How Do Mass Attachments Affect Wave Reflection and Transmission on a String?

[piano.lisa]

Homework Statement

Consider an infinitely long continuous string with tension \tau. A mass M is attached to the string at x=0. If a wave train with velocity \frac{\omega}{k} is incident from the left, show that reflection and transmission occur at x=0 and that the coefficients R and T are given.
Consider carefully the boundary condition on the derivatives of the wave functions at x=0. What are the phase changes for the reflected and transmitted waves?

Homework Equations

i. R = sin^2\theta
ii. T = cos^2\theta
iii. tan\theta = \frac{M\omega^2}{2k\tau}
iv. \psi_1(x,t) = \psi_i + \psi_r = Ae^{i(\omega t - kx)} + Be^{i(\omega t + kx)}
v. \psi_2(x,t) = \psi_t, however, I do not know what this is.

** note ** \psi_i is the incident wave, \psi_r is the reflected wave, and \psi_t is the transmitted wave

The Attempt at a Solution

I am used to dealing with situations where the string is of 2 different densities, therefore, \psi_t will have a different value for k than \psi_i. However, in this case, the densities are the same on either side of the mass, and the only obstruction is the mass. If I knew how to find an equation for \psi_2(x,t), then I could potentially solve the rest of the problem.
Thank you.

 

[piano.lisa]

I still haven't reached any solution to my problem.

Any help is appreciated.

 

[v0id]

EDIT: Ignore this post. The result leads nowhere.

According to my calculations, this is true: \frac{d^2\psi_1}{dt^2}(0,t) = \frac{d^2\psi_2}{dt^2}(0,t)

Do you see why?

Hint: Apply Newton's (2nd?) Law to the central mass and find an expression for the net force on M

 

[v0id]

I realize this is due in about ~1/2 hour, but the second boundary condition is given by:
M\frac{d^2\psi_1}{dt^2} = M\frac{d^2\psi_2}{dt^2} = \tau \left( \frac{d\psi_1}{dx} - \frac{d\psi_2}{dx}\right) (0,t)