D'Alembert question - boundary conditions parts

[Ratpigeon]

Homework Statement

I have a general wave equation on the half line
u_tt-c²u_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 $\int$^x+ct_x-ctβ(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...

 

[gabbagabbahey]

Homework Statement

I have a general wave equation on the half line
u_tt-c²u_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 $\int$^x+ct_x-ctβ(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...

Are you sure you copied down the problem statement correctly? It would make a lot more sense if $\beta$ were an even function, and your boundary condition was $u_t(0,t)=c\eta u_x(0,t)$ for some constant $\eta$.