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Inhomogeneous damped wave equation

  1. Dec 9, 2014 #1
    1. The problem statement, all variables and given/known data
    Solve
    [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
    [tex]u(0,t)=u(l,t)=0[/tex]
    [tex]l<\pi[/tex]
    Initial conditions
    [tex]u(x,0)=u_t(x,0)=0[/tex]
    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
    [tex]f(x)=a\cos(kx)+b\sin(kx)[/tex]
    and the time dependent part auxilliary equation is
    [tex]\lambda^2+2\mu\lambda+\omega^2=0[/tex]
    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

    stevendaryl

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    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]A_{tt}(x,t)+2A_{t}(x,t)-A_{xx}(x,t)=0[/itex]

    [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

    stevendaryl

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    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]
     
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