1-D Wave Equation  (String Vibrations)                   

  

 

Example:   Find the displacement, u(x,t), of the vibrating string given by the 1-D wave

                   equation

 

                             ∂2u/∂t2   =   4 ∂2u/∂x2       0  <  x  <  π,    t  >  0

 

     Subject to the boundary conditions       u(0,t)   =   u(π,t)  =  0

 

     Along with the initial conditions           u(x,0)  =  sin x     and     ∂u(x,0)/∂t   =   1           

 

 

Strategy:  Start by assuming separation of variables

 

                                   u(x,t)  =  X(x) T(t)

 

Substitution of this expression into the above wave equation gives

 

                            X(x) dT2/dt2  =  4 d2X/dx2

 

Then by division     (d2X/dx2)/X  =  (dT2/dt2)/ 4T   =   ˗  λ  =  separation constant

 

So                d2X/dx2  +  λ X  =  0     and    dT2/dt2 + 4 λT   =  0

 

 

Start with the eigenvalue problem for X(x).

 

                             d2X/dx2  +  λ X  =  0       subject to the boundary conditions

 

                            X(0)  =  X(π)  =  0

 

 

Case 1    λ = 0     d2X/dx2  =  0   or   X(x)  =  Ax + B

 

X(0) =  0  =  B  and   X(π)  =  Aπ =  0  so  A = 0   There are no eigenvalues for this case.

 

 

Case 2    λ < 0     Let  λ  =  ˗   α2  ,  α > 0

 

                 d2X/dx2  -  α2  X  =  0      

 

So    X(x)  =  A cosh αx  +  B sinh αx      and 

 

   X(0)  =  0  =  A,   X(π)  =  0  =  B sinh απ   now  sinh απ   ≠  0   so  B = 0

 

Result:  No eigenvalues for this case.

 

  

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