Energy, power and intensity of a harmonic wave

The main characteristic of a wave is that it transfers energy from one point to another. For example, the energy of the seismic waves created during an earthquake cause damages in buildings, roads, bridges, and so on.

The energy transported by a sound wave causes the eardrum to vibrate, and our brain then interprets this vibration as sound.

The energy transported by a wave depends on both its frequency and its amplitude. In this page we are going to derive an expression for the energy transported by a transverse harmonic wave propagating along a rope.

The total energy of a particle of mass m undergoing simple harmonic motion is given by:

where A is the amplitud of the simple harmonic motion and ω its angular frequency.

When a harmonic wave travels along a rope, each mass element of the rope Δm (of length Δx) oscillates vertically undergoing simple harmonic motion, and the energy propagates horizontally:

The mass linear density μ of the rope is:

And substituting into the expression of the total energy in a harmonic simple motion, the energy of each mass element is then:

On the other hand, calling v the wave speed:

And substituting into the expression above we get:

Power

The time-averaged power of a wave is defined as the energy transported by the wave per unit of time, and can be found by taking the total energy given by the expression above divided by the time it takes for this energy to pass a point of the rope:

The SI unit for power is watts (W). 1 W = 1 J/s

This expression is also valid for a rope of cross-section S. We simply have to write the linear mass density μ in terms of the density ρ:

And substituting into the expression for the time-averaged power:

Intensity

The intensity is defined as power per unit area:

This expression can also be used to find the intensity of a sound wave.

The SI unit for intensity is watts per square meter (W/m²).

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