A string of linear density ρ terminates in a small ring of mass m. A horizontal bar is passed through the ring, perpendicular to the direction of the string at rest, and the string is stretched to a tension T. Friction between the ring and the bar generates a resistive force –bvy, where vy is the transverse velocity of the ring and b is a constant. A harmonic wave of angular frequency ω is directed along the string towards the ring. To what value of T should the string be stretched to minimise the energy flux in the reflected wave, and what is the energy reflection coefficient at this tension? (Assume that ω m/b <<1).
F = ma (Newton's Second Law)
y(x,t)i = Aei(ωt-kx) (mathematical description of incident wave)
y(x,t)r = Bei(ωt+kx) (mathematical description of reflected wave)
Energy Reflection Coefficient = Reflected Energy/Incident Energy
The Attempt at a Solution
My initial thoughts are, firstly, that there is a geometrical boundary condition: the displacement of the incident and reflected wave at the ring must be continuous:
yi(x=0) = yr(x=0) (letting the position of the ring be at x=0);
therefore, A=B. (?)
Secondly, the net transverse force acting on the ring must be the combined transverse components of the tension in the string from both the incident and reflected wave balanced by the friction between the ring and rod:
I tried substituting in the mathematical description of the incident and reflected wave, but it doesn't seem to rearrange into anything immediately useful that can be plugged into the power equation.
What do you think?