Wave equation in cylindrical coordinates - different expression?

[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/c_T²⋅∂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

 

[MiSo]

An attachment...

 

[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.

 

[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...

 

[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}