Waves and oscillations

Why this matters

Anything that is pulled back toward a resting position tends to oscillate: a mass on a spring, a pendulum, a building in an earthquake, a car on its suspension, the current in a tuned radio circuit. And when an oscillation spreads out through a material or through space, it becomes a wave: sound, light, radio, ocean swell, seismic waves. Engineers design to avoid unwanted vibration; physicists and chemists use waves (light, X-rays, ultrasound) to measure the world. One small equation, v = f·λ, connects most of it.

Describing an oscillation

• Amplitude A: the largest distance from the rest position.
• PeriodThe time for one full cycle of an oscillation, T = 1 / f. T (s): the time for one full cycle.
• FrequencyThe number of cycles per second, measured in hertz. f (hertz, Hz): cycles per second. f = 1 / T.
• Angular frequency ω = 2π·f (rad/s).

A swing that takes 2.0 s for a full back-and-forth has f = 0.5 Hz. Household mains alternates at 50 or 60 Hz, so one cycle lasts 20 or about 16.7 ms.

Simple harmonic motion

When the restoring force is proportional to the displacement (F = −k·x, Hooke's law for a spring), the motion is simple harmonicMotion in which the restoring force is proportional to displacement, so position follows a sine wave.: the position follows a sine wave,

x(t) = A · sin(ω·t)

For a mass m on a spring of stiffness k:

T = 2π · √(m / k)

A 0.50 kg mass on a 200 N/m spring: T = 2π × √(0.0025) = 2π × 0.050 = 0.314 s, so f ≈ 3.2 Hz. The amplitude does not appear: big or small, the oscillation takes the same time.

For a simple pendulum of length L, for small swings (under about 15°):

T = 2π · √(L / g)

A 1.00 m pendulum has T = 2π × √(1.00 / 9.81) = 2.01 s. Notice that the mass is not in the formula. You will use this in the lab lesson to measure g.

Resonance

Every oscillating system has a natural frequency. Push it periodically at that frequency and each push adds to the motion, so the amplitude can grow large: that is resonanceLarge growth in amplitude when a system is pushed periodically at its natural frequency.. You use it to pump a playground swing; tuned radio circuits use it to pick one station. Engineers must keep machinery, bridges and buildings away from resonating with wind, footsteps, engines or earthquakes, and that analysis is specialist work that is reviewed by licensed engineers.

Waves

A wave carries energy from place to place without carrying the material along with it. A cork on a pond bobs up and down as ripples pass; it does not travel with them.

• Transverse waves: the medium moves perpendicular to the wave's direction (a wave on a rope, light).
• Longitudinal waves: the medium moves back and forth along the direction of travel (sound, which is a series of compressions and rarefactions in air).

The wavelengthThe distance between two neighbouring crests of a wave, λ. λ (lambda) is the distance between two neighbouring crests. In one period T the wave moves forward one wavelength, so its speed is

v = λ / T = f · λ

The speed is set by the medium, not by the source. Sound travels at about 343 m/s in air at 20 °C, about 1,480 m/s in water and around 5,000–6,000 m/s in steel. Light in a vacuum travels at c = 3.00 × 10⁸ m/s.

Worked examples

Concert A. A tuning fork vibrates at 440 Hz in air:

λ = v / f = 343 / 440 = 0.780 m

FM radio. A station at 100 MHz (1.00 × 10⁸ Hz):

λ = c / f = 3.00 × 10⁸ / 1.00 × 10⁸ = 3.00 m

Wi-Fi. At 2.4 GHz, λ = 3.00 × 10⁸ / 2.4 × 10⁹ = 0.125 m, about 12.5 cm. Antenna sizes are often a simple fraction of the wavelength, which is why Wi-Fi antennas are a few centimetres long.

Same frequency, new medium. When sound goes from air into water the frequency stays the same (it is set by the source), but the speed rises, so the wavelength grows: 440 Hz in water gives 1480 / 440 = 3.36 m.

Superposition

When two waves overlap, their displacements add. Crest on crest gives a bigger wave (constructive interferenceThe adding of overlapping waves: crest on crest is constructive, crest on trough is destructive.); crest on trough cancels (destructive interference). Noise-cancelling headphones play an inverted copy of incoming sound to cancel it.

In code

import math

def spring_period(mass, k):
    return 2 * math.pi * math.sqrt(mass / k)

print(round(spring_period(0.50, 200), 3))   # 0.314

Common mistakes

• Mixing up period and frequency. They are reciprocals: 4 Hz means a 0.25 s period.
• Forgetting the unit prefixes. MHz is 10⁶ Hz and GHz is 10⁹ Hz.
• Thinking a higher frequency travels faster. In air, all audible sound travels at about the same speed; higher frequency just means shorter wavelength.
• Using the pendulum formula for big swings. Beyond about 15° the period grows slightly.

Quick recap

• f = 1 / T and ω = 2πf.
• Spring: T = 2π√(m/k). Pendulum (small swings): T = 2π√(L/g).
• Wave speed v = f·λ; the medium sets v, the source sets f.
• Sound in air ≈ 343 m/s; light ≈ 3.00 × 10⁸ m/s.

Try it

In the Do tab, write a wavelength function and test it on sound, radio and light.