Elastic waves in very thin plate

Question

Elastic waves in structural elements are described by \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \left( \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 v}{\partial x \partial y} + \frac{\partial^2 w}{\partial x \partial z} \right) \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \left( \frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial z^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial y} + \frac{\partial^2 w}{\partial y \partial z} \right) \\ \rho \frac{\partial^2 w}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 w}{\partial z^2} + \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) + (\lambda+\mu) \left( \frac{\partial^2 u}{\partial x \partial z} + \frac{\partial^2 v}{\partial y \partial z} \right) \end{align}

In a thin plate, these equations result in lamb-waves. However, imagine that we make the plate much thinner such that we end up in a very thin sheet. Is it then possible to assume that there is no variation over the thickness ($\partial / \partial z =0$)? If there is no variation over the thickness (z-direction) of the thin sheet, then the equations become \begin{align} \rho \frac{\partial^2 u}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 u}{\partial x^2} + \mu \frac{\partial^2 u}{\partial y^2} + (\lambda+\mu) \frac{\partial^2 v}{\partial x \partial y} \\ \rho \frac{\partial^2 v}{\partial t^2} &= (2 \mu + \lambda) \frac{\partial^2 v}{\partial y^2} + \mu \frac{\partial^2 v}{\partial x^2} + (\lambda+\mu)\frac{\partial^2 u}{\partial x \partial y} \\ \rho \frac{\partial^2 w}{\partial t^2} &= \mu \left( \frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} \right) \end{align}

Question: Is this a real case? Is this form of the equations used for real systems/structures (if yes, which ones)? Does it have a specific name like plane stress, antishear strain, quasi 2D elastodyanmics ...?

Answer 1

I've not looked at your specific equations, but in general if you take a plate and make it very thin, that's called a membrane. A membrane is to a plate what a string is to a beam. It has no bending stiffness, and can only carry load through in-plane tension. You can definitely have traveling waves in membranes. E.g. a drum head is a membrane. See https://en.wikipedia.org/wiki/Vibrations_of_a_circular_membrane for just one example.