Bending beams and deforming plates
We next consider a (damped) beam equation:
\[\pdd{u}{t}+ C\pd{u}{t}=D^2 \frac{\partial^4 u}{\partial x^4} uQ,\]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 gravitylike 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 and select Neumann for $u$, $v$, and $w$. This will use the following ‘free end’ boundary conditions: → Boundary conditions
Deformable plates
Let’s next look at the (damped) plate equation,
\[\pdd{u}{t}+ C\pd{u}{t}=D^2 \nabla^4 uQ,\]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 firstorder 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.