Mathematical Methods in the Physical Sciences · Problem 8.8

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · green-functions-wave-equation

Problem

If a stretched string vibrates under the action of a transverse force (per unit length) F(x,t)F(x,t), the wave equation for the displacement y(x,t)y(x,t) is ρ∂2y∂t2=T∂2y∂x2+F(x,t),\rho \frac{\partial^2 y}{\partial t^2} = T \frac{\partial^2 y}{\partial x^2} + F(x,t), where ρ\rho is the density of the string (mass per unit length) and TT is the tension. (Compare Chapter 13, Section 4, when F=0F = 0, v2=T/ρv^2 = T/\rho.) Let F(x,t)=−Tf(x)sin⁡ωtF(x,t) = -Tf(x)\sin\omega t, and y(x,t)=y(x)sin⁡ωty(x,t) = y(x)\sin\omega t to obtain (8.7) if, for simplicity, ω2/v2=ω2ρ/T=1\omega^2/v^2 = \omega^2\rho/T = 1.

Related theory

Browse the problem library