Boundary Values & Fourier Series · Superchallenge problem

A string on $0 \le x \le 2$ with fixed ends is made of two pieces joined at $x = 1$: the left piece obeys $u_{tt} = u_{xx}$ and the heavier right piece obeys $u_{tt} = \dfrac14u_{xx}$, and $u$ and $u_x$ are continuous at the joint. Find the fundamental (lowest) angular frequency $\omega$ of this string, rounded to four decimal places.