What's with this separation of variables business?

[Ai52487963]

Electromagnetism just got weird. REALLY weird. Everything was going great until we hit this new chapter on separation of variables. I don't remember doing this kind of stuff in my DiffEqs class.

Frankly, I'm feeling overwhelmed. I have a midterm at the end of this week, and I feel as though if I were to be tested on boundary conditions of electric potentials, then I'm doomed for sure. Multipole expansions make more sense to me and seem far, far less hand-wavey than separation of variables does.

Is there a better way of understanding separation of variables than all this wacky math stuff? This n and m business combined with double summations is getting out of hand. Fast.

 

[malawi_glenn]

I don't think separation of variables are handwaveying, maybe it is just your teacher who gives you this impression? Separation of variables is a great way to solve Partial Differential equations.

We did have sep. of var. both in our transform methods - math class and in mathematical methods of physics class.

 

[gabbagabbahey]

Perhaps you can be more specific about which aspect(s) of SOV you find hand-wavy?

 

[Ai52487963]

Perhaps you can be more specific about which aspect(s) of SOV you find hand-wavy?

SOV made sense to me in my differential eqns class perfectly, but in my book, Introduction to Electrodynamics by Griffiths, he goes over an example of an infinitely long rectangular pipe with three sides grounded and one end of the pipe at V.

He goes through boundary conditions and applies Laplace's equation and arrives at some double summation for V(x,y,z) involving nearly a million arbitrary constants. I guess its the part of SOV involving the part where he takes his X(x), Y(y) and Z(z) equations and applies the boundary conditions to them is where I get lost.

 

[gabbagabbahey]

Are you referring to Example 3.5 in the 3rd edition?

Are you okay with the derivation up to the point where he has

$$X\frac{d^2 X}{d x ^2}=C_1 , \quad Y\frac{d^2 Y}{d y ^2}=C_2, \quad Z\frac{d^2 Z}{d z ^2}=C_3$$ with $$C_1+C_2+C_3=0$$

Or is there anything up until thet point that you don't understand?