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Advanced Mathematics in Seismology
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Dr. Quakelove or: How I Learned To Stop Worrying And Love The Wave Equation
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When Am I Ever Going To Use This Stuff? Wave Equation Complex Analysis Diffusion Equation Linear Algebra
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The 1-D Wave Equation k k 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)] mmm F = m ü(x,t)
![The 1-D Wave Equation k k 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)] mmm F = m ü(x,t)](http://images.slideplayer.com./39/10871524/slides/slide_4.jpg "The 1-D Wave Equation k k 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)] mmm F = m ü(x,t)")
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The 1-D Wave Equation M = N mL = N hK = k / N
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The 1-D Wave Equation
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Solution to the Wave Equation ► Use separation of variables:
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Solution to the Wave Equation ► Now we have two coupled ODEs: ► These ODEs have simple solutions:
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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
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And in case you don’t believe the math Pure harmonic solutions Harmonic and exponential solutions
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The 3-D Vector Wave Equation ► We can decompose this into vector and scalar potentials using Helmholtz’s theorem: where
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The 3-D Vector Wave Equation P-waves! S-waves!
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Applications in the real world
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ShakeOut/1906 Simulations
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