Suppose I have a PDE to describe the vibration $u(x,t)$ of a string of length $L$ with fixed ends: $$ \begin{cases} u_{tt} - a^2 u_{xx} = f(x,t), & 0<x<L, \, t>0, \\ u(0,t)=u(L,t)=0, & t \ge 0, \\ u(x,0)=\phi_0(x), \, u_t(x,0)=\phi_1(x), & 0\le x \le L. \end{cases} $$ Using separation of variables, I can find its general solution to the homogeneous equation: $$ u(x,t)=\sum_{k=1}^{\infty} \left( A_k \cos \frac{k \pi at}{L} + B_k \sin \frac{k \pi at}{L} \right) \sin \frac{k \pi x}{L}, $$ where $A_k$ and $B_k$ can be found by the initial conditions with the Fourier expansion.
I'm trying to understand how to create only one mode to have standing wave.
- When $\phi_0(x)=\phi_1(x)=0$, is there a way to describe a force (like a sine function) acting at $x=0$ of the string, and the wave will be traveling to the right? What is the solution formula?
Can I change the boundary to the sine-wave function of $t$ for one end to achieve this? I tried $A \sin \frac{k \pi at}{L}$ but the answer has many modes.
Should I switch back to d'Alembert formula?
- Is there a way to write a solution of the wave traveling to the right, hitting the other end, and coming back? I looked up, and it seemed like people just manually switch $A \sin(kx-wt)$ to $A \sin(kx+wt+\pi)$. Can we do it automatically in math (e.g., when $t$ is large, the function will change) by separation of variables method?
Thank you. I'm sorry I'm not good at physics at all.
EDIT: I tried to do C2 and C3 on page 3 as in this article: article