Homework

1. The Lagrange density for a string of mass density $\mu$ and tension $\tau$ is ${\cal L}(y,\dot{y},y';x,t)={1\over 2}\mu\dot{y}^2-{1\over 2}\tau y'^2$. What is the momentum conjugate to the coordinate $y(x)$? Calculate the Hamiltonian density. What is the Lagrange equation?
2. Transform the wave equation in $y(x,t)$: $\tau y''-\mu\ddot{y}=0$, into an equation for $y(u,v)$ where $u=x-ct$ and $v=x+ct$. What should $c$ be to simplify the equation? What is the most general solution for $y(u,v)$?
3. If we pluck a fixed end string of length $L$ by pulling the point at $x={L\over 4}$ to a height $h$, $(y({L\over 4},0)=h)$, with the rest of the string forming two lines meeting at that point, what is the formula for $y(x,t)$ at a later time?