A progressive wave with a uniform velocity v, angular frequency ω, wavelength λ and wave number k that is travelling in the positive x direction is given by:
y₁(x, t) = displacement of particle at x at time t = A Sin (k x - ω t)
k = 2π/λ
Let another progressive wave of equal magnitude, frequency, wave number also travel in the same medium, but in the negative x direction.
y₂ (x, t) = A Sin (k x + ω t)
When both waves interfere, the resultant displacement of particles is:
y(x, t) = A [ Sin (k x+ ω t) + Sin (k x - ω t) ]
= (2 A Sin k x ) Cos ωt
The effect of two equal waves in opposite directions is that the amplitude at each value of x is fixed in time. Amplitude of the vibrating particles in the wave = 0 when k x = n π, where n is an integer. Amplitude is maximum when k x = (n + 1/2) π.
Since all the elements in the medium oscillate with same phase, move all synchronously and have fixed positions in space, they are called standing or stationary waves.
nodes = points where amplitude of oscillation = 0, ie., k x = n π
k = 2 π/λ x = n λ / 2 here n = 0,1,2,3,4,5...
antinodes = points where amplitude = maximum : ie., x = (n +1/2) π
x= (n + 1/2) λ/2 here n = 0,1,2,3,4
Suppose the second wave is a wave reflected from an obstruction that the progressive wave encounters on the medium. It has a phase difference of π.
y₁ (x, t) = A Sin (k x + ω t)
y₂ (x, t) = A Sin (k x - ω t - π) = A Sin (ω t - k x)
Superposition gives, y(x, t) = A Sin ωt * Cos kx
Thus standing waves of same nature will result. Nodes and antinodes will be interchanges as compared to the above.
Nodes x = (n+1/2) λ/2
anti nodes at x = n λ/2