We put a small ring of mass m on the long (infinite) string. The ring slides without friction. How does the ring move, when we send by a string the sine wave? (I need mathematical equation of moving.) How does the wave scatter on the ring? (I need the amplitude of scattering waves, or reflection coefficient R).
I need mathematical procedure (differential equations) -some steps to show me how can we find the solution of this problem. And also some limits (when m goes against 0 or infinity) are welcome (not necessary).
Let's denote the local height of the string as y(x). The derivative of y(x), dy/dx, is assumed to be small and this is thus approximately the angle of the string with the positive x-direction.
If you forget about the ring for a moment, you should have no difficulties deriving the wave equation for the string. If you focus on a small piece of the string at position x of length dx, then the force exerted by the on this piece is:
T [dy/dx(x+dx)-dy/dx(x)]= T d^2y/dx^2 dx (1)
where T is the tension of the string.
By Newton's second law this must equal
rho dx d^2y/dt^2 (2)
where rho is the ...
The physics of this problem is explained in detail. The problem is reduced to straightforward algebra.