# Derive the equation of stationary wave and deduce the conditionfor nodes and antinodes.

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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

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