Hi, i wonder if anybody can help me to understand how these equations were linked. I found this in phd thesis and could not understand after hard trying. The author reasons by using kronecker delta transformation whatever but i have no idea where the term omega (nat freq) come from. This is related to acoustic cavity mode analysis.

I'm not saying someone where will surely be able to help, but is hard to answer anything without knowing what functions are which or what the constants resemble. With respect to what inner product is what orthogonal to what? Etc.

Dear Sir, i'm sorry for being very brief earlier, i attach again the whole derivation from the thesis.
The purple colour is where i dont understand and the red text is the question.

I dont know how to paste the equations here so i just put the file

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