Wave Energy and Intensity

A wave is a travelling package of energy. Drop a pebble in a pond and the ripples carry energy outward — enough, far away, to bob a floating leaf up and down. Pluck a guitar string and the energy of your finger becomes a sound wave that crosses the room and rattles your eardrum. Nothing material makes the trip: the water stays put, the air stays put, the string stays put. What flows outward, tirelessly, is energy. This page is about how much energy a wave carries, and what happens to it as the wave spreads out.

The whole page rests on one sentence, worth reading twice: the energy a wave carries is proportional to the square of its amplitude. Double the amplitude and you do not double the energy — you quadruple it. Below we will see why that square appears, define the everyday quantity that measures how "strong" a wave feels — its intensity — and follow one beautiful consequence: why a source that spreads its energy over a growing sphere gets fainter as the square of the distance, the famous inverse-square law.

Energy on a wave: why amplitude gets squared

Take the cleanest example: a long string under tension, carrying a travelling wave. Each little element of the string, of mass dm, is doing simple harmonic motion — sliding up and down as the wave passes. It therefore carries the same two kinds of energy a mass on a spring does: kinetic energy from its motion, and potential energy from the stretching of the string.

For simple harmonic motion of angular frequency \omega and amplitude A, an element moving up and down has a maximum speed v_{\max} = \omega A. Its energy is set by that peak speed:

dE = \tfrac{1}{2}\,dm\,v_{\max}^2 = \tfrac{1}{2}\,dm\,\omega^2 A^2.

Look at what appears: an A^2 and an \omega^2. The potential-energy part, from the tension stretching the string as it curves, scales exactly the same way — so the total does too. Because the speed of a wave is v_{\max}=\omega A and energy goes as speed squared, the amplitude and frequency both come in squared. This is not special to strings; it is true for sound, for light, for water — every wave.

Energy scales as amplitude squared. For a mechanical wave the energy per unit length (and the power it transports) is proportional to A^2 and to \omega^2: E \propto \omega^2 A^2.
Power along a string. A sinusoidal wave of amplitude A and angular frequency \omega on a string of linear mass density \mu, travelling at wave speed v, transports average power \;\bar{P} = \tfrac{1}{2}\,\mu\, v\, \omega^2 A^2.
The take-home. Loudness, brightness and wave energy all track A^2 — doubling the amplitude means four times the energy.

No — they add, and it is worth seeing why they scale the same way. A string element's kinetic energy is largest as it whips through the middle of its swing, where its speed peaks at \omega A; that gives the \tfrac12 dm\,\omega^2 A^2 we wrote down. Its potential energy comes from the local stretching of the string, which is greatest where the string is most steeply sloped — and for a sinusoid that steepest slope is itself proportional to A, so the stored energy goes as A^2 too.

A curious feature of a travelling wave: unlike a mass on a spring, the kinetic and potential energies of a string element peak together, not out of phase — both are largest at the zero-crossing and both vanish at the crest. Either way, both pieces carry the same \omega^2 A^2 dependence, so their sum does as well. The A^2 law is airtight.

Intensity: energy per second, per square metre

"How much energy" is not quite what your ear or your eye actually registers. What matters is how much energy arrives, each second, on a given patch of area — a sunbather cares about the power falling on their skin, not the Sun's total output. That quantity has a name: intensity.

Definition. Intensity is the average power crossing a unit area held perpendicular to the wave: I = \dfrac{P}{A_\text{area}}, measured in watts per square metre, \text{W/m}^2.
Still an amplitude-squared law. Since power goes as A^2, so does intensity: I \propto A^2. Quadruple the intensity and you have only doubled the amplitude.

(Careful with the two meanings of A in physics: here A_\text{area} is a geometric area, while A alone is the wave's amplitude. We keep them apart with the subscript.) To make it concrete: bright sunlight at the ground delivers about 1000\ \text{W/m}^2; a comfortable conversation reaching your ear is around 10^{-6}\ \text{W/m}^2. Intensity is the number that spans that enormous range.

The inverse-square law

Now for the payoff. Imagine a small source — a lamp, a firecracker, a lonely star — that radiates its power P equally in all directions into empty three-dimensional space. Draw an imaginary sphere of radius r centred on the source. All of the power must pass through that sphere (energy has nowhere else to go). The sphere's surface area is 4\pi r^2, so the intensity there — power spread over that area — is

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

The area grows as r^2, so the same power is smeared ever more thinly: intensity falls off as 1/r^2. Go twice as far away and the light is not half as bright but a quarter as bright. Three times as far, a ninth. This is the inverse-square law, and it governs the fading of sound, light, gravity and radio alike.

Intensity. For an isotropic point source of power P, \;I(r) = \dfrac{P}{4\pi r^2} \propto \dfrac{1}{r^2}.
Amplitude. Because I \propto A^2, and I \propto 1/r^2, the amplitude itself falls only as A \propto \dfrac{1}{r} — one power of r, not two.

Worked examples

Example 1 — moving twice as far away. A loudspeaker sounds at intensity I_1 = 0.08\ \text{W/m}^2 when you stand r_1 = 2\ \text{m} away. You walk back to r_2 = 4\ \text{m}. Since I \propto 1/r^2, the ratio of intensities depends only on the ratio of distances:

\frac{I_2}{I_1} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{4}\right)^2 = \frac{1}{4}.

I_2 = \tfrac14 \times 0.08\ \text{W/m}^2 = 0.02\ \text{W/m}^2.

Doubling the distance quartered the intensity — exactly as the law promises.

Example 2 — intensity at a given range. A small siren radiates P = 50\ \text{W} of acoustic power uniformly in all directions. What is the intensity 120\ \text{m} away? Spread that power over a sphere of radius r = 120\ \text{m}:

I = \frac{P}{4\pi r^2} = \frac{50}{4\pi (120)^2} = \frac{50}{4\pi \times 14400} \approx 2.8\times 10^{-4}\ \text{W/m}^2.

Faint, but a quiet night siren is still easily audible at that range.

Example 3 — an amplitude change. You turn up an amplifier until the vibration amplitude of the speaker cone triples. By what factor does the energy — and hence the emitted intensity — increase? Energy goes as A^2, so

\frac{E_2}{E_1} = \left(\frac{A_2}{A_1}\right)^2 = 3^2 = 9.

Three times the amplitude is nine times the energy — not three.

The single most common slip is to say the amplitude also drops as 1/r^2. It does not. It is the intensity — the energy flux — that obeys the inverse-square law, because intensity carries the square of the amplitude. Since I \propto A^2 and I \propto 1/r^2, matching the powers of r gives

A^2 \propto \frac{1}{r^2} \quad\Longrightarrow\quad A \propto \frac{1}{r}.

So the wave's amplitude fades only as 1/r — one power of the distance. Go ten times further and the intensity is a hundredth, but the amplitude is merely a tenth. Keep the two laws straight: square for energy, single power for amplitude.

Your ear copes with intensities spanning a factor of a trillion — from the faintest whisper it can detect to the threshold of pain. Writing those out in \text{W/m}^2 is clumsy, so sound is measured on a compressed logarithmic scale, the decibel:

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

Because it is logarithmic, every 10\ \text{dB} means a factor of ten in intensity: a 60\ \text{dB} conversation carries a thousand times the intensity of a 30\ \text{dB} whisper. And notice how the inverse-square law reads in decibels — doubling your distance quarters the intensity, which is a drop of 10\log_{10}(1/4) \approx -6\ \text{dB}. That handy rule, "−6 dB per doubling of distance", is the inverse-square law wearing a logarithm.