Q.Define plane progressive wave. Derive expressions for the displacement of a plane progressive wave in different forms.

A plane progressive wave is one that carries energy continuously through a medium, with every particle performing SHM of the same amplitude and frequency but a progressively changing phase; its displacement can be written as y(x,t) = A sin(omega t - kx), and equivalently in several other standard forms.

Definition: A plane progressive (travelling) harmonic wave is a wave in which every particle of the medium executes simple harmonic motion of the same amplitude and frequency, but each particle's oscillation is progressively out of phase (delayed) relative to the particle behind it, as the wave's disturbance travels continuously through the medium in a fixed direction, carrying energy without any change of form (undamped, non-dispersive medium assumed).

Derivation of the displacement expression: Consider a wave travelling in the +x direction. A particle at the origin (x = 0) undergoes SHM: y(0,t) = A sin(omega t), where A is amplitude and omega = 2pi/T is the angular frequency. A particle at position x experiences the same disturbance, but delayed by the time x/v it takes the wave to travel there (v being wave speed):

y(x,t) = A sin[omega (t - x/v)]

Using v = omega/k, where k = 2pi/lambda is the propagation constant (wave number), this becomes:

y(x,t) = A sin(omega t - kx)

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