2.3 — Waves and the Wave Equation

Flick one end of a long rope and a hump runs down it to the far end. Something clearly travelled — you can time it, and it arrives. But no piece of rope went anywhere. Mark a spot with tape and it moves up and then down and ends exactly where it started. The rope stayed; the shape travelled.

That is what a wave is: a disturbance that moves through a medium, carrying energy and momentum, while the medium itself only moves locally and returns. Everything else in this chapter is bookkeeping on that one sentence.

The two kinds, and the one that has no medium

Transverse waves displace the medium perpendicular to the direction of travel. A rope, a guitar string, the ripples on a pond, and light.

Longitudinal waves displace the medium along the direction of travel, producing alternating compressions and rarefactions. Sound in air is the important one, and a slinky pushed rather than flicked shows it clearly.

A few waves are both. Water waves make surface particles trace circles — forward at the crest, backward in the trough — which is why a floating bottle bobs and shuffles rather than being carried out to sea. Earthquake waves come in both kinds and their different speeds are what lets seismologists locate an epicentre: the longitudinal P-wave arrives first at about 6 km/s, the transverse S-wave follows at about 3.5 km/s, and the gap between them gives the distance.

And light is the exception to the opening sentence: it is transverse, and it needs no medium at all. Chapter 4.7 derives it as a self-sustaining ripple in the electric and magnetic fields, and Chapter 6.1 tells the story of the nineteenth century's long, doomed hunt for the medium it was assumed to need.

The vocabulary, and the one equation connecting it

For a repeating wave:

• Wavelength \lambda — the distance between successive identical points, crest to crest. Metres.
• Period T — the time for one full cycle to pass a fixed point. Seconds.
• Frequency f = 1/T — cycles per second, in hertz.
• Amplitude A — the maximum displacement from rest.
• Wave speed v — how fast the shape travels.

In one period, the wave advances exactly one wavelength. Speed is distance over time, so:

\boxed{v = \frac{\lambda}{T} = f\lambda}

This is the most-used equation in the whole of wave physics, and the thing to understand about it is which quantity is fixed by what:

• The source sets the frequency. A 440 Hz tuning fork produces 440 pressure pulses per second, and nothing downstream can change that count. When sound passes from air into water, its frequency is identical — the pulses have to keep arriving at the same rate or they would pile up at the boundary.
• The medium sets the speed. Sound travels at 343 m/s in air and 1480 m/s in water, and that is a property of air and water.
• The wavelength is whatever those two force it to be. That 440 Hz note has \lambda = 343/440 = 0.78 m in air and 1480/440 = 3.36 m in water.

Getting this backwards is the single most common wave mistake. Frequency travels with the wave; wavelength changes at every boundary.

It is also the reason a swimming pool looks shallower than it is (Chapter 5.1) and the reason your voice sounds different underwater.

The mathematical description of a travelling wave

Suppose at time zero the rope has some shape y = f(x). A moment later the whole shape has slid to the right by a distance vt without changing form. A shape shifted right by d is written f(x-d), so:

y(x,t) = f(x-vt)

Any function of the combination (x-vt) is a wave moving right at speed v, and any function of (x+vt) moves left. That is the general statement, and the sine wave is just the most useful special case of it.

For a sine wave:

y(x,t) = A\sin\!\left[\frac{2\pi}{\lambda}(x-vt)\right]

The 2\pi/\lambda is there so that advancing x by one wavelength advances the sine's argument by exactly 2\pi, which is one full cycle. That combination is used constantly and gets its own name, the wave number:

k = \frac{2\pi}{\lambda}

Using v = \omega/k (which is just v = f\lambda rewritten, since \omega = 2\pi f and k = 2\pi/\lambda):

\boxed{y(x,t) = A\sin(kx-\omega t+\phi)}

This one expression contains two different oscillations, and separating them is worth doing explicitly.

Freeze time — take a photograph. Then \omega t is a constant and y varies sinusoidally with x. You see the wave's shape in space, repeating every \lambda.

Freeze position — watch one point. Then kx is a constant and y varies sinusoidally with t. That point is doing simple harmonic motion, exactly as in Chapter 2.1, with period T.

Every particle in the medium is a simple harmonic oscillator. What makes it a wave rather than a room full of independent oscillators is that each one starts a little later than its neighbour, and that progressive delay is what the kx term encodes.

The speed of a particle in the medium is not the speed of the wave, and they are unrelated numbers:

v_{\text{particle}} = \frac{\partial y}{\partial t} = -A\omega\cos(kx-\omega t)

so v_{\text{particle, max}} = A\omega, which depends on amplitude. The wave speed v = \omega/k does not depend on amplitude at all. A loud sound and a quiet sound at the same pitch travel at exactly the same speed, which is the only reason an orchestra sounds like an orchestra at the back of the hall.

Deriving the wave equation from a piece of string

Everything above described waves. Now derive that they must exist, from Newton's second law alone.

Take a string under tension F with mass per unit length \mu. Look at a very short segment of length \Delta x that has been displaced sideways, so it is slightly curved.

