Past exam of the mathematical methods II course of the University of Oxford / 2014 resit / 10 / a / Solution

Interpret “held” and then “released” as zero initial velocity. In the small-slope wave equation on a string model, the elastic energy is $U=(T/2){\int }_0^L{y}_x^{2}dx$ and the initial kinetic energy is zero. The given profile hasThe two cosine functions are orthogonal on $[0,L]$ and each has squared integral $L/2$. ThusThe implicit leading displacement amplitude is one in the source's chosen length units. If that amplitude is written explicitly as $A$, all energies here and below acquire a factor $A^2$.