Equation of Stationary Wave

The equation of a simple harmonic wave travelling with velocity v = ω/k in a medium is

y1 = – a sin (ωt – kx)

On reflection from a denser medium, suppose the wave travels along the same line along X–axis in the opposite direction with phase change of π. The equation of the reflected wave is

y2 = a sin (ωt – kx)

Thus, owing to the superposition of the two waves, the resultant displacement at a given point and time is

y = y1 + y2

y = a sin (ωt – kx) – a sin (ωt – kx)

y = –2a sin kx cos ωt

Put –2a sin kx = A

y = A cos ωt

This equation represents a resultant wave of angular frequency ω and amplitude 2a sin kx. This is the equation of stationary wave. The amplitude of the resultant wave, oscillates in space with an angular frequency ω, which is the phase change per metre.

At such points where kx = mπ = mλ/2, sin kx= sin mπ = 0. Hence A = 0.

The points where the amplitude is zero are referred to as nodes. At these points ∆y/∆x = maximum, that is strain is maximum. The spacing between two nearest points is λ/2.

At those points where kx = (2m + 1)π/2 or x = (2m + 1)λ/2 × λ/2π = (2m + 1)λ/4, sin kx = sin (2m + 1)π/2 = ±1. Hence A is maximum. At these points the strain ∆y/∆x is zero. The spacing between two such neighboring points is λ/2. These points where the amplitude is maximum but strain is zero are referred to as antinodes.