The equation of a wave gives the position of a particle that follows along the path of the wave at any moment of time. Take the equation of the progressive wave as y = f(t) where the position y is given as a function of time t. The velocity of a moving particle is the rate of change of its position or the displacement with respect to time. The instantaneous velocity is the first derivative of the position function with respect to time, v = dy/dt = f'(t). Acceleration is the rate of change of velocity or the seconds derivative of the position function, a = d^2y/dt^2 = f''(t).
To determine the maximum acceleration we need to find the derivative of acceleration with respect to time, equate it to 0 and solve the resulting equation. If the solution is t = a, the value of f'''(a) should be negative to ensure that the point is one where the acceleration is maximum. Once the value of t has been determined it can be replaced in the expression for the acceleration a = f''(t) to arrive at the maximum acceleration.