Trying to start my homework and stuck on this first problem...
A point mass M is concentrated at a point on a string of characteristic
impedance pc. A transverse wave of frequency w moves in the positive x direction and is partially reflected and transmitted at the mass. The boundary conditions are that the string displacements just to the left and right of the mass are equal and that the difference between the transverse forces just to the left and right of the mass equal the mass times its acceleration.
Then the problem goes on to stating variables for the wave amplitudes and how I have to show that the reflected amplitude ratio and the transmitted amplitude ratio equal these values that it set.
Okay, the problem I seem to be having with this problem is the part where it says "the difference between the transverse forces just to the left and right of the mass equal the mass times its acceleration". As I tried to work out the boundary conditions, it seems to be to be saying that T(d/dx(yi + yr)) - T(d/dx(yt)) = Ma where T is tension, d/dx is partial differentation with respect to x, yi is the incident wave, yr is the reflected wave, yt is the transmitted wave and a is acceleration of the mass. The part which I seem to be stuck on is the Ma part. How exactly am I suppose to determine what the acceleration of the mass is in terms of the wave equations? It can't just be the second derivatives of my wave equations since those are the accelerations of the waves themselves and not of the mass. Is there some
relation here that I'm missing or am I totally off in writing down the equation for the boundary condition?
