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
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Advanced Advanced Mathematics in Mathematics in
SeismologySeismology
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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
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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
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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
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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
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The 1-D Wave EquationThe 1-D Wave Equation
2
22
2
2 ),(),(
x
txuc
t
txu
M
KLc
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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
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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
)(
)(
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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),(
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And in case you don’t believe the And in case you don’t believe the mathmath
Pure harmonic solutions Harmonic and exponentialsolutions
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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
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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!
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Applications in the real worldApplications in the real world
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Applications in the real worldApplications in the real world
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Applications in the real worldApplications in the real world
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Applications in the real worldApplications in the real world
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ShakeOut/1906 SimulationsShakeOut/1906 Simulations