When two progressive waves having the same amplitude, wavelength and speed propagate in opposite directions through the same region of a medium, their superposition under certain conditions creates a stationary interference pattern called a stationary wave.
Consider two simple harmonic progressive waves, of the same amplitude A, wavelength A and frequency n = ω/2π, travelling on a string stretched along the x-axis in opposite directions. They may be represented by y1 = A sin (ωt – kx) (along the + x-axis) and … (1)
y2 = A sin (ωt + kx) (along the – x-axis) …. (2) where k = 2π/λ is the propagation constant.
By the superposition principle, the resultant displacement of the particle of the medium at the point at which the two waves arrive simultaneously is the algebraic sum y = y1 + y2 = A [sin (ωt – kx) + sin (ωt + kx)] Using the trigonometrical identity,
y = 2A sin ωt cos (- kx)
= 2A sin ωt cos kx
[∵ cos(- kx) = cos(kx)]
= 2A cos kx sin ωt … (3)
∴ y = R sin ωt, … (4)
where R = 2A cos kx. … (5)
Equation (4) is the equation of a stationary wave.
