Topic 3: C. Wave behaviour

Waves carry energy and information through a medium or space without permanently moving the matter itself. This theme builds from a single oscillating particle in simple harmonic motion up to the rich behaviours of waves: how they bend, overlap, and shift in frequency. Mastering the core relationships here unlocks optics, sound, and much of modern communication.

Simple harmonic motion

Simple harmonic motion (SHM) describes any oscillation where the restoring force on an object is directly proportional to its displacement from equilibrium and always points back toward that equilibrium. A mass on a spring and a swinging pendulum (for small angles) are the classic examples. Because the force grows with displacement, the object accelerates most at the extremes and moves fastest as it passes through the centre, where the net force is zero. The motion repeats with a fixed period regardless of amplitude, a property called isochronism. SHM matters for waves because every particle in a wave executes this same back-and-forth motion; the wave is simply the pattern that emerges when many such oscillators are coupled together and slightly out of step with one another.

The wave model and wave characteristics

A wave is a travelling disturbance that transfers energy from one place to another. Several quantities describe it. The amplitude is the maximum displacement from equilibrium and relates to the energy carried. The wavelength, given the symbol lambda, is the distance between two consecutive points that are in phase, such as crest to crest. The period T is the time for one complete oscillation, and the frequency f is the number of oscillations per second, measured in Hz, with f equal to 1 divided by T. Two points are in phase when they move in step, and out of phase when their motions differ. The wavefront is an imaginary line joining points that are all in phase, and a ray is an arrow drawn perpendicular to the wavefront showing the direction of travel.

The wave equation (worked example)

The speed of a wave links its frequency and wavelength through the wave equation: v equals f times lambda. The speed is set by the medium, so if frequency rises in a given medium the wavelength must shrink to keep the product constant. Worked example: a sound wave in air has a frequency of 500 Hz and the speed of sound in air is 340 m/s. Find its wavelength. Rearranging the equation gives lambda equals v divided by f, so lambda equals 340 m/s divided by 500 Hz, which is 0.68 m. As a check, multiplying back gives 500 Hz times 0.68 m equals 340 m/s, confirming the answer. Always make sure the speed value you use belongs to the same medium the wave is travelling in.

Transverse versus longitudinal waves

Waves are grouped by the direction in which the medium oscillates relative to the direction the wave travels. In a transverse wave the oscillations are perpendicular to the direction of energy transfer; ripples on water and all electromagnetic waves are transverse, and these can be polarised because the oscillation has a definite orientation. In a longitudinal wave the oscillations are parallel to the direction of travel, producing regions of compression where particles bunch together and rarefaction where they spread apart; sound in air is the standard example. A useful contrast: transverse waves have crests and troughs, while longitudinal waves have compressions and rarefactions, but both still obey the same wave equation and carry energy without net transport of matter.

Reflection and refraction (Snell's law)

When a wave meets a boundary it can be reflected, transmitted, or both. In reflection the angle of incidence equals the angle of reflection, with both angles measured from the normal, the line drawn perpendicular to the surface. Refraction is the bending that occurs when a wave passes into a new medium and changes speed; its frequency stays the same but its wavelength and speed change. Snell's law states that n1 times sin(theta1) equals n2 times sin(theta2), where n is the refractive index of each medium and theta is the angle from the normal. A wave slowing down, for instance light entering glass, bends toward the normal, while a wave speeding up bends away from it. If a wave moving into a faster medium meets the boundary beyond the critical angle, total internal reflection occurs and no light escapes.

Diffraction

Diffraction is the spreading of waves as they pass through a gap or around an obstacle. The effect is most pronounced when the size of the gap is comparable to the wavelength: a narrow slit causes the emerging wavefronts to spread out in nearly circular arcs, while a gap much wider than the wavelength lets the wave pass with little spreading. This is why sound, with its relatively long wavelengths, bends readily around doorways and corners so you can hear someone in the next room, whereas light, with its tiny wavelength, casts comparatively sharp shadows. Diffraction confirms the wave nature of light and underlies the behaviour seen at single slits, double slits, and diffraction gratings.

Interference and the double-slit experiment

When two waves overlap, their displacements add together, a process called superposition. Where crests meet crests the waves reinforce, producing constructive interference and a larger amplitude; where a crest meets a trough they cancel, producing destructive interference. Stable patterns require coherent sources, meaning a constant phase relationship and the same frequency. In the classic double-slit experiment, light passing through two narrow slits produces a series of bright and dark fringes on a screen. A bright fringe forms where the path difference from the two slits is a whole number of wavelengths, and a dark fringe forms where it is an odd number of half-wavelengths. The fringe spacing s is given by s equals lambda times D divided by d, where D is the slit-to-screen distance and d is the slit separation.

Standing waves and resonance

A standing wave forms when two waves of equal frequency and amplitude travel in opposite directions and superpose, typically when a wave reflects back on itself. Unlike a travelling wave, a standing wave does not transport energy along its length. It has nodes, points that never move, and antinodes, points of maximum oscillation, fixed in place. On a string fixed at both ends, only certain wavelengths fit, giving a fundamental frequency and a series of higher harmonics. Resonance occurs when a system is driven at one of its natural frequencies, causing the amplitude to grow dramatically; this is how musical instruments produce loud, sustained notes and why matching driving frequencies to natural ones must be managed carefully in engineering.

The Doppler effect

The Doppler effect is the change in observed frequency that occurs when a wave source and an observer move relative to each other. When the source approaches, the wavefronts bunch up ahead of it, so the observer measures a higher frequency and shorter wavelength; when the source recedes, the wavefronts stretch out and the observed frequency drops. The pitch change you hear as an ambulance siren passes is the everyday example. The same principle applies to light: galaxies moving away from us show a redshift toward longer wavelengths, evidence used to study the expanding universe, and police radar measures vehicle speed from the frequency shift of reflected waves.