Modal Structure of Plate Boundaries and Klein Bottle Reverberation

Modal Approach
Rectangles with Various Boundary Conditions
- Dirichlet, Neumann
- Periodic, Twisting (i.e., “Phase-conjugating”)
Using Topological Identification
Application to
- Physical Modelling of Reverberation on diverse objects
- where some of them are even Physically Impossible to Exist
- e.g., Klein Bottle, …

Rectangle with Dirichlet BC

Schematics

Quotient space

\phi_{m,n}(x,y)=\dfrac{2}{\sqrt{L_xL_y}}\sin\!\left(\dfrac{m\pi x}{L_x}\right)\sin\!\left(\dfrac{n\pi y}{L_y}\right)

Rectangle with Neumann BC

Schematics

Quotient space

\phi_{m,n}(x,y)=\sqrt{\frac{1}{L_x^{(2)}L_y^{(2)}}}\cos\!\left(\dfrac{m\pi x}{L_x}\right)\cos\!\left(\dfrac{n\pi y}{L_y}\right)

Torus

Schematics

Quotient space

\phi_{m,n}(x,y)=\dfrac{1}{\sqrt{L_xL_y}}\exp\!\left[2\pi i\!\left(\dfrac{mx}{L_x}+\dfrac{ny}{L_y}\right)\right]

\phi_{m,n}(x,y)=\phi_{m,n}(x+L_x,y)=\phi_{m,n}(x,y+L_y)

Möbius Strip

Schematics

Quotient space

\phi_{m,n}(x,y)=\sqrt{\dfrac{2}{L_xL_y}}\sin\!\left(\dfrac{m\pi x}{L_x}\right)\exp\!\left(\dfrac{i\pi ny}{L_y}\right)

\phi_{m,n}(x,y)=-\phi_{m,n}(x+L_x,y+L_y)

Klein Bottle

Schematics

Quotient space

\phi_{m,n}(x,y)=\sqrt{\frac{2}{L_x^{(1)} L_y}}\cos\!\left[\pi\!\left(\dfrac{2mx}{L_x}-\dfrac{n}{2}\right)\right]\exp\!\left(\dfrac{i\pi ny}{L_y}\right)

\phi_{m,n}(x,y)=-\phi_{m,n}(x+L_x,y+L_y)

Real Projective Plane (RP2)

Schematics

Quotient space

\phi_{m,n}(x,y)=\frac{2}{\sqrt{L_x^{(1)} L_y^{(1)}}}\cos\!\left[\pi\!\left(\dfrac{mx}{L_x}-\dfrac{m+n}{2}\right)\right]\cos\!\left[\pi\!\left(\dfrac{ny}{L_y}+\dfrac{m+n}{2}\right)\right]