# Physical Interpretation of the Initial Conditions for the Wave Equation

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?

• Also $\frac{\partial \psi(x,t)}{\partial t}|_{t=0}$ is NOT the wave transversal velocity. This is what you say in the question. Jan 15, 2022 at 14:54