1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Inhomogeneous damped wave equation

  1. Dec 9, 2014 #1
    1. The problem statement, all variables and given/known data
    [tex]u_{tt}(x,t)+2u_{t}(x,t)-u_{xx}(x,t)=18\sin\left(\dfrac{3\pi x}{l}\right)[/tex]
    [tex]\omega=\frac{\pi n c}{l}[/tex]
    Boundary conditions
    Initial conditions
    2. Relevant equations

    The general inhomogeneous damped wave equation is
    [tex]u_{tt}(x,t)+2\mu u_{t}(x,t)-c^{2}u_{xx}(x,t)=p(x,t)[/tex]
    3. The attempt at a solution
    By separation of variables the position dependent part is
    and the time dependent part auxilliary equation is
    which takes one of three expressions depending on the sign of [itex]\mu-\omega[/itex]

    So from what is given I would say [itex]\mu=1[/itex] and [itex]c^2=1[/itex] and with
    [itex]\omega=\frac{\pi n c}{l}[/itex] and [itex]l<\pi[/itex] then [itex]\mu<\omega[/itex] which is weak damping

    [tex]g_{n}(t)=\left(A_{n}\cos\left(\Omega t\right)+B_{n}\sin\left(\Omega t\right)\right)e^{-\mu t}[/tex]

    where [tex]\Omega=\sqrt{\omega^{2}-\mu^{2}}[/tex]

    Is this correct so far?
  2. jcsd
  3. Dec 9, 2014 #2


    User Avatar
    Staff Emeritus
    Science Advisor

    I think you're on the right track, but separation of variables is mostly useful for homogeneous partial differential equations. Here's a trick to solving the inhomogeneous case:

    You want a solution to [itex]u_{tt}(x,t)+2u_{t}(x,t)-u_{xx}(x,t)=18\sin\left(\dfrac{3\pi x}{l}\right)[/itex]

    Try writing [itex]u(x,t) = A(x,t) + B(x)[/itex] and choose [itex]A[/itex] and [itex]B[/itex] so that:


    [itex]-B_{xx}(x) = 18\sin\left(\dfrac{3\pi x}{l}\right)[/itex]

    Then the approach you're describing will give you a solution to [itex]A(x,t)[/itex]
  4. Dec 10, 2014 #3
    Hi, thanks, that's great but we are asked to solve the homogeneous part by separation of variables first. Is what I have done ok? Is it weak damping?
  5. Dec 10, 2014 #4


    User Avatar
    Staff Emeritus
    Science Advisor

    Yes, you have the solution to the homogeneous equation (except for the constants [itex]A_n[/itex] and [itex]B_n[/itex]). Comparing with your original equation, [itex]\mu = 1[/itex] and [itex]c=1[/itex]
  6. Dec 10, 2014 #5
    Thanks, I wanted confirmation I had the right damping.
  7. Dec 18, 2014 #6
    So with the boundary conditions the general solution of the homogeneous equation is
    [tex]u(x,t)=\sum_{n=1}^{\infty}b_{n}\sin(kx)\left(A_{n}\cos\left(\Omega t\right)+B_{n}\sin\left(\Omega t\right)\right)e^{- t}[/tex]
    My course text is a bit confusing but to get the particular solution should I start with
    [tex]u(x,t)=\sin\left( k_nx \right)g(t) [/tex] and then solve
    [tex]g^{\prime\prime}+2g^\prime+\omega^2_ng=18\sin\left( \frac{3\pi x}{l} \right)[/tex]
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted