# Flexural wave question

1. Mar 18, 2006

### psv

Whenever I've seen solutions to elastic or viscoelastic stress strain relations and resulting wave equations, the solutions are only acoustic, shear, love, reyleigh waves etc. Recently I saw that flexural waves are governed not by the 2nd order wave equation I'm used to seeing, but by a 4th order different one derived from bending moments. So my question then is when you say that some object is a kelvin-voigt solid and assume the corresponding continumm mechanics equations, are flexural waves a solution or do they need to be considerred somewhat seperatly?

2. Mar 19, 2006

### Claude Bile

If you assume a non-linear restoring force, then this permits higher order solutions to the wave equation.

Claude.

3. Mar 20, 2006

### psv

Do you, or anyone else happen to know any good links or books/papers concerning this, my searches have so far come up with little.

Thnx

4. Mar 20, 2006

### Cyrus

Sorry to be off topic here, what is this about? Is this for an advanced material science class? Is it theory of elasticity?

5. Mar 21, 2006

Claude.

6. Jun 1, 2006

### psv

cyrusabdollahi: no, this was just my own interest mostly, although since I should know more about elasticity maybe you could say that area

Claude: yes, I have access to journals and books, and I was mostly interested in anything where it is put in terms as plain as you did about higher order solutions resulting from non-linear restoring forces. Sorry I didn't mean for people to go do journal searches for me, I was just hoping someone might know a better book. I looked in some classic engineering books by timoshenko and fung, but they seemed to present there elastic equations and those governing flexure seperatly.

I guess my main confusion lies in the the fact that I usually think forces, u, v in a continuum as linear, so bending moments sort of throw me for a loop when I am trying to tie everything together. (there has to be a pun in there)

7. Jun 4, 2006

### Claude Bile

The non-linear restoring force is usually experessed as a polynomial of some description. Since the wave equation is linear, you can solve each term in the polynomial separately - this is why some solutions are labelled 'second-order' or 'fourth-order', for example, because those solutions correspond to the inclusion of the second and fourth terms in the polynomial.

Textbooks solve each part separately because it is much simpler (and far less confusing) than tackling the whole solution at once. In addition, each set of solutions (first, second, third order etc) often have a distinct physical manifestation, so it is advantageous to obtain an distinct, separate equation for each physical effect, rather than have them all conglomerated into one super-equation.

Claude.