A stretched string of length carries transverse waves at speed and obeys the wave equation

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 stretched string of length L=1.2 mL = 1.2\,\mathrm{m} carries transverse waves at speed c=150 m s−1c = 150\,\mathrm{m\,s^{-1}} and obeys the wave equation ∂2y/∂t2=c2 ∂2y/∂x2\partial^2y/\partial t^2 = c^2\,\partial^2y/\partial x^2. Both ends are tied down, y(0,t)=y(L,t)=0y(0,t) = y(L,t) = 0. ((a)) Substitute y(x,t)=X(x)G(t)y(x,t) = X(x)G(t), separate the equation, and write the two ordinary equations in terms of a separation constant λ\lambda. ((b)) Impose the boundary conditions and give the allowed frequencies f1f_1, f2f_2, f3f_3 (the three lowest).

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