4 Homogeneous Wave PDE in 1D (plugged string)

 4.1 Boundary conditions both homogeneous. One end Neumann, other end Dirichlet
 4.2 Boundary conditions both homogeneous. One end Neumann, other Dirichlet. Damping present.

4.1 Boundary conditions both homogeneous. One end Neumann, other end Dirichlet

Solve, 0 < x < L

∂2u-= c2∂2u-   0 < x < L,t > 0
∂t2     ∂x2

Boundary conditions, t > 0

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Initial conditions, t = 0

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Use c = 4,h = 0.1,L = 1.

Solution,      ---24h---   ( (2n−1) )      (2n−-1)π
Cn = ((2n−1)π)2 sin    6  π ,λn =   2L

        ∑∞
u(x,t) =   Cn cos(λnct)sin(λnx)
        n=1

  

4.2 Boundary conditions both homogeneous. One end Neumann, other Dirichlet. Damping present.

Solve, 0 < x < L

∂2u    ∂u     ∂2u
--2-+ b---= c2 --2-   0 < x < L,t > 0
∂t     ∂t     ∂x

Boundary conditions, t > 0

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Initial conditions, t = 0

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Use c = 4,h = 0.1,L = 1.  Consider three cases for damping, b = 0.5πLc,b = πcL ,b = 2πcL

Solution case (underdamped) b = 0.5πLc  .

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        ∞∑        (                     )
u(x,t) =    Cne− b2t  cos(βnt)+  -b-sin (βnt)  sin(λnx)
        n=1                  2βn

  

Solution case (critical damped)

b = πLc  .

           (            )                    (                     )
             −b2 t b  −b2 t          ∑∞     − b2t          -b--
u (x,t) = C1 e   + 2 te     sin(λ1x)+    Cne     cos(βnt)+ 2βn sin(βnt) sin(λnx)
                                   n=2

  

Solution case (overdamped damped)

     πc
b = 2 L  . First mode only overdamped, rest underdamped.

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