Introduction to Waves

Eighty thousand people are sitting in the stands at Eden Gardens during a day-night India-Australia Test. A moment passes. Suddenly the crowd on one side leaps to its feet, arms up, then sits down; a heartbeat later the next section leaps; then the next. A visible band of motion ripples across the stadium, travels the full circumference in about twenty seconds, and arrives back where it started. Nobody moved from their seat. A pattern of standing-up-and-sitting-down moved around the stadium at roughly 12 metres per second.

That pattern is a wave. It carried something real — energy (the metabolic work each person did standing up and sitting down), timing information, a visual signal readable from the opposite stand. But it did not carry people. Each spectator returned to their own seat the instant their turn was over. The wave was made of their motion, not their position.

Now shift the scene. Walk to the Juhu Beach chowpatty late in the evening and watch a wave crest roll in from the Arabian Sea. Throw a chunk of driftwood twenty metres offshore. Does the wood ride the wave all the way in? No. It bobs up and down as the crest passes, and it drifts toward the beach only very slowly — mostly because of the tide and the wind, not because the wave "carried" it. The water molecules beneath the crest trace small elliptical loops and then return to nearly where they started. The shape of the water surface moved; the water did not.

Pluck the middle of a sitar string at a baithak in Pune. A vibration runs out from your finger toward each end of the string, reflects off the bridge, and comes back. Something has travelled along the string at a measurable speed. But if you stain a tiny dot on the string with a pen, the dot stays exactly where you put it on the string. It oscillates up and down; it does not slide sideways. The dot never moves along the string. Only the disturbance does.

These three scenes — the stadium wave, the sea wave, the string wave — have the same underlying structure. They are waves: travelling disturbances that carry energy without carrying matter. This article works out what that sentence means, classifies the main kinds of waves, and names the four numbers that describe any periodic wave.

You will need the language of simple harmonic motion from the previous part of this syllabus — the same mathematics that describes a pendulum also describes each individual particle of a wave medium. A wave is, in a very literal sense, what you get when every point in a long chain of SHM oscillators is set up to oscillate with a phase that shifts smoothly as you move along the chain.

What a wave is — and what it is not

Start from the cleanest example: a pulse on a rope. Hold one end of a long rope (say a 5-metre length of clothesline used for drying sarees on a Mumbai terrace); the other end is tied to a post. Give your end a sharp, single flick — up and then back down. A hump appears near your hand. A fraction of a second later the hump is halfway along the rope. Half a second after that, it has reached the post and bounced back.

What exactly is travelling?

It is not the rope. If you mark a point on the rope with chalk, that point stays in place along the rope's length — it moves upward as the pulse arrives, and downward as the pulse passes, but it does not translate horizontally. The rope is the medium. It does not travel.

It is not energy in any one piece of rope. Each piece of rope receives kinetic energy as the pulse arrives, holds it for the duration of the pulse, and then returns to rest. The energy moves through the rope.

What travels is the pattern of displacement — the shape of the hump. At one instant the hump is centred at x = 1 m; later, at x = 2 m; later, at x = 3 m. The pattern slides along the rope at a definite speed v, carrying energy with it. That is a wave.

The disturbance can be a single pulse (one flick of the rope), a train of pulses, or a continuous sinusoidal oscillation. All three are waves. When the disturbance is a sinusoid repeating in both time and space, the wave is called periodic or, more specifically, a sinusoidal (or harmonic) wave.

What moves and what doesn't

The defining claim — "energy travels, matter does not" — is worth staring at for a moment, because it is the one that beginners get wrong.

The stadium example makes this vivid. The people do not run around the stadium; they stand up and sit down. But watching from above, you see a band of raised arms moving at 12 m/s. The "wave" is the band; the "medium" is the seated crowd.

