Exercises

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? Explain.

02

Divided wave

Using the solution form provided in the example, characterize the future evolution of the wave with initial condition $y(x,0)=f(x)$ and $y_t(x,0)=0$ with $f(x)=\{\begin{array}{ll}2& x\in [-1,1]\\ 0& \text{else}.\end{array}$ Draw a picture of your answer.