By Bahaa Saleh
Describing and comparing the elemental ideas and strategies of subsurface sensing and imaging, creation to Subsurface Imaging is a transparent and entire remedy that hyperlinks conception to quite a lot of real-world functions in medication, biology, safety and geophysical/environmental exploration. It integrates the various sensing suggestions (acoustic, electrical, electromagnetic, optical, x-ray or particle beams) by means of unifying the underlying actual and mathematical similarities, and computational and algorithmic tools. Time-domain, spectral and multisensor equipment also are lined, when the entire worthwhile mathematical, statistical and linear structures instruments are given in beneficial appendices to make the ebook self-contained. that includes a logical combination of idea and purposes, a wealth of colour illustrations, homework difficulties and various case stories, this can be compatible to be used as either a path textual content and as a certified reference.
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1-10 The acoustic spectrum. Impedance. For a harmonic acoustic wave of angular frequency ω the velocity and pressure perturbations obey the Helmholtz equation with wavenumber k = ