Boundary Values & Fourier Series · Superchallenge problem

A string $0 \le x \le \pi$ with fixed ends obeys $u_{tt} = u_{xx}$. It is plucked at its midpoint into the triangle of height $1$ ($u(x,0) = \dfrac{2x}{\pi}$ on $\left[0, \dfrac\pi2\right]$, $\dfrac{2(\pi - x)}{\pi}$ on $\left[\dfrac\pi2, \pi\right]$) and released from rest. The energy $\dfrac12\displaystyle\int_0^\pi(u_t^2 + u_x^2)\,dx$ splits among the normal modes. What fraction of it is carried by the fundamental mode $\sin x\cos t$?