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 λ 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,
Sin C + sin D = 2 sin\((\frac{C−D}{2})\) cos\((\frac{C−D}{2})\)
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.
