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## Homework Statement

I have a general wave equation on the half line

u

_{tt}-c

^{2}u

_{xx}=0

u(x,0)=α(x)

u

_{t}(x,0)=β(x)

and the boundary condition;

u

_{t}(0,t)=cηu

_{x}

where

**α**is α extended as an odd function to the real line (and same for

**β**)

I have to find the d'alembert solution for x>=0; and show that in general it doesn't exist for η=-1

## Homework Equations

The d'alembert solution is

u(x,t)=1/2(

**α**(x+ct)+

**α**(x-ct))+1/2c [itex]\int[/itex]

^{x+ct}

_{x-ct}

**β**(y) dy

for x>ct

## The Attempt at a Solution

I know that to restrict it to the whole of the x>=0, t>=0 region, I need to use the boundary condition; but I get that

u(0,t)=0 because

**α**and

**β**are odd, which makes

**α**(ct)+(-ct)

and the integral from -ct to ct of

**β**(y) zero; and so u_t(0,t) is zero which is supremely not useful...