Incident, reflected, transmitted waves

[v_pino]

Homework Statement

Suppose you send an incident wave of specified shape, g_I(z - v_1 * t ) , down string
number 1. It gives rise to a reflected wave, h_R(z + v_1 *t ) , and a transmitted wave,
g_T(z - v_2 *t). By imposing the boundary conditions, find h_R and g_T.

Homework Equations

I know that the boundary conditions are such that the first derivative and second derivative of the sum of the waves on one side is equal to the of the other side.

The Attempt at a Solution

I know that from boundary condition 1, gI(-v_1 *t) + h_R(v_1 *t) = g_T(-v_2 *t).

How do I proceed from this? The hint given is that dg_I/dz = (-1/v_1)*(dg_I / dt). I've tried chain rule with that but I can't get the minus sign.

 

[kuruman]

Are you saying that you cannot derive the hint? It is indeed an application of the chain rule. Let u = z - v₁t and use \frac{\partial g}{\partial t}=\frac{\partial g}{\partial u}\frac{\partial u}{\partial t}=-v_1\frac{\partial g}{\partial u} and similarly with respect to z then put it together.