wave equation in Einstein Notation

[cwb736]

Hi everybody, here is something I have seen in a geophysics text. I thought I was proficient using the einstein notation convention but I cannot follow this expansion of the wave equation. Note, I have not included the entire wave equation here, it is just the portion of the expansion which I am having trouble following

 

∂i[ λδij∂kUk ] = ∂iλ∂kUk + λ∂i∂kUk

 

here i,j,k are indices. δ is the kronacker delta such that δij=1 when i=j and zero otherwise. ∂ is the differential operator. U is displacement. λ is the Lame parameter whic is a constant.

 

I would have thought that

 

∂i[ λδij∂kUk ] = ∂iλ∂kUk

 

Can someone please explain where the second term comes from!

 

thanks very much

[timo]

1) In TeX, so the term can be read better. You mean [math] \partial_i [ \lambda \delta_{ij} \partial_k U_k ] = \partial_i \lambda \partial_k U_k + \lambda \partial_i \partial_k U_k[/math], right?

 

2) First off: The expression is not correct. On the left-hand side there is an open (=non-contracted) index j and on the right-hand side there is an open index i. EDIT: The position of the indices (upper or lower index) sometimes (e.g. in relativistic physics) is important. I have dropped that here - writing the terms with correct positions of the indices is straightforward.

 

3) I am not completely sure about the convention what to derive. I'd expect that [math] \partial_x AB := \partial_x (AB) [/math] but the parentheses in your example seem to indicate otherwise. So I think it is the chain rule (plus the correction of the indices) that took place there: [math] \partial_i [ \lambda \delta_{ij} \partial_k U_k ] =

\underbrace{\partial_i \delta_{ij}}_{=\partial_j} [ \lambda \partial_k U_k ] = (\partial_j \lambda)(\partial_k U_k) + \lambda (\partial_j \partial_k U_k)[/math]. Since you say lambda is a constant that term would then simplify to [math]\underbrace{(\partial_j \lambda)}_{=0}(\partial_k U_k) + \lambda (\partial_j \partial_k U_k) = \lambda \partial_j \partial_k U_k[/math].

 

4) I do spontaneously not see what the expression are supposed to represent. So keep in mind that what I said above is purely algebraic, not physical.

Edited December 23, 2009 by timo

[cwb736]

Hi Timo,

 

Thanks for the reply. This equation is a portion of the Seismic Wave Equation. The non-trivial solutions to this differential equation describes both the P-waves and S-waves that are commonly referred to in discussions of Earthquakes. As soon as I read your reply I realized my mistake. The Lambda in the equation is a constant for a simple Earth Seismic Model, which is the most common convention in Seismology, and using this model then that one term would indeed be zero. However, in the derivation I was reading, the Earth Model was more complex(and accurate) and in this model Lambda is a function of depth and therefore a gradient of Lambda is non-zero.

 

 

 

Thanks again.

Edited December 24, 2009 by cwb736