Energy and intensity of waves

Waves carry energy

A wave transports energy without permanently transporting matter over the same distance. In a string wave, each small segment of string moves up and down while energy travels along the string. In a sound wave, air molecules oscillate locally while sound energy travels outward. In light, electromagnetic fields carry energy through space.

The energy carried by a wave usually depends strongly on amplitude. Larger-amplitude waves carry more energy.

Energy in a sinusoidal wave on a string

For a sinusoidal wave on a string,

$y(x,t)=Acos(kx-omega t)$

the energy includes kinetic energy of moving string elements and potential energy associated with stretching the string. For small transverse waves, the average power transmitted by the wave is proportional to

$P_{avg}propto mu omega^2 A^2 v$

where $mu$ is linear mass density, $omega$ is angular frequency, $A$ is amplitude, and $v$ is wave speed.

The key dependence is

$P_{avg}propto A^2$

Doubling amplitude quadruples the average power, all else equal.

Intensity

Intensity is power per unit area:

I=rac{P}{A_{area}}

The area symbol here represents area, not wave amplitude. To avoid confusion, some texts use $S$ for area.

Intensity measures how concentrated the wave energy flow is. Bright light has greater intensity than dim light. Loud sound has greater intensity than quiet sound.

Spherical spreading

If a source radiates uniformly in all directions, its energy spreads over a sphere. The surface area of a sphere is

$A_{sphere}=4pi r^2$

If the source power is $P$, the intensity at distance $r$ is

I=rac{P}{4pi r^2}

This is the inverse-square law for intensity. Double the distance, and intensity becomes one fourth as large.

Amplitude and intensity

For many linear waves,

$Ipropto A^2$

This means amplitude decreases as a wave spreads. For spherical spreading, since intensity decreases like $1/r^2$, amplitude decreases approximately like $1/r$.

This is why sound and light become weaker with distance even when no energy is absorbed by the medium.

Absorption

Real media may absorb wave energy, transforming it into thermal energy or other forms. Sound is absorbed by walls, curtains, and air. Light is absorbed by materials, heating them or exciting electrons. Seismic waves lose energy as they travel through Earth.

Absorption reduces intensity beyond geometric spreading.

Energy density

Wave energy can also be described as energy per unit volume, called energy density. For waves moving with speed $v$, intensity is related to average energy density $u_{avg}$ by

$I=u_{avg}v$

This says the energy passing through area each second depends on how much energy is stored per volume and how fast it moves.

Sound intensity and hearing

Human hearing covers an enormous range of intensities. Because of this, sound levels are often measured on a logarithmic decibel scale rather than a linear intensity scale. That topic appears later in the sound module.

For now, the important point is that physical intensity is energy flow per area, while perceived loudness is a biological response.

The big idea

Waves carry energy, and intensity measures the rate of energy flow per area. In many waves, power and intensity scale with amplitude squared. As waves spread outward, intensity decreases because energy is distributed over larger areas. Absorption and damping further reduce wave energy in real media.