Given:

Y_1 = Asin[k(x + ct)] ... (i) and Y_2 = Asin[k(x - ct)] ... (ii)

By the principle of superposition, the resultant displacement of the particle is given by

$Y = Y_1 + Y_2$ $Y = A[\sin\{k(x + ct)\} + \sin\{k(x - ct)\}]$

By the formula

$\sin C + \sin D = 2 \sin \frac{C + D}{2} \cdot \cos \frac{C - D}{2}$ We have $y = 2 A \sin \frac{kx + kct + kx - kct}{2} \cdot \cos \frac{kx + kct - kx + kct}{2}$ $y = 2 A \sin kx \cdot \cos kct$ For first antinode $\sin kx_1 = 1$ $\sin kx_1 = \sin \frac{\pi}{2}$ $kx_1 = \frac{\pi}{2} \ldots (iii)$ For second antinode $\sin kx_2 = -1$ $\sin kx_2 = \sin \frac{3\pi}{2}$ $kx_2 = \frac{3\pi}{2} \ldots (iv)$ $\therefore$ The distance between adjacent antinodes $kx_2 - kx_1 = \frac{3\pi}{2} - \frac{\pi}{2}$

$\cdot$ $\cdot$ $\cdot$ The distance between adjacent antinodes

$kx_2 - kx_1 = \frac{3\pi}{2} - \frac{\pi}{2}$ $k(x_2 - x_1) = \pi$ $\Delta x = \frac{\pi}{k}$