Physical Interpretation of the Initial Conditions for the Wave Equation

Question

Say you have the following wave eq:

$$ \frac{\partial^2\psi(x,t)}{\partial t^2} = \frac{W}{M}\frac{\partial^2\psi(x,t)}{\partial x^2}$$

In my textbook, it says that if you have the following initial conditions for travelling waves:

$$ \psi(x,0) $$

$$ \frac{\partial\psi(x,0)}{\partial t} $$

i.e., if you know the wave shape at t=0 and the (in this particular case) transversal velocity at t=0, that's enough to determine a unique solution. If you consider the wave as made up of two different travelling waves of equal but opposite speed $\psi_A(x-|v|t)$ and $\psi_B(x+|v|t)$, you can do the maths to find that the second equation implies:

$$ \frac{\partial\psi(x,0)}{\partial t} = |v|\left[\frac{d\psi_B}{dx}-\frac{d\psi_A}{dx}\right] $$

I understand the mathematics entireley, but I don't understand the physics of this result. How does knowing the transversal velocity at t=0 give a unique solution? How is the result obtained helpfull towards understanding how this happens?

EDIT: And how does this let us in any way shape or form determine the specific, unique solution given those boundary contitions?

Answer 1

Think of every point in space as a harmonic oscillator: specifying its initial position and velocity fully determines its solution (sinc ethe oscillator is described by a second order ODE, a general soluton of which has two unknown constants).