Progressive wave energy per unit volume

Let a plane progressive wave in a medium of density ρ be y = A sin(ωt − kx), with speed v = ω/k.

Kinetic energy per unit volume

Particle velocity ∂y/∂t = Aω cos(ωt − kx).

KE per volume = ½ ρ (∂y/∂t)² = ½ ρ A² ω² cos²(ωt − kx)

Potential energy per unit volume

The energy of deformation per unit volume is ½ (elastic constant) × (strain)², where the strain is ∂y/∂x = −Ak cos(ωt − kx) and the elastic constant is ρv² (for a string, the tension is T = μv²).

PE per volume = ½ ρ v² (∂y/∂x)² = ½ ρ v² A² k² cos²(ωt − kx)
              = ½ ρ A² ω² cos²(ωt − kx)       (since vk = ω)

So at every point KE and PE are equal and in phase.

Variation and averages

• Both depend on (ωt − kx), so they change with time (and position) between zero and ½ρA²ω²; the energy of a given particle is not constant. Particles at the mean position have maximum KE and PE; at the extremes both are zero.
• Averaging over one period, <cos²(ωt − kx)> = ½, so

<KE per volume> = ¼ ρ A² ω² ,   <PE per volume> = ¼ ρ A² ω²

These are constants, independent of time and position.

• The total average energy density is ½ ρ A² ω², a constant, and the two averages are equal.

 energy |  .  .  .  .   KE and PE: same curve, 2ω frequency
 density|.' '.' '.' '.   (always >= 0)
        |-------------- average = ¼ρA²ω² each
        |______________ t