Hi,

I'm having a bit of trouble understanding D'Alembert's solution to the wave equation.

The Problem

$\displaystyle \displaystyle \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2}$

Or, in short notation:

$\displaystyle \displaystyle u_{tt} = c^2 u_{xx} $

Find $\displaystyle \displaystyle u(x,t) $ subject to the initial conditions:

$\displaystyle \displaystyle u(x,0) = f(x) $

$\displaystyle \displaystyle u_t(x,0) = g(x) $

The Solution (the part I understand)

The first stage is to express the equation in characteristic coordinates, which, once performed, gives:

$\displaystyle \displaystyle u_{\xi \eta} = 0 $

Here, $\displaystyle \displaystyle \xi = x + ct $ and $\displaystyle \displaystyle \eta = x-ct $.

Integrating the equation w.r.t both variables gives us the simple general solution:

$\displaystyle \displaystyle u(\xi, \eta) = F_1(\xi)+F_2(\eta) $

Or, back in cartesian coordinates:

$\displaystyle \displaystyle u(x,t) = F_1(x+ct) + F_2(x-ct) $ (1)

Now, differentiating this, we get:

$\displaystyle \displaystyle u_t(x,t) = cF_1'(x+ct) - cF_2'(x-ct) $ (2)

Applying the initial conditions to equations (1) and (2), we get:

$\displaystyle \displaystyle u(x,0) = F_1(x) +F_2(x) = f(x) $ (3)

$\displaystyle \displaystyle u_t(x,0) = cF_1'(x) - cF_2'(x) = g(x) $ (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:

$\displaystyle \displaystyle \int_a^x g(\tau)d\tau = cF_1(x) - cF_2(x) $

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:

$\displaystyle \displaystyle \int_a^x g(\tau)d\tau = \int_a^x (cF_1'(\tau) - cF_2'(\tau)) d\tau$

$\displaystyle \displaystyle = \biggg(cF_1(\tau)-cF_2(\tau)\biggg)^x_a $

$\displaystyle \displaystyle = cF_1(x) - cF_1(a) -cF_2(x) + cF_2(a) $

How do they manage to conclude that $\displaystyle \displaystyle cF_2(a) - cF_1(a) = 0 $??