I am having an insane amount of trouble figuring out this problem that I solved probably ten years ago. Googling leads to solutions that make use of D'Alembert's formula, but that doesn't work for this.
I have a semi-infinite string with a frequency-dependent velocity (that might be complex). The initial condition is that the string is flat with no velocity anywhere. I want to grab hold of one of the ends and DISPLACE that end according to a sinusoidal function with frequency $\omega_0.$
So here are my equations:
Wave equation: $-\omega^2\hat{\psi}(x,\omega)=v^2(\omega)\frac{\partial^2 \hat{\psi}(x,\omega)}{\partial x^2}.$
Initial Conditions: $\psi(x,0)=0$ and $\partial_t \psi(x,0)=0.$
Boundary Condition: $\psi(0,t)=a \sin(\omega_0 t).$
Domain: $x\geq 0$ and $t\geq 0.$
My approach is to find the general solution for the differential equation (which has two unknown functions in terms of omega) and the fourier transformed versions of the initial and boundary conditions. The hope is to be able to find the fourier transformed solution for $\hat{\psi}$ and numerically find the inverse transform. However, this is leading me to either finding $\hat{\psi}(x,\omega)=0$ which violates the boundary condition or finding solutions which violate my initial conditions. Any advice on how to proceed? Thanks!