Reflection and transmission wave on three joined strings
Suppose we have a system of three joined strings, of different linear mass densities and subjected to a constant tension force, such that the velocity of propagation is $v_1$ in the string 1 (at $x <0$) and string 3 (at $x> L$), and $v_2$ on string 2 (from $x = 0$ to $x = L$). A plane wave of wave number $k_1$, frequency $ω$, and amplitude $A$ propagates to the right on string 1.
We have to calculate the reflected wave's final amplitude $A_r$ (on string 1), and the transmission wave amplitude $A_t$ on string 3, supposing that at first there is only one incident wave moving towards the positive X axis on string 1. 453