Solutions to the Acoustic Wave Acoustic Wave Equation

Solutions to the Acoustic Wave Equation Outline Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength Spherical Wave Green’s function, asymptotic Green’s function Harmonic Motion: Phase P = Acos(wt) Time (s) w=2pi/T Period T = sec/cycle 1/T=f = cycle/s 2pi/T=w=radians/s Phase Time (s) Harmonic Ripples: P = Acos(kx) Wavenumber k=2pi/ Wavelength x Plane Wave Solution Faster Velocities = Stiffer Rocks κ ; c = ρ P = P c Phase=O (1) i(kx-wt) (2) P = Ae dO dt Plug (2) into (1) 2 wavenumber (k - w ) P = c2 dx/dt =w/k = kdx/dt –w =0 angular frequency pi k = w k ==2 w c c Wavefront = Line of constant phase Wavelength=shortest distance between adjacent peaks 2D Plane Wave Solution Phase P = Ae = Ae i(k x + k z - (2) wt) x z i(k r - wt) k = (k , k z ) x (2) = |k|(sin , cos ) k Equation of a line: k r = cnst k is Perpendicular to Wavefront z r k k = |r||k| cos(O) = constant Any pt along line phase is cnst O r (x,z) x 2D Plane Wave Solution Phase P = Ae = Ae i(k x + k z - (2) wt) x z i(k r - wt) k = (k , k z ) x x= sinO z (2) = |k|(sin , cos ) k z= cosO Equation of a line: k r = cnst = 2pi/|k| Time (s) Apparent Velocity dx/dt=apparent V x = x dz/dt=apparent V z = z z T x T = T sinO Time (s) Examples: dx/dt = v/sinO v/sinO=v x= z = O=90 Time (s) Examples: dx/dt = v/sinO x z = = O=0 = 0.0 Time (s) 0.45 0.0 X (ft) 250 Outline Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength Spherical Wave Green’s function, asymptotic Green’s function Energy of an Acoustic Wave du Work Performed: dy dz F=Pdzdy dx dW = (Pdzdy)du = PdV W = (Pdzdy)du = PdV Hooke’s Law dP = κ dV/V but dV = VdP/κ W = V PdP =V P Spherical Wave in Homogeneous Medium P= Ae i(k r - wt) r (2) satisfies Geometrical speading c P = 2 P except at origin Ray is traced such that it is always Perpendicular to wavefront r 2 r= x + y + z r is distance between pt source and observer at (x,y,z) Outline Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength Spherical Wave Green’s function, asymptotic Green’s function Spherical Wave in Heterogeneous Medium P= Ae i(wτ - wt) (2) satisfies P = c 2 P except at origin Geometrical speading w i kr i= (kc)r/ci = wτ ii i r Time taken along ray Valid at high w and smooth media Summary P = Ae i(kx-wt) V = T = ; cos O Vz = z ; ρ k k=|k|(sin O, cos O) z κ x = V = x sin O x Summary k=|k|(sin O, cos O) = p w w Slowness Vector Motivation: Spatial aliasing X< x Geophone sampling interval p ... Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength Spherical Wave Green’s function, asymptotic Green’s function Energy of an Acoustic. ..Outline Plane Wave: Dispersion relationship, freq., wavelength, wavenumber, slowness, apparent velocity, apparent wavelength Spherical Wave Green’s function, asymptotic Green’s function... Ripples: P = Acos(kx) Wavenumber k=2pi/ Wavelength x Plane Wave Solution Faster Velocities = Stiffer Rocks κ ; c = ρ P = P c Phase=O (1) i(kx-wt) (2) P = Ae dO dt Plug (2) into (1) 2 wavenumber (k -