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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Advanced Mathematics in Seismology

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Advanced mathematics in seismology

Advanced Mathematics in Seismology


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


The 1 d wave equation2

The 1-D Wave Equation


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!


Applications in the real world

Applications in the real world


Applications in the real world1

Applications in the real world


Applications in the real world2

Applications in the real world


Applications in the real world3

Applications in the real world


Shakeout 1906 simulations

ShakeOut/1906 Simulations


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