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we just call this equation the model equation, since by varying the coefficients you can model tons of stuff, from vibrations in any which direction, heat transfer through conduction or convection, probably lots more stuff but that's the bulk of what we focus on.
Its just a good overall exercise since it includes second and fourth order spatial derivatives and first and second time derivatives.
k can be any number of things, but my major is ESM(engineering science and mechanics), some places call it Applied Mechanics, in either case, i haven't touched most forms of chemistry since freshman year. So no, kA is not a chemical property. In heat transfer problems, k*A would be thermal conductivity of the material*cross sectional area. In solid mechanics and dynamics, k would be a spring constant, or a radius of gyration (neither of those showed up in the examples i typed in)In solid mechanics problems, rho*A would be density of the material times area, E*A would be Young's Modulus * area. Etc.
There's lots more about the Finite Element method than i can ever write in a CD post, i'd highly recommend taking a course in it in college. I believe most companies use it fairly extensively as a more realistic way of modeling things that don't perfectly match text books (hopefully any of the Professional Engineers in here can back me up on that) I just took the Fundamentals of Engineering exam (all 8 hours of it) on saturday, so i've got at least 5 years to go before I can be Scott England, P.E. and now at least 2 years of grad school before i'll ever hit a real company extensively.
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