Vibrating string problem

[jtrol]

the problem is: Solve the vibrating string problem when there is damping proportional to the velocity. With 0 initial velocity and initial displacement function f(x), the boundary value problem is d^2u/dt^2=(c^2)(d^2u/dx^2)-a(du/dt). 0<x<L, t>0. u(0,t)=u(L,t)=0, u(x,0)=f(x), du/dt(x,0)=0

I assumed u(x,t) would take the form F(x)G(t), substituted this form into the displacement eqn and separated the variables to get F''(x)/F(x)=-k^2=G''(t)+aG'(t)/c^2k^2G(t).

I then rearranged to get F''(x)+k^2F(x)=0 which i know how to solve

Also, I got G''(t)+aG'(t)+(ck)^2G(t)=0
this appears to be second order linear homogeneous with constant coefficients, but I don't know the value of the coefficients so I don't know what form the soln of this eqn needs to be in.

 

[HallsofIvy]

Just using the coeffients as you are given them, the "characteristic equation" for this is \(\displaystyle r^2+ ar+ (ck)^2= 0\).

By the quadratic formula, that has roots \(\displaystyle \frac{-a\pm\sqrt{a^2- 4c^2k^2}}{2}\). The kind of solution to that equation, and so the solution to the differential equation, will depend upon the "discriminant", \(\displaystyle a^2- 4c^2k^2\). If \(\displaystyle a^2- 4c^2k^2> 0\) the characteristic equation has two real roots, \(\displaystyle r_1\) and \(\displaystyle r_2\) and so the general solution to the differential equation is of the form \(\displaystyle G(t)= C_1e^{r_1t}+ C_2e^{r_2t}\). This iis the "over damped" case in which the high damping causes the solution to go to 0 without any "vibration". If \(\displaystyle a^2- 4c^2k^2< 0\), the characteristic equation has two conjugate solutions, \(\displaystyle a\pm bi\) and the general solution to the differential equation is of the form \(\displaystyle G(t)= e^{at}(C_1cos(bt)+ C2sin(bt)\). This is the "under damped" case where we have some vibration before the solution damps to 0. Finally, if \(\displaystyle a^2- 4c^2k^2= 0\), the characteristic equation has a double root, r, so the general solution is of the form \(\displaystyle G(t)= C_1e^{rt}+ C_2te^{rt}\). This is the "critically damped case" where the string moves just once past 0, then immediately damps to 0.