Equation of the wave is represented by:

y = A sin (ωt – kx)

Differentiating it with respect to t, we get

O = A cos (ωt – kx) [ω – k.dx/dt]

And the velocity of wave,

dx/dy = ω/k

Similarly the velocity of the particle

dx/dy = ωA cos (ωt – kx)

And the acceleration of a point particle is given by

d²y/dt² = ω² A sin (ωt – kx)    ….(i)

Differentiating the y with respect to x

dx/dy = -Ak cos (ωt – kx)

and d²y/dx² = -Ak² sin (ωt – kx)   …(ii)

From eqn. (i) and eqn. (ii) we get

d²y/dx² = v² d²y/dx²