Eigenvalues of a Structurally Damped Wave Equation
Analyze the stability of solutions of a partial differential equation by examining its eigenvalues. All eigenvalues of a stable system have negative real parts.
Compute the first 100 values of and such that on the unit disk and on the unit circle.
Use the preceding solutions to solve the structurally damped wave equation with on the unit disk by looking for solutions of the form . Smaller values of correspond to solutions that decay more quickly.
Visualize the effect of the damping parameter on . Values of accumulate at , and if , then there are nonreal eigenvalues on the circle of radius centered at .