Hi,
I'm having a bit of trouble understanding D'Alembert's solution to the wave equation.
The Problem
Or, in short notation:
Find subject to the initial conditions:
The Solution (the part I understand)
The first stage is to express the equation in characteristic coordinates, which, once performed, gives:
Here, and .
Integrating the equation w.r.t both variables gives us the simple general solution:
Or, back in cartesian coordinates:
(1)
Now, differentiating this, we get:
(2)
Applying the initial conditions to equations (1) and (2), we get:
(3)
(4)
The Solution (part I don't understand)
Now, the book I am using then makes the following statement:
If we integrate equation (4), we get:
To be honest, this is the only part I don't understand. Where do the other terms which results from the integration disappear to? i.e., by my working, I would get:
How do they manage to conclude that ??
From (1) you can see that F1 and F2 are both functions of x and t. Therefore when integrating equation 4 wrt to x the resulting constants of integration must be a function of t.
I have attached my version using your initial conditions. This should help you find the general solutio u(x,t)