Now watch the actual motion of the crest of the wave you set up on the rope. If the pulse travels at v = 10 m/s, then in 0.5 s the pulse has moved 5 m. But the piece of rope at x = 5 m did not move 5 m horizontally — it moved perhaps 5 cm vertically upward, and then back down. The displacement of the medium is tiny compared to the distance the disturbance travels.

Energy does travel, though

Pick a point on the rope and watch what happens as the pulse arrives. Before the pulse: the point is at rest. During the pulse: the point accelerates upward, reaches maximum speed at the halfway point of the pulse, decelerates, stops at peak displacement, then accelerates back downward, and returns to rest. During that entire episode, it had kinetic energy and (if the rope has tension) stored some potential energy. After the pulse passes, all that energy is gone from this point — it has been handed off to the next point along the rope. The wave transfers energy from one piece of medium to the next by passing the motion down the line.

That is why you can hear someone speak in a Rajasthan haveli from ten metres away. Their vocal chords wiggled a tiny bit of air next to their mouth; that air compressed the air next to it; which compressed the air next to it; and after about 0.03 seconds, enough pressure variation reaches your eardrum to make it wiggle in turn, and you hear a word. No single parcel of air travelled from the speaker to you. A pattern of compressions did.

Mechanical waves vs electromagnetic waves

Every wave either does or does not require a medium.

Mechanical waves require a medium — a material substance whose particles can be displaced and pulled back by inter-particle forces. A sound wave needs air or water or steel. A string wave needs a string. A water wave needs water.

Electromagnetic (EM) waves — light, radio, microwaves, X-rays, gamma rays — need no medium. They are oscillations of the electric and magnetic fields themselves and propagate through perfect vacuum at the universal speed c \approx 3 \times 10^{8} m/s. The sun's light reaches Earth across 150 million km of vacuum; radio signals from the Chandrayaan orbiter reach Bengaluru across vacuum. That is why you can see the Moon: EM waves do not need anything between you and the source.

This distinction matters because every concept introduced below — wavelength, frequency, amplitude, speed, v = f\lambda, reflection, superposition — applies to both kinds of waves. What differs is the underlying physics: for mechanical waves, you trace the motion of physical particles; for EM waves, you trace the oscillations of the fields. But the mathematics is identical. That is the reason you can learn waves on a string first and then pick up optics and radio with very little extra effort.

There is also a third category — matter waves (the quantum-mechanical wavefunction) — that applies to electrons, neutrons, and even small atoms. The de Broglie wavelength \lambda = h/p is a genuine wave phenomenon, confirmed in electron diffraction. Those are for a later article. The language and equations in this one still apply.

Transverse and longitudinal — two ways the medium can wiggle

Ask yourself: when a wave passes through a medium, in which direction do the particles move, relative to the direction the wave is going? There are exactly two clean possibilities, and almost every physical wave is one or the other.

Transverse waves

In a transverse wave, particle displacement is perpendicular (transverse) to the direction of propagation.

The wave on a sitar string is the paradigm: the wave runs along the string (say, in the +x direction), and each point of the string moves up and down (in the \pm y direction). The displacement is at 90° to the travel direction.

Transverse waves require a medium that resists shear — that can be deformed sideways and snap back. Solids can; so can surfaces of liquids (held in shape by surface tension); so can strings under tension. Gases cannot sustain a shear deformation (layers of gas simply slide past one another without any restoring force), so sound cannot travel through a gas as a transverse wave.

Examples: a wave on a sitar or sarangi string; a wave on the taut leather of a tabla surface; surface ripples on the Ganga (approximately); light and all other electromagnetic waves; S-waves (secondary waves) in an earthquake.

Longitudinal waves

In a longitudinal wave, particle displacement is parallel to the direction of propagation.

Sound is the paradigm. A loudspeaker cone pushes forward, compressing the air directly in front of it; that compressed layer pushes the layer in front of it; and a pulse of compression travels forward. As the cone retracts, the air behind it expands into a rarefaction — a region of lower density. The alternating compressions and rarefactions move forward at the speed of sound; the individual air molecules jiggle forward and back a fraction of a millimetre about their average positions. Particle motion and wave motion are along the same line.

