# Physical interpretation of Neumann boundary conditions for wave equation on a disk?

If you take consider the wave equation on a disk $D$, then if we use Dirichlet boundary conditions, it means the wave function is fixed at $0$ on the boundary of the disk, and if we consider the eigenvalues of the system, this can be visualized physically as a vibrating drumhead.

If we use Neumann boundary conditions then the normal derivative of the wave function is fixed at $0$. How can we visualize this physically? The eigenvalues for the Dirichlet boundary conditions on a disk represented a vibrating drum. What physical phenomenon do the eigenvalues for the Neumann boundary conditions on a disk represent?

The Neumann boundary condition implies that no momentum can flow off the disk, while Dirichelet boundary conditions stop motion at the edge, allowing momentum to be exchanged. In a drum, momentum can flow off the skin and Vibrational energy can be transported to the wooden walls of the drum.

I am not entirely sure, but the surface of an ideal liquid in a cylindric container seems to be a disk fulfilling Neumann boundary conditions.

The Neumann boundary condition is rarely relevant on drums or other membrane-based instruments, since it implies the membrane is not tightly fixed to the frame, which makes it very hard to produce a meaningful sound on that instrument. The same argument also goes for stringed instruments.

More commonly it's used to describe wind instruments, where it signifies that the end is open and that the mean displacement of air particles on that end during tone production is at a local maximum, i.e. $\partial_x \psi(0,t)=\partial_x \psi(L,t)=0$, as illustrated in the picture below (taken from here), which is exactly the Neumann boundary condition, as opposed to the Dirichlet condition which is equivalent to a closed pipe end.

This, of course, is only illustrated in the case of one dimension, but similar reasoning applies to the more general 3D case. Also, a similar picture holds for membrane instruments, but, as I said, they are mainly bound by the Dirichlet condition, so my answer focuses on wind instruments.

The Neumann boundary condition is also relevant for vibrating rigid (elastic) plates (rather than membranes) that are clamped at the centre point and free to vibrate at the edges. Such vibrating discs or plates produce the well-known Chladni patterns.