoying or harming people. When vibrations travel, they become waves, which carry energy without carrying matter, as sound, seismic waves and the ripples on a vibrating string. This lesson develops simple harmonic motion, its energy, damping and resonance, then the properties of mechanical waves, standing waves and harmonics, sound levels in decibels and the Doppler effect. MATH-201 treats the differential equations of vibration in more depth.

Simple harmonic motion

A mass m on a spring of stiffness k, displaced and released, experiences a restoring force −kx (Lesson 4). Newton's second law gives \(m\ddot{x} = -kx\), whose solution is a sinusoidal oscillation:

The amplitude A and phase φ depend on how the motion starts; the angular frequency ω depends only on the system. This is simple harmonic motion (SHM). The velocity \(-A\omega\sin(\omega t + \phi)\) is greatest as the mass passes through equilibrium, and the acceleration \(-\omega^2x\) is greatest at the extremes. Stiffer systems vibrate faster, heavier ones slower.

Other systems behave the same way for small motions. A simple pendulum of length L has \(\omega = \sqrt{g/L}\), independent of the mass and, for small swings, of the amplitude, which is why pendulums kept time for three centuries. (For larger swings the period grows, as MATH-201 shows.) A pendulum with a 1 s swing each way (period 2 s) needs \(L = g/\pi^2 = 0.994\) m, the length of a grandfather clock's pendulum. Torsional pendulums, floating ships rolling in waves, and the air in a bottle when you blow across it are further examples.

Energy in SHM

With no friction, mechanical energy is conserved, and it moves back and forth between kinetic and elastic potential energy (Lesson 6):

At the extremes, all the energy is potential; at the centre, all kinetic, so the maximum speed is \(v_{max} = A\omega\). The energy is proportional to the square of the amplitude, a result that carries over to waves.

Damping and resonance

Real oscillations die away because friction, air resistance and internal losses remove energy: they are damped. With light damping, the system oscillates at nearly its natural frequency while the amplitude decays; with heavy damping, it returns slowly without oscillating; critical damping, the boundary between them, returns to rest fastest without overshoot, which is what door closers, vehicle shock absorbers (approximately) and measuring instruments aim for.

When a system is pushed by a periodic force, it eventually vibrates at the forcing frequency. If that frequency is close to the natural frequency, the response can be very large: resonance. With light damping, the amplitude at resonance can be tens of times the static deflection the same force would cause. Resonance is useful in radio tuning, musical instruments and ultrasonic cleaners, and dangerous in structures and machines. Rotating machinery is balanced and mounted so that its running speed is far from the natural frequencies of its supports; tall buildings and slender bridges are checked for wind and footfall excitation.

London's Millennium Bridge is a well-known example: on its opening day in 2000, crowds made it sway sideways, because the bridge's lateral natural frequency, about 1 Hz, matched the sideways sway of walking, and as the bridge moved, people unconsciously fell into step with it, adding to the force. The bridge was closed and fitted with dampers. The famous collapse of the Tacoma Narrows Bridge in 1940, by contrast, was not simple resonance but an aerodynamic instability (flutter) in which the wind fed energy into the twisting motion, a self-excited oscillation of the kind described in MATH-201.

Mechanical waves

A wave is a travelling disturbance that carries energy through a medium without carrying the medium along. In a transverse wave, such as a wave on a string, the medium moves perpendicular to the direction of travel; in a longitudinal wave, such as sound, it moves back and forth along it. Seismic waves include both: the faster P waves are longitudinal and the slower S waves transverse, and the delay between their arrivals tells seismologists how far away an earthquake was.

A sinusoidal wave has a wavelength λ (the distance between crests), a frequency f (crests passing per second) and a speed

The speed is set by the medium, not by the source. On a string with tension T and mass per unit length μ, \(v = \sqrt{T/\mu}\). In a fluid, the speed of sound is \(v = \sqrt{B/\rho}\), with B the bulk modulus (Lesson 9): about 343 m/s in air at 20 °C, 1480 m/s in water. In a solid bar it is \(\sqrt{E/\rho}\), about 5000 m/s in steel, which is why ultrasonic testing can locate a flaw from the timing of its echo.

Superposition and standing waves

When two waves meet, the displacements add (the principle of superposition). Waves in step reinforce each other (constructive interference); waves half a wavelength out of step cancel (destructive interference), the principle behind noise-cancelling headphones. A wave reflected from a fixed end of a string travels back and combines with the incoming wave. At certain frequencies the combination is a standing wave: a pattern that oscillates in place, with nodes that never move and antinodes of maximum motion between them.

On a string of length L fixed at both ends, standing waves fit only when a whole number of half-wavelengths fits the length, \(L = n\lambda/2\). The allowed frequencies are the harmonics:

These are exactly the natural frequencies and mode shapes found for a taut cable in MATH-201. A pipe open at both ends has the same harmonic series for sound; a pipe closed at one end has only the odd harmonics, \(f_n = nv/(4L)\) with n odd. Every structure has a set of natural frequencies and mode shapes in the same way, which is why engineers compute them before checking for resonance.

Sound intensity and decibels

The intensity of a wave is the power it carries per unit area, in W/m². Human hearing spans an enormous range, from about 10⁻¹² W/m² (the threshold of hearing) to 1 W/m² (painful), so sound is measured on a logarithmic scale, the sound level in decibels:

Every 10 dB is a tenfold increase in intensity, and 3 dB a doubling. Two equal sources together are therefore only 3 dB louder than one. For a small source radiating equally in all directions, intensity falls as \(1/r^2\), so the level drops by 6 dB for each doubling of distance. Occupational exposure limits are typically 85 dB(A) averaged over an 8-hour day in most Canadian jurisdictions and US guidance, with permitted time halving for each 3 dB increase (some US regulations use 90 dB(A) and 5 dB); HSE-241 covers hearing conservation.

The Doppler effect

When a source of sound moves toward a listener, successive wave crests are emitted closer together, so the listener hears a higher frequency; when it moves away, a lower one. For a source moving at speed \(v_s\) directly toward or away from a stationary listener, with sound speed v:

using the minus sign for approach. For an ambulance siren at 700 Hz approaching at 25 m/s, \(f' = 700 \times 343/318 = 755\) Hz, falling to \(700 \times 343/368 = 652\) Hz as it passes. Doppler shifts of reflected waves measure speed: radar speed guns, Doppler ultrasound of blood flow, and the acoustic Doppler current profilers that measure river and ocean currents for hydrometric surveys.