Problems

01

Traveling waves

Verify that the functions $y(x,t)=\ell (x+vt)$ (left-traveling wave) and $y(x,t)=r(x-vt)$ (right-traveling wave) solve the wave equation. Here $\ell (u)$ and $r(u)$ are arbitrary ‘shape functions’ for these waves. Verify that the left-traveling wave travels to the left, and the right-traveling wave travels to the right. How fast do they go?

02

Plucked band

A rubber band fixed at $x=0$ and $x=\pi$ is plucked by stretching the center point to a height of $1$ and releasing it. At the point of release, the band shape is an isosceles triangle with base width $\pi$ and height $1$.

Find the function $y(x,t)$ that describes the future evolution of the band, written as a sum of normal modes. Is $y(x,t)$ a standing wave? (If yes, prove it; if no, demonstrate that it cannot be one.)

You should assume the band satisfies the BVP given by $v^2\frac{{\partial }^{2}y}{\partial x^2}=\frac{{\partial }^{2}y}{\partial t^2}$ and $y(0,t)=0=y(\pi ,t)$ for all $t$. To find your solution $y(x,t)$, first write down the initial shape function $y(x,0)=F(x)$ at the moment of release, and then compute its Fourier series in $x$. (Review and modify your work in Problem 14-01.) Any solution is a sum of normal modes. Using the theory of orthogonality of Fourier terms, solve for the coefficients of the normal modes in your solution.