Longitudinal waves can propagate through any medium that resists compression — solids, liquids, gases — because every such medium has some bulk modulus. That is why sound travels through air (343 m/s at room temperature), through water (1500 m/s), and through a steel rail (5000 m/s).

Examples: sound in air (and in any fluid or solid); pressure pulses in a long pipe; P-waves (primary waves) in an earthquake, the fastest seismic waves to reach a seismograph; the compression wave you create by shoving one end of a long Slinky.

A mixed case: water surface waves

Waves on the surface of a body of water — the familiar rolling crests at Marina Beach — are neither purely transverse nor purely longitudinal. Each water parcel at the surface traces a roughly circular loop as a wave passes: forward at the crest, downward on the leading edge, backward at the trough, upward on the trailing edge. This combined motion is sometimes called orbital. For the purpose of introducing waves, it is enough to say: real-world surface waves mix both modes, and that mixing is why throwing a ball of wet sand into the sea produces ripples that look qualitatively different from ripples on a taut string.

A summary table

The four numbers of a periodic wave

Single pulses are easy to visualise but hard to do calculations with. The workhorse of wave physics is the sinusoidal wave — a disturbance that repeats sinusoidally in both space (at any fixed instant, a snapshot looks like \sin) and time (at any fixed point, the displacement oscillates like \sin). Four numbers characterise it:

Amplitude A

The amplitude A is the maximum displacement of a particle from equilibrium. For a string wave, it is the height of a crest above the rest-position of the string, in metres. For a sound wave, it is the maximum overpressure, in pascals. Amplitude is always positive and has whatever units the displacement has.

Amplitude tells you how loud or bright or strong the wave is. The energy carried by a wave scales as A^2 — double the amplitude and the wave carries four times the energy. That is why a violin bowed harder ("forte") delivers four times the energy of a violin bowed half as hard, for the same note.

Wavelength \lambda

The wavelength \lambda (Greek letter lambda) is the spatial period of the wave — the distance over which the pattern repeats. If you take a snapshot of the wave (freeze time, photograph the string), \lambda is the distance from one crest to the next crest, or equivalently from one trough to the next, or from any point to the next point at the same phase.

Units: metres.

For visible light, \lambda ranges from about 380 nm (violet) to 750 nm (red). For FM radio, \lambda is a few metres (100 MHz ↔ 3 m). For audible sound in air, \lambda ranges from about 1.7 cm (20 kHz) to 17 m (20 Hz).

Frequency f and period T

The time period T is the temporal period — the time for a single particle of the medium to complete one full oscillation. Units: seconds.

The frequency f = 1/T is the number of oscillations per second. Units: hertz (Hz), which is \text{s}^{-1}.

These are paired, as in SHM: T tells you how long one cycle takes; f tells you how many cycles happen per second. Because they are reciprocals, they carry identical information.

A note on the angular frequency \omega = 2\pi f: the factor of 2\pi converts "cycles per second" to "radians per second," and is the natural variable inside sines and cosines. Everything below can equally be written in terms of \omega.

Speed v

The speed v is how fast a fixed phase point (for instance, a crest) moves through space. It is sometimes called the phase velocity.

Units: metres per second.

For sound in air at 25 °C, v \approx 346 m/s. For light in vacuum, v = c = 2.998 \times 10^8 m/s. For a wave on a sitar string under typical tension, v is a few hundred m/s — varies by string.

The linking equation: v = f\lambda

These four numbers are not independent. Three are free; the fourth is forced. The constraint is the famous relation

It is not a law of physics — it is a definition in disguise. Here is why it must hold.

In one time period T, a particular crest of the wave travels forward by exactly one wavelength. ("By construction," because the wave is periodic: after one temporal period, the pattern has shifted so the next crest stands where the last one was.)

