# A Wave equation in cylindrical coordinates - different expression?

1. Sep 19, 2016

### MiSo

Hello to everybody,
I am solving some examples related to wave equation of shear horizontal wave in cylindrical coordinates (J.L Rose: Ultrasolic Waves in Solid Media, chapter 6), which is expressed as follows:

2u=1/cT2⋅∂2u/∂t2

The Laplace operator in cylindrical coordinates can can be derived in the form of (which I have verified to myself):
2=∂2/∂r2+1/r⋅∂/∂r+1/r2⋅∂2/∂θ2+∂2/∂x2

Prof. Rose uses in his book following expression of Laplace operator:
2=1/r⋅∂/∂r⋅(r⋅∂/∂r)+1/r2⋅∂/∂θ2+∂2/∂x2

Can anyone please give be an explanation how to get such an expression? I am quite confused.

Mike

2. Sep 19, 2016

### MiSo

An attachment...

3. Sep 19, 2016

### the_wolfman

The two expression are equivalent. Prof. Rose equations is a little more compact. You use the chain rule on the first term to verify that they are the equivalent.

4. Sep 19, 2016

### MiSo

Thank you very much for you reply!

You mean to make chain rule on this term? 1/r⋅∂/∂r⋅(r⋅∂u/∂r) or ∂2u/∂r2
I ask because when I´ll make a chain rule on ∂2u/∂r2 term, I think, that I have to obtain a Laplace operator for cartesian coord. system...

5. Sep 19, 2016

### the_wolfman

$\frac{1}{r} \frac{\partial}{\partial r} \left( r \frac{\partial u}{\partial r} \right) = \frac{1}{r} \left( r \frac{\partial^2 u}{\partial r^2} + \frac{\partial r}{\partial r} \frac{\partial u}{\partial r} \right) = \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r}$