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Advanced Mathematics in Seismology

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Advanced Mathematics in Seismology

Dr. Quakelove

or:

How I Learned To Stop Worrying

And Love The Wave Equation

Wave Equation

Diffusion Equation

Complex Analysis

Linear Algebra

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)

M = N m

L = N h

K = k / N

- Use separation of variables:

- Now we have two coupled ODEs:
- These ODEs have simple solutions:

- The general solution is:
- Considering only the harmonic component:
- The imaginary part goes to zero as a result of boundary conditions

Harmonic and exponential

solutions

Pure harmonic solutions

- We can decompose this into vector and scalar potentials using Helmholtz’s theorem:

where

P-waves!

S-waves!