D'Alembert Formula

Hi,

I'm having trouble with this question:

I know that D'Alembert's solution is

u(x,t) = F(x-ct) + G(x+ct)

, but I get stuck shortly after starting the second part.

Can anyone help? I'd really appreciate it.

I'm going to change your

u(x,t) = F(x-ct) + G(x+ct)

u(x,t) = A(x-ct) + B(x+ct)

, since there are already

F

and

G

in the question. Just makes things clearer :smile:

From initial conditions you have

u(x,0)=A(x)+B(x)=F(x)

and

u_t(x,0)=-cA'(x)+cB'(x)=G(x)

.

Differentiating the first equation wrt x gives

A'(x)+B'(x)=F'(x)

.

You can use this, together with the penultimate equation above, to find

A

and

B

in terms of

F

and

G

.

Hope this helps :smile:

(edited 12 years ago)

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