- #1

piano.lisa

- 34

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## Homework Statement

Consider an infinitely long continuous string with tension [tex]\tau[/tex]. A mass [tex]M[/tex] is attached to the string at x=0. If a wave train with velocity [tex]\frac{\omega}{k}[/tex] 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.**[tex]R = sin^2\theta[/tex]

**ii.**[tex]T = cos^2\theta[/tex]

**iii.**[tex]tan\theta = \frac{M\omega^2}{2k\tau}[/tex]

**iv.**[tex]\psi_1(x,t) = \psi_i + \psi_r = Ae^{i(\omega t - kx)} + Be^{i(\omega t + kx)}[/tex]

**v.**[tex]\psi_2(x,t) = \psi_t[/tex], however, I do not know what this is.

** note ** [tex]\psi_i[/tex] is the incident wave, [tex]\psi_r[/tex] is the reflected wave, and [tex]\psi_t[/tex] 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, [tex]\psi_t[/tex] will have a different value for k than [tex]\psi_i[/tex]. 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 [tex]\psi_2(x,t)[/tex], then I could potentially solve the rest of the problem.

Thank you.