Formation of stationary waves by analytical method:

Consider two identical progressive waves of equal amplitude and frequency travelling along X axis in opposite direction. They are given by

y₁ = A sin \(\frac{2π}{λ}\) (vt - x) along positive X-axis     .........(1)

y₂ = A sin \(\frac{2π}{λ}\) (vt + x) along positive X-axis     .........(2)

The resultant displacement ‘y’ is given by the principle of superposition of waves,

Y = y₁ + y₂                    .....(3)

_{y =}  A sin \(\frac{2π}{λ}\) (vt - x) +  A sin \(\frac{2π}{λ}\) (vt + x)

By using,

sin C + sin D = 2sin [\(\frac{C+D}{2}\)] cos [\(\frac{C-D}{2}\)]

We get,

y = 2A sin [\(\frac{2π}{λ}\) \((\frac{vt-x+vt+x}{2})\)] cos  [\(\frac{2π}{λ}\) \((\frac{vt-x+vt+x}{2})\)]

= 2A sin \((\frac{2πvt}{λ})\) cos \((\frac{2π}{λ}{(-x)})\)

∴ y = 2Asin 2πnt cos \((\frac{2πx}{λ})\)     .....[∴ n = \(\frac{v}{λ}\) and cos (-θ) = cosθ]

∴ y = 2Asin \((\frac{2πx}{λ})\) sin 2πnt

Let R = 2Acos \((\frac{2πx}{λ})\)

∴ y = Rsin (2πnt)    ...........(4)

But, ω = 2πn

∴ y = R sin ωt        ............(5)

Equation (5) represents the equation of S.H.M. Hence, the resultant wave is a S.H.M. of amplitude R which varies with x.

The absence of x in equation (5) shows that the resultant wave is neither travelling forward nor backward. Therefore it is called as stationary wave.

Amplitude at node is minimum, i.e., 0.

∴ R_min = 0

Since R = 2A cos \((\frac{2πx}{λ})\)

∴ cos \((\frac{2πx}{λ})\) = 0

∴ \(\frac{2πx}{λ}\) = \(\frac{π}{2},\frac{3π}{2},\frac{5π}{2},.......\)

∴ x = \(\frac{λ}{4},\frac{3λ}{4},\frac{5λ}{4},.......\)

At antinodes: R = ± 2A

∴ cos \((\frac{2πx}{λ})\) = ± 1

∴ \((\frac{2πx}{λ})\) = 0, π, 2π,3π,.............nπ

∴ 0, \(\frac{λ}{2}\), λ, \(\frac{3λ}{2}........\)

∴ Distance between first node and adjacent

antinode = x_n - x_an = \(\frac{λ}{4}\)- 0 = \(\frac{λ}{4}\) 

Thus, the distance between a node and an adjacent antinode is \(\frac{λ}{4}\)