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Solutions to the Acoustic Wave Equation

Solutions to the Acoustic Wave Equation. Outline. Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength. 2. Spherical Wave. 3. Green’s function, asymptotic Green’s function. Harmonic Motion:. w=2pi/T. T = sec/cycle. Phase.

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Solutions to the Acoustic Wave Equation

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  1. Solutions to the Acoustic Wave Equation

  2. Outline • Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength. 2. Spherical Wave. 3. Green’s function, asymptotic Green’s function.

  3. Harmonic Motion: w=2pi/T T = sec/cycle Phase P = Acos(wt) Period Time (s) 1/T=f = cycle/s 2pi/T=w=radians/s

  4. Harmonic Ripples: k=2pi/ Phase P = Acos(kx) Wavenumber Time (s) Wavelength x

  5. Plane Wave Solution (1) Phase=O (2) i(kx-wt) P = Ae angular frequency k = w wavenumber k k = w c = 2 pi c = 2 c 2 2 (k - w ) P = 0 r c .. 2 2 P c ; P = 2 dO = kdx/dt –w =0 dt dx/dt =w/k Faster Velocities = Stiffer Rocks Plug (2) into (1) Wavefront = Line of constant phase Wavelength=shortest distance between adjacent peaks

  6. 2D Plane Wave Solution i(k x + k z - wt) (2) P = Ae x z i(k r - wt) (2) = Ae Equation of a line: k r = cnst k = (k , k ) = |k|(sin , cos ) x z k Phase

  7. k is Perpendicular to Wavefront = |r||k| cos(O) k = constant Any pt along line phase is cnst O k r r z x (x,z)

  8. 2D Plane Wave Solution i(k x + k z - wt) (2) P = Ae x z = = cosO sinO i(k r - wt) (2) = Ae Equation of a line: k r = cnst k = (k , k ) = |k|(sin , cos ) x z k = 2pi/|k| z x z Phase

  9. Apparent Velocity dz/dt=apparent V dx/dt=apparent V = = z x x z T T sinO = T z x Time (s)

  10. Examples: dx/dt = v/sinO Examples: dx/dt = v/sinO=v = = O=90 z x Time (s)

  11. Examples: dx/dt = v/sinO = = = O=0 z x Time (s)

  12. X (ft) 0.0 250 0.45 Time (s) 0.0

  13. Outline • Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength. 2. Spherical Wave. 3. Green’s function, asymptotic Green’s function.

  14. Energy of an Acoustic Wave du F=Pdzdy W = (Pdzdy)du = PdV k Hooke’s Law dP = dV/V 2 = P V W = PdP V k 2 k dy Work Performed: dz dx dW = (Pdzdy)du = PdV but dV = VdP/k

  15. Spherical Wave in Homogeneous Medium i(k r - wt) (2) Ae satisfies P = r except at origin Geometrical speading .. 2 2 2 r = x + y + z 1 2 P c P = 2 2 3 4 Ray is traced such that it is always Perpendicular to wavefront r r is distance between pt source and observer at (x,y,z)

  16. Outline • Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength. 2. Spherical Wave. 3. Green’s function, asymptotic Green’s function.

  17. Spherical Wave in Heterogeneous Medium i(wt - wt) (2) Ae P = satisfies except at origin Geometrical speading w Time taken along ray .. 1 2 P c P = 2 2 3 4 i i kr = (kc)r/c = wt i i i i r Valid at high w and smooth media

  18. Summary 1. V = T k 2. k=|k|(sin O, cos O) i(kx-wt) P = Ae k = r ; = z x cos O sin O ; V = V = x z x z T T

  19. Summary Slowness Vector p 3. k=|k|(sin O, cos O) = p w w 4. Motivation: Spatial aliasing X < x 2 Geophone sampling interval

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