Harmonic Wave Density

During wave motion, the particles of the medium are put in state of motion. This implies not only kinetic energy of motion but also potential energy of deformation of the medium. Thus, during wave motion, the medium is furnished with an energy density, energy per unit volume. This energy is supplied by the source which initially produces disturbance or wave and it transported through the medium in the form of wave motion. In fact, a progressive wave transports both momentum and energy as it travels.

Let us calculate the total energy density associated with a progressive harmonic wave travelling along X-axis,

y = A sin (ω t – kx)

Consider a thin slab of the medium at point x, having unit area of cross-section and thickness dx. The mass of this element is ρ dx, where ρ is density of the medium. At time t, the displacement of particles at position x, is given by above equation. The instantaneous velocity of the particles is



Assuming that velocity of all the particles in the above infinitesimally thin slab is same, the kinetic energy of the slab is given as



Dividing by the volume dx of the slab, we define the local kinetic energy density at point x as

dK/dx = ½ ρ ω² A² cos² (ω t – kx)

The potential energy of the slab is obtained from calculating the work done in displacing the slab by amount y. The force  acting on the slab (when its displacement is y) is



= – ρ dx ω² y

The work done in moving the slab further by dy is

dW = F dy

Hence, total work done is displacing the slab from y = 0 to distance y is given by


= – ½ ρ dx ω² y²

The potential energy of the slab is negative of above work done, i.e.

dU = ½ ρ dx ω² y² = ½ ρ dx ω² A² sin² (ω t – kx)

The local potential energy density is defined as

dU/dx = ½ ρ ω² A² sin² (ω t – kx)



= ½ ρ ω² A²

Note that both kinetic and potential energy densities are oscillating functions of space and time. However, the average kinetic and potential energy densities in space (or time) are constants and equal. For example, if the average is taken over a distance of one wavelength in space, we find


Since we are considering unit area of cross-section, the above expression gives the average kinetic energy density in space; the average is taken over a distance of at least one wavelength. Similarly, we have



Note that



where t is taken as constant and k = 2π/λ.

Thus, the total energy density, E_av = ½ ρ ω² A², is a constant. The energy is proportional to squares of frequency and amplitude of the wave; it also depends linearly on the inertia of the medium, i.e. density of the medium.

To take an example, consider the transverse waves on a stretched string. Now ρ denotes the mass per unit length of the string. In order to produce one extra wave (of wavelength λ) on the string, the source must supply an energy

E = ( E_av ) λ = ½ ρ ω² A² λ

This is the energy associated with each harmonic wave moving in the medium.

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