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Advanced Mathematics in Seismology. Dr. Quakelove. or: How I Learned To Stop Worrying And Love The Wave Equation. When Am I Ever Going To Use This Stuff?. Wave Equation. Diffusion Equation. Complex Analysis. Linear Algebra. The 1-D Wave Equation. F = k[u(x,t) - u(x-h,t)].
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Dr. Quakelove or: How I Learned To Stop Worrying And Love The Wave Equation
When Am I Ever Going To Use This Stuff? Wave Equation Diffusion Equation Complex Analysis Linear Algebra
The 1-D Wave Equation F = k[u(x,t) - u(x-h,t)] F = k[u(x+h,t) – u(x,t)] k k m m m u(x-h,t) u(x,t) u(x+h,t) F = m ü(x,t)
The 1-D Wave Equation M = N m L = N h K = k / N
Solution to the Wave Equation • Use separation of variables:
Solution to the Wave Equation • Now we have two coupled ODEs: • These ODEs have simple solutions:
Solution to the Wave Equation • The general solution is: • Considering only the harmonic component: • The imaginary part goes to zero as a result of boundary conditions
And in case you don’t believe the math Harmonic and exponential solutions Pure harmonic solutions
The 3-D Vector Wave Equation • We can decompose this into vector and scalar potentials using Helmholtz’s theorem: where
The 3-D Vector Wave Equation P-waves! S-waves!
