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. 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

Dr. Quakelove

or:

How I Learned To Stop Worrying

And Love The Wave Equation


When am i ever going to use this stuff
When Am I Ever Going To Use This Stuff?

Wave Equation

Diffusion Equation

Complex Analysis

Linear Algebra


The 1 d wave equation
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 equation1
The 1-D Wave Equation

M = N m

L = N h

K = k / N



Solution to the wave equation
Solution to the Wave Equation

  • Use separation of variables:


Solution to the wave equation1
Solution to the Wave Equation

  • Now we have two coupled ODEs:

  • These ODEs have simple solutions:


Solution to the wave equation2
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
And in case you don’t believe the math

Harmonic and exponential

solutions

Pure harmonic solutions


The 3 d vector wave equation
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 equation1
The 3-D Vector Wave Equation

P-waves!

S-waves!