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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).

Please provide a 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).

See the attached file.


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.