DanielO_o said:
I have to solve the 1-dimensional wave equation using D'Alembert's solution.
I know the solution has the forum 1/2[f(x-ct) + f(x+ct)] + 1/2C*integral(g(s)ds)
and i have two intial conditions f(x) and g(x)... do i just plug in (x-ct) where i see x etc...?
also what is g(s) in the integral?
Thanks
Dan
You've written this very confusingly! You have f used in at least two different ways and g used in two different ways.
What you are giving as the "D'Alembert solution" is too specialized.
The
general D'Alembert solution to the wave equation is [itex]\phi (x,t)= F(x+ ct)+ G(x- ct)[/itex] where F and G can be
any two twice differentiable functions of a single variable. Then [itex]\phi_t(x,t)= cF'(x+ct)- cG'(x- ct)[/itex].
If one initial condition is [itex]\phi_t(x,0)= 0[/itex], then it follows that [itex]F'(x)= G'(x)[/itex] so F and G differ by a constant- in that case, [itex]\phi (x,t)= (1/2)(F(x+ct)+ F(x-ct))+ constant[/itex].
But that is not the case here. (It is also confusing to say just "initial values f(x) and g(x)" since functions are not "conditions". I assume you mean [itex]\phi (x,0)= f(x)[/itex] and [itex]\phi_t(x,0)= g(x)[/itex].)
Then you have [itex]\phi (x,0)= F(x)+ G(x)= f(x)[/itex] and [itex]\phi_t(x,0)= cF'(x)- cG'(x)= g(x)[/itex]. Solve those two equations for F and G.