# Advanced Mathematics in Seismology - PowerPoint PPT Presentation

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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)

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

P-waves!

S-waves!