A string of length carries transverse waves at speed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Oscillations, Waves & Acoustics · Superposition, boundaries and interference

Mathematical Methods for Physics, Ch Separation of variables in Cartesian, cylindrical and spherical geometry

Problem

A string of length L=0.9 mL = 0.9\,\mathrm{m} carries transverse waves at speed c=180 m s−1c = 180\,\mathrm{m\,s^{-1}}. It is tied down at x=0x=0, so y(0,t)=0y(0,t)=0, but its other end at x=Lx=L is attached to a frictionless ring that can slide freely along a transverse rail, so that no transverse force acts there and ∂y/∂x(L,t)=0\partial y/\partial x(L,t) = 0. ((a)) State the two ordinary equations obtained by separating the wave equation with y=X(x)G(t)y = X(x)G(t), using a separation constant λ\lambda. ((b)) Find the allowed values of λ\lambda from the two boundary conditions, and give the three lowest allowed frequencies f1f_1, f2f_2, f3f_3.

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