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A ring on an vibrating string

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).


Solution Preview

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 ...

Solution Summary

The physics of this problem is explained in detail. The problem is reduced to straightforward algebra.