- #1

- 117

- 15

## Homework Statement

Suppose you send an incident wave of specified shape, ##g_I(z-v_1t)##, down string number 1. It gives rise to a reflected wave, ##h_R(z+v_1t)##, and a transmitted wave, ##g_T(z-v_2t)##. By imposing the appropriate boundary conditions [see below], find ##h_R## and ##g_T##.

## Homework Equations

The appropriate boundary conditions as given in the book:

##f(0^-,t)=f(0^+,t)## and ##\frac{\partial f}{\partial z}|_{z=0^-} = \frac{\partial f}{\partial z}|_{z=0^+}##.

## The Attempt at a Solution

The net displacement of the strings is ##f=g_I(z-v_1t)+h_R(z+v_1t)## for ##(z<0)## and ##f=g_T(z-v_2t)## for ##(z>0)##. Using the first boundary condition: ##g_I(-v_1t)+h_R(v_1t)=g_T(-v_2t)##. For the second condition, I think that the definition of a derivative as a limit is needed. Thus, [tex]\lim_{\Delta z→0}\frac{g_I(-v_1t)-g_I(-v_1t-\Delta z) + h_R(v_1t)-h_R(v_1t-\Delta z)}{\Delta z} = \lim_{\Delta z→0}\frac{g_T(-v_2t-\Delta z)-g_T(-v_2t)}{\Delta z}[/tex]

Rearranging and plugging in the first boundary condition, [tex]\lim_{\Delta z→0}\frac{2g_T(-v_2t)-g_I(-v_1t-\Delta z)-h_R(v_1t-\Delta z)-g_T(-v_2t-\Delta z)}{\Delta z}=0[/tex]

Now I'm stuck. I tried to evaluate the last equation at different times and using the first constraint to attempt to eliminate either ##h_R## or ##g_T##. For instance, evaluating at ##t=\frac{\Delta z}{v_1}## and noting that ##h_R(0) = g_T(0)-g_I(0)## by the first constraint I got: [tex]\lim_{\Delta z→0}\frac{2g_T(\frac{v_2}{v_1}t)-g_T(0)-g_T(-\frac{v_2}{v_1}\Delta z-\Delta z)+g_I(0)-g_I(-2\Delta z)}{\Delta z} = 0[/tex]

I don't see where this leads me and the same happens when I try to evaluate at different times. Any ideas on how to progress from here?

Any suggestions/comments will be greatly appreciated!