Solving Klein Gordon’s equation

[Safinaz]

Homework Statement

I try to solve Klein Gordon’s equation for specific boundary and initial conditions

Relevant Equations

The Klein Gordon’s equation for a masses scalar is given by :
$\left( \frac{\partial^2}{\partial t^2} - \frac{\partial^2}{\partial x^2} \right) \phi (x, t) = 0$………(1)

My solution:

Let $\phi (x, t) = F(x) A(t)$, then Eq. (1) becomes

$\frac{1}{A(t)} \frac{\partial^2}{\partial t^2} - \frac{1}{F(x)} \frac{\partial^2}{\partial x^2} = 0$

So that : $\frac{\partial^2}{\partial t^2} = k^2 ~A (t)$, and $\frac{\partial^2}{\partial x^2} = k^2 ~F (x)$.

Leads to :
$\phi(t,x) = ( c_1 e^{kt} + c_2 e^{-kt} ) ( c_3 e^{kx} + c_4 e^{-kx} )$

Assuming BC and IC :

$bc={\phi[t,0]==1,(D[\phi[t,x],x]/.x->Pi)==0}$
$ic={\phi[0,x]==0,(D[\phi[t,x],t]/.t->0)==1}$

BC leads to $c_3 = c_4= 1/2$ and the IC leads to to $c_1=- c_2= 1/2$.

Ending up by :
$\phi(t,x) = \frac{1}{4} e^{k(t-x)} - \frac{1}{4} e^{-k(t-x)}……… (2)$

So any help are these steps correct till Eq. (2) ? And how to determine $k$?

 

[Frabjous]

You just wrote down the scalar wave equation, not the Klein-Gordon equation.