Question #42873

A long, taut string of linear mass density $\mu$ and tension $T$ lies along the $x$-axis. A small bead of mass $M$ is fixed to the string at the origin ($x = 0$). Two harmonic waves are incident from the negative $x$-direction, given by the superposition $y_i(x,t) = A \cos(\omega_1 t - k_1 x) + A \cos(\omega_2 t - k_2 x)$, where $k_1$ and $k_2$ are the respective wave numbers. The angular frequencies are $\omega_1 = \frac{2T}{Mv}$ and $\omega_2 = \frac{2\sqrt{3}T}{Mv}$, where $v = \sqrt{\frac{T}{\mu}}$ is the wave speed. The waves are partially transmitted into the region $x > 0$. The transmitted wave can be written as $y_t(x,t) = A_1 \cos(\omega_1 t - k_1 x + \phi_1) + A_2 \cos(\omega_2 t - k_2 x + \phi_2)$, where $\phi_1$ and $\phi_2$ are the phase constants introduced at the boundary. Assuming the phase constants are chosen in the principal range $(-\pi, \pi]$, what is the magnitude of the relative phase shift $|\phi_1 - \phi_2|$ between the two transmitted components?