So in time T, the crest moves distance \lambda. Its speed is therefore

Why: the crest must cover one wavelength in one period — that is what "one period" means for a spatially periodic wave. Everything else is arithmetic.

Equation (1) lets you find the third quantity whenever any two are known. Given v = 343 m/s (sound in air) and f = 440 Hz (the musical note "A" on a tanpura), the wavelength is \lambda = v/f = 0.78 m — about the width of a steel almirah.

The interactive explorer

Drag the frequency slider below and watch the wavelength change in response. The wave speed v is fixed (so you can see what equation (1) does), the amplitude is fixed, and the frequency is free to vary. As you raise f, the wavelength \lambda = v/f shrinks — the pattern crowds up. As you lower f, \lambda stretches.

The travelling sinusoid — watching it move

All of the above is about a snapshot. What does the wave look like over time? A sinusoidal wave travelling in the +x direction at speed v has the form

where k = 2\pi/\lambda is the wave number (radians per metre) and \omega = 2\pi f is the angular frequency. The phase kx - \omega t + \varphi_0 is what advances as the wave moves.

Watch a travelling sinusoid of fixed amplitude and wavelength move across the screen:

Worked examples

Common confusions

• "The medium travels with the wave." The single most common misconception. Nothing in the medium travels with the wave over macroscopic distances. Each particle oscillates about a fixed equilibrium and then returns. If the medium itself travelled, a pond would empty every time a ripple crossed it.

• "Sound is a transverse wave." No — sound in fluids (air, water) is strictly longitudinal, because fluids cannot sustain a shear stress. Sound in solids can have both longitudinal and transverse components (P-waves and S-waves in seismology), but what your ears hear is almost always longitudinal pressure variation in air.

• "Higher frequency means higher speed." No. v is set by the medium, not by the wave. A 50 Hz note and a 5000 Hz note in the same air both travel at the same speed (≈ 343 m/s at 20 °C). Changing f changes \lambda in inverse proportion; the product f\lambda = v stays constant. (This is true for non-dispersive media; in a dispersive medium — glass, for instance — different frequencies can travel at different speeds, but that is a more advanced topic.)

• "Amplitude affects frequency." No. Pluck a guitar string harder and you make the note louder (larger A), not higher-pitched. The frequency is set by the length, tension, and mass density of the string. Amplitude and frequency are independent degrees of freedom of a wave.

• "v = f\lambda is a law of physics." It is a geometric identity that any periodic wave automatically satisfies — "one wavelength per period" restated. What is a law of physics is the separate expression for v in terms of the medium's properties (e.g., v = \sqrt{T/\mu} for a string, v = \sqrt{B/\rho} for sound in a fluid). Those relations, which you will meet in upcoming articles, tell you what determines v from the medium.

• "If two waves have the same wavelength, they have the same frequency." Only if they are in the same medium. Light of wavelength 500 nm in vacuum has f = c/\lambda \approx 6 \times 10^{14} Hz. Sound of wavelength 500 nm in air has f = 343/(5\times10^{-7}) \approx 7 \times 10^8 Hz — six orders of magnitude higher. Different media, different speeds, different f for the same \lambda.

Where this leads next

• The Wave Equation — the partial differential equation every classical wave satisfies, and the sinusoidal solution y = A\sin(kx - \omega t + \varphi_0) that encodes v = f\lambda exactly.
• Speed of Waves on a String — why v depends on tension and mass density, derived from Newton's second law applied to a small element of string.
• Principle of Superposition — when two waves meet, their displacements add, giving constructive or destructive interference.
• Reflection and Transmission of Waves — what happens when a wave meets a boundary (fixed end, free end, or a change of medium).
• Sound Waves — Nature and Propagation — the specific case of longitudinal waves in air, and why v_\text{sound} depends on temperature but (at fixed T) not on pressure.