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)]. - PowerPoint PPT PresentationTRANSCRIPT
Advanced Advanced Mathematics in Mathematics in
SeismologySeismology
Dr. QuakeloveDr. Quakelove
or:or:
How I Learned To Stop How I Learned To Stop WorryingWorrying
And Love The Wave EquationAnd Love The Wave Equation
When Am I Ever Going To Use This When Am I Ever Going To Use This Stuff?Stuff?
Wave Equation
Compl
ex A
nalysis
Diffusion Equation
Linear Algebra
The 1-D Wave EquationThe 1-D Wave Equation
kk
u(x-h,t) u(x,t) u(x+h,t)
F = k[u(x,t) - u(x-h,t)] F = k[u(x+h,t) – u(x,t)]
m m m
F = m ü(x,t)
),(),(),(),(),(
2
2
thxutxutxuthxukt
txum
The 1-D Wave EquationThe 1-D Wave Equation
M = N m L = N h K = k / N
2
2
2
2 ),(),(2),(),(
h
thxutxuthxu
M
KL
t
txu
The 1-D Wave EquationThe 1-D Wave Equation
2
22
2
2 ),(),(
x
txuc
t
txu
M
KLc
Solution to the Wave EquationSolution to the Wave Equation
►Use separation of variables:Use separation of variables:)()(),( tTxXtxu
22
22
2
2
2
22
2
2
2
22
2
2
)(
)(
)(
)(
1
)()(
)()(
),(),(
dx
xXd
xX
c
dt
tTd
tT
dx
xXdtTc
dt
tTdxX
x
xuc
t
txu
Solution to the Wave EquationSolution to the Wave Equation
►Now we have two coupled ODEs:Now we have two coupled ODEs:
►These ODEs have simple solutions:These ODEs have simple solutions:
)()(
)()(
22
2
2
2
2
2
tTdt
tTd
xXcdx
xXd
titi eBeBtT
eAeAxX cxi
cxi
21
21
)(
)(
Solution to the Wave EquationSolution to the Wave Equation
►The general solution is:The general solution is:
►Considering only the harmonic Considering only the harmonic component:component:
►The imaginary part goes to zero as a The imaginary part goes to zero as a result of boundary conditionsresult of boundary conditions
)(4
)(3
)(2
)(1),( c
xcx
cx
cx titititi eCeCeCeCtxu
cxcxti tiAtAAetxu c
x
sincos),(
And in case you don’t believe the And in case you don’t believe the mathmath
Pure harmonic solutions Harmonic and exponentialsolutions
The 3-D Vector Wave The 3-D Vector Wave EquationEquation
uuKu 2
3
►We can decompose this into vector We can decompose this into vector and scalar potentials using Helmholtz’s and scalar potentials using Helmholtz’s theorem:theorem: u
22
22
3
4K
where
The 3-D Vector Wave The 3-D Vector Wave EquationEquation 22 22
2
223
22
21
332211exp),(
kkk
xkxkxktiAtx
k
xktiBtx
exp),(
P-waves!
S-waves!
Applications in the real worldApplications in the real world
Applications in the real worldApplications in the real world
Applications in the real worldApplications in the real world
Applications in the real worldApplications in the real world
ShakeOut/1906 SimulationsShakeOut/1906 Simulations