# Help to understand the use of a double-Laplacian in this vibrating disk problem

I photocopied a section of a chapter in a book, now can't flip back and start from the beginning.

Vibrations of Shells and Plates, 2nd Edition Werner Soedel:

5.6 CIRCULAR PLATES VIBRATING TRANSVERSELY

$$D \nabla^4u_3 \ + \ \rho h \frac{\partial^2u_3}{\partial t^2} \ = \ 0 \ \ \ \ \ \text{ (Eq. 5.6.1)}$$

where

$$\nabla^2(\cdot) \ = \ \frac{\partial^2(\cdot)}{\partial r^2} \ + \ \frac{1}{r} \frac{\partial(\cdot)}{\partial r} \ + \ \frac{1}{r^2} \frac{\partial^2(\cdot)}{\partial r^2} \ \ \ \ \ \ \text{ (Eq. 5.6.2)}$$

...$U_3(r,\theta)$ being a time-independent solution at a natural frequency...

$$D \nabla^4U_3 \ - \ \rho h \omega^2 U_3 \ = \ 0 \ \ \ \ \ \text{ (Eq. 5.6.4)}$$

$$\lambda^4 \ = \ \frac{\rho h \omega^2}{D} \ \ \ \ \ \text{ (Eq. 5.6.5)}$$

$$(\nabla^2 \ + \ \lambda^2)(\nabla^2 \ - \ \lambda^2)U_3 \ = \ 0 \ \ \ \ \ \text{ (Eq. 5.6.6)}$$

I think that Eq. 5.6.2 is just the Laplacian in cylindrical coordinates without the $\hat{z}$ part, and $D$ is just a constant.

Should I be afraid of the $\nabla^4(u_3)$ for some reason? Is it just the Laplacian simply applied twice - e.g. $\nabla^2(\nabla^2(u_3))$ or are there further implications?

Since the usual wave equation just uses the Laplacian once, Eq. 5.6.1 is not the wave equation. Is there a different name for that form?

Yes, $\nabla^4$ is just the standard scalar Laplacian applied twice. This is because you are considering an isotropic almost rigid structure (think of the table of a violin) instead of an isotropic elastic structure (think of the membrane of a drum). If you deduce the the equation of small deformation of these structures, in view of the different relation between stresses and deformations, you find that the double Laplacian in place of the standard Laplacian. All that is known as the Kirchhoff-Love's plate theory which also considers the non-isotropic case. I do not know if the equation you are considering has a name...