Drumhead

{ DRUMHEAD.PDE }

{ *******************************************************************

   This example illustrates the use of FlexPDE in Eigenvalue problems, or

   Modal Analysis.

   *******************************************************************

   The two-dimensional initial-boundary value problem associated with the

   scalar wave equation can be written as

       c^2*del2(u) - dtt(u) = 0

   with accompanying initial values and boundary conditions

       u = f(s,t)              on S1

       dn(u) + a*u = g(s,t)    on S2.

   If we assume that solutions have the form

       u(x,y,t) = exp(i*w*t)*v(x,y)

   then the equation becomes

       del2(v) + lambda*v = 0

   with lambda = (w/c)^2, and with boundary conditions

       v = 0                   on S1

       dn(v) + a*v = 0         on S2.

   The values of lambda for which this system has a non-trivial solution

   are known as the eigenvalues of the system, and the corresponding solutions

   are known as the eigenfunctions or vibration modes of the system.

   In this problem, we determine the eight lowest-energy vibrational modes of

   a circular drumhead, clamped on the periphery.

   This problem can be solved analytically.  The solutions are of the form

       v = Jn(r*jnm)*exp(i*n*theta),

   where Jn is the Bessel function of order n,

         jnm is the m-th root of Jn.

   The eigenvalues are then just the sequence of jnm^2. In increasing order, or

       5.7832, 14.682, 14.682, 26.375, 26.375, 30.471, 40.706, 40.706

   With errlim=0.001, FlexPDE in the current test gives

       5.7832, 14.682, 14.682, 26.377, 26.377, 30.476, 40.718, 40.720

}

title "Vibrational modes of a drumhead"

select

    modes=8            { -- Define the number of vibrational modes desired.

                            The appearance of this selector tells FlexPDE

                            to perform an eigenvalue calculation, and to

                            define the name LAMBDA to represent the eigenvalue }

Variables

    u

definitions

    k = 1

equations             { -- the eigenvalue equation }

    div(K*grad(u)) + lambda*u   = 0

boundaries

    Region 1

       start(0,-1)

       value(u) = 0

       arc(center=0,0) angle 360

monitors              { -- repeated for all modes }

    contour(u)

plots                 { -- repeated for all modes }

    contour(u)

    surface(u)

end