The forces. Tension pulls at both ends, along the string. Because the segment is curved, the two tensions point in slightly different directions and do not cancel. Let the string make angle \theta_1 with the horizontal at the left end and \theta_2 at the right.

The net vertical force is:

F_y = F\sin\theta_2 - F\sin\theta_1

Small-amplitude approximation. For small displacements the angles are small, so \sin\theta\approx\tan\theta, and \tan\theta is exactly the slope of the string, \partial y/\partial x. So:

F_y \approx F\left[\left(\frac{\partial y}{\partial x}\right)_{x+\Delta x} - \left(\frac{\partial y}{\partial x}\right)_x\right]

Recognise the bracket. A difference of a quantity at two nearby points, divided by the gap between them, is a derivative. So the bracket equals \dfrac{\partial^2y}{\partial x^2}\Delta x in the limit:

F_y = F\frac{\partial^2y}{\partial x^2}\Delta x

Newton's second law. The segment's mass is \mu\Delta x and its vertical acceleration is \partial^2y/\partial t^2:

\mu\Delta x\frac{\partial^2y}{\partial t^2} = F\frac{\partial^2y}{\partial x^2}\Delta x

The \Delta x cancels from both sides — which is what makes this a statement about the string at a point rather than about one particular chosen segment:

\boxed{\frac{\partial^2y}{\partial t^2} = \frac{F}{\mu}\frac{\partial^2y}{\partial x^2}}

This is the wave equation, first written down by Jean le Rond d'Alembert in 1747.

What it says, and why the speed falls out

Compare it with the standard form of the wave equation:

\frac{\partial^2y}{\partial t^2} = v^2\frac{\partial^2y}{\partial x^2}

Matching term for term:

\boxed{v = \sqrt{\frac{F}{\mu}}}

Chapter 1.1 guessed this from dimensional analysis and could not supply the numerical constant. Here it is derived, and the constant is exactly 1.

Read it physically. More tension means faster — a stiffer restoring force snaps each bit of string back sooner. More mass per length means slower — more inertia to shift. It is the same \sqrt{\text{stiffness}/\text{inertia}} pattern as \omega = \sqrt{k/m} in Chapter 2.1, and for the same reason.

Your fingers already know both halves. Tighten a guitar string with the machine head and the pitch goes up, because v rose and \lambda is fixed by the string's length. The low strings on a guitar are the thick ones, because raising \mu lowers v.

Why the solution is a travelling wave. Substitute the general form y = f(x-vt) into the equation. By the chain rule, each t-derivative brings out a factor -v, so \partial^2y/\partial t^2 = v^2f'', while each x-derivative brings out 1, so \partial^2y/\partial x^2 = f''. Then v^2f'' = v^2f'', which is true for any shape f. So the equation does not merely permit waves; it permits waves of any shape whatsoever, travelling without distortion. That is why a complicated pluck holds its shape as it runs down a string.

The wave equation is one of the three great partial differential equations of physics, and it shows up in places that have nothing to do with string:

Wave	The equation describes	Speed
String	Transverse displacement	\sqrt{F/\mu}
Sound	Pressure variation	\sqrt{B/\rho}
Light	Electric and magnetic field	1/\sqrt{\mu_0\varepsilon_0}
Water (deep)	Surface height	\sqrt{g\lambda/2\pi}
Quantum particle	The wavefunction	(Chapter 7.3)

The third row is the astonishing one, and Chapter 4.7 derives it: Maxwell wrote down four equations about electricity and magnetism, combined them, got the wave equation out, and the speed that fell out of two constants measured with batteries and wires was the measured speed of light.

Sound waves, and the speed of sound

Sound is a longitudinal pressure wave. Its speed comes from the same \sqrt{\text{stiffness}/\text{inertia}} pattern, with the bulk modulus of Chapter 1.10 as the stiffness:

v = \sqrt{\frac{B}{\rho}}

Newton did this calculation in the Principia and got 298 m/s against a measured 340. He was 15% low and knew it, and could not find the error. It stood for over a century.

Laplace found it in 1816. Newton had used the isothermal bulk modulus, assuming the gas stays at constant temperature as it is compressed. But a sound wave compresses and rarefies air hundreds of times a second — far too fast for heat to flow out of the compressed regions. The compressions are therefore adiabatic: they heat up, which makes the air stiffer than the isothermal calculation assumed. The correction is a factor \gamma, the ratio of specific heats, which is 1.4 for air (Chapter 3.3 derives where that comes from):

v = \sqrt{\frac{\gamma P}{\rho}}

Now the numbers. At 0 °C, P = 101\,325 Pa and \rho = 1.293\ \text{kg/m}^3:

v = \sqrt{\frac{1.4\times101\,325}{1.293}} = \sqrt{\frac{141\,855}{1.293}} = \sqrt{109\,710} = 331\ \text{m/s}

Which matches measurement.

Using the ideal gas law P/\rho = RT/M (Chapter 3.2) gives the more useful form:

v = \sqrt{\frac{\gamma RT}{M}}

Sound speed depends on temperature and not on pressure. That is unexpected — raising the pressure of a gas raises its density in exactly the same proportion, so the ratio is unchanged. Climb a mountain and the air is thinner in both stiffness and density, and sound travels at the same speed as at sea level, provided the temperature is the same. What does change it is temperature: the practical formula is

v \approx 331 + 0.6\,T_{\text{C}}\ \text{m/s}

so at 20 °C it is 343 m/s.

That temperature dependence causes a real effect you have heard. On a still night the ground cools and the air just above it is colder than the air higher up, so sound travels faster higher up. The upper part of each wavefront outruns the lower part, bending the wave back down towards the ground instead of letting it escape upwards. That is why distant sounds carry unnaturally far at night, and why the same sounds vanish on a hot afternoon when the gradient is the other way round.

Energy, intensity, and why sound fades

A wave carries energy. Each bit of medium is an oscillator with energy \propto A^2 (Chapter 2.1), so:

\text{energy} \propto A^2\omega^2

Intensity is the power crossing unit area, in watts per square metre:

I = \frac{P}{A_{\text{area}}}

For a point source radiating equally in all directions, the power spreads over a sphere of area 4\pi r^2:

\boxed{I = \frac{P}{4\pi r^2}}

An inverse-square law, for exactly the same geometric reason as the inverse-square laws of gravity and electrostatics: the surface area of a sphere grows as r^2 and the total flowing through it is fixed. Double your distance from a speaker and the intensity drops to a quarter.

Since I \propto A^2, the amplitude itself falls as 1/r.

Human hearing spans an absurd range — from 10^{-12}\ \text{W/m}^2 (a mosquito at three metres, the threshold of hearing) to about 1\ \text{W/m}^2 (pain). That is twelve orders of magnitude, which is why loudness is measured logarithmically:

\beta = 10\log_{10}\!\left(\frac{I}{I_0}\right)\ \text{dB}, \qquad I_0 = 10^{-12}\ \text{W/m}^2

Two consequences of that logarithm that people get wrong constantly:

+10 dB is ten times the intensity, and sounds roughly twice as loud subjectively.

+3 dB is double the intensity. So two identical speakers are 3 dB louder than one, not twice as loud. Getting a sound to seem twice as loud needs ten speakers.

Sound	dB	Intensity (W/m²)
Threshold of hearing	0	10^{-12}
Whisper	30	10^{-9}
Conversation	60	10^{-6}
Busy road	80	10^{-4}
Rock concert	110	10^{-1}
Pain, immediate damage	130	10

Hearing damage is cumulative and depends on intensity times time. The usual workplace rule allows 8 hours at 85 dB, and halves the allowed time for every 3 dB after that — which is exactly the doubling of intensity. At 100 dB that leaves about 15 minutes, which is less than one song at a concert.

The numbers worth carrying

Quantity	Value	Where it comes from
Speed of sound in air at 20 °C	343 m/s	\sqrt{\gamma RT/M}, derived above
Speed of sound in air at 0 °C	331 m/s	same, at 273 K
Speed of sound in water	1480 m/s	water's bulk modulus is far larger
Speed of sound in steel	~5000 m/s	stiffer still
Threshold of hearing	10^{-12}\ \text{W/m}^2 = 0 dB	the reference I_0 in the decibel
Threshold of pain	10\ \text{W/m}^2 = 130 dB	thirteen orders of magnitude up
Human hearing range	20 Hz – 20 kHz	narrows with age from the top
Concert pitch A	440 Hz	international agreement, 1939
Middle C	262 Hz	440 divided by 2^{9/12}
Ratio of specific heats for air, \gamma	1.4	Chapter 3.3 derives it

Where this shows up in your life

Ultrasound imaging is this chapter used deliberately. A transducer sends a pulse at 2–15 MHz into the body and listens for echoes from boundaries between tissues of different density. The time delay gives depth, since v \approx 1540 m/s in soft tissue, and the resolution is limited by the wavelength: at 5 MHz, \lambda = 1540/5\times10^6 = 0.3 mm, which is the size of the smallest feature it can resolve. Higher frequency gives finer detail and less penetration, which is the trade-off every ultrasound operator makes when choosing a probe.

Sonar and echolocation are the same idea in water and air. A bat's calls run up to 200 kHz, giving \lambda \approx 1.7 mm, fine enough to resolve a mosquito. That is not a coincidence — it is the resolution a bat needs, and evolution pushed the frequency until it was available.

And the reason you can tell where a sound came from is that your two ears are about 20 cm apart, so a sound from the side arrives at one ear roughly 0.2/343 = 0.6 milliseconds before the other. Your brain resolves that delay to about 10 microseconds, which corresponds to a few degrees of angle. For high frequencies it uses the loudness difference instead, because your head casts an acoustic shadow once the wavelength is smaller than your head — which happens above about 1.5 kHz.

What the next chapter fixes

One wave travelling in one direction is the easy case. What happens when two waves occupy the same place at the same time is where the interesting physics is: they can cancel completely, they can lock into a standing pattern that does not travel at all, and they can beat against each other in a slow throb. Chapter 2.4 does superposition, standing waves, the normal modes that decide what note an instrument plays, and why a guitar and a piano playing the same note sound nothing alike.