Given, y1=A‌sin[k(x+ct)].....(i) and ‌‌y2=A‌sin[k(x−ct)].....(ii) By the principle of superposition, the resultant displacement of the particle is given by y=y1+y2 y=A[sin{k(x+ct)}+sin{k(x−ct)}] By the formula
sin‌C+sin‌D=2‌sin‌
C+D
2
⋅cos‌
C−D
2
We have
y=2A‌sin‌
kx+kct+kx−kct
2
⋅cos‌
kx+kct−kx+kct
2
Y=2A‌sin‌k‌x⋅cos‌k‌c‌t For first antinode sin‌k‌x1=1 sin‌k‌x1=sin‌
Ï€
2
kx1=‌
Ï€
2
....(iii) For second antinode sin‌k‌x2=−1 sin‌k‌x2=sin‌
3Ï€
2
kx2=‌
3Ï€
2
......(iv) ∴ The distance between adjacent antinodes kx2−kx1=‌