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:
$$M\frac{d^2\psi_1}{dt^2} = M\frac{d^2\psi_2}{dt^2} = \tau \left( \frac{d\psi_1}{dx} - \frac{d\psi_2}{dx}\right) (0,t)$$