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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

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