We next consider a (damped) beam equation:

\[\pdd{u}{t}+ C\pd{u}{t}=-D^2 \frac{\partial^4 u}{\partial x^4} u-Q,\]

initially with fixed boundary conditions given by

\[u=0 \quad \text{and} \quad \pdd{u}{x} = 0\]

along the boundary. The constant $D$ represents the relative size of the domain and its material properties (e.g. stiffness), $C>0$ is a damping constant, and $q>0$ a gravity-like force.

  • Load the interactive simulation.

  • Click to push down on the beam, creating a localised depression which creates ripples in the beam nearby.

  • Now go to  Boundary conditions and select Neumann for $u$, $v$, and $w$. This will use the following ‘free end’ boundary conditions:

\[\pdd{u}{x}=0 \quad \text{and} \quad \frac{\partial^3 u}{\partial x^3} = 0.\]

Deformable plates

Let’s next look at the (damped) plate equation,

\[\pdd{u}{t}+ C\pd{u}{t}=-D^2 \nabla^4 u-Q,\]

with fixed boundary conditions given by

\[u=0 \quad \text{and} \quad \nabla^2 u = 0\]

along the boundary.

  • Load the interactive simulation.

  • Initially the plate is deformed to a value of $u=-4$ everywhere, representing an initial deformation which instantaneously snaps to the fixed condition of $u=0$ at the edges, which gives rise to compression waves which propagate inwards.

  • Click to compress the plate downward locally, and observe waves propagating from this disturbance.

Numerical notes

As in previous examples, we must write the second time derivative using a system of first-order equations. We also have to use an algebraic equation to represent the biharmonic term:

\[\begin{aligned}\pd{u}{t}&=v+DD_c\nabla^2 u,\\ \pd{v}{t} &= -D \nabla^2 w -Cv -Q,\\ w &= D \nabla^2u, \end{aligned}\]

which is the plate equation for $D_c=0$. The parameter $D_c$ is used to prevent spurious oscillations as seen in the wave equation.

3D deformations

  • Load the interactive simulation and click! This is the same simulation as in the 2D plate equation above, but shown in 3D with a view that can be rotated.