Wave on a string / impedance problem

[Darren93]

We have looked fairly extensively at waves on a string without anything on them, however there is a portion in my notes about an infinite string with a mass in the middle. Essentially the setup is that x=0 we have a mass. Here the waveform on each side must be equal as the string is continuous, that I get. However then we look at force considerations and we seem to place a negative sign on the B term and I have no idea where that comes from. Does anybody see why that would be in the attached picture?

 

[UVCatastrophe]

boundary conditions -- waves are 2nd order partial differential equations so you need two boundary conditions, on f(x) and f'(x). the minus sign is from a derivative. you'll need to invoke none other than Newton's 2nd law for the mass in between.

your professor assigned this problem to teach you about boundary conditions. one always requires that the function remain continuous across a boundary, but the derivative may be discontinuous. the analogous problem in quantum mechanics (which you will certainly encounter, if you learn quantum mechanics) is the delta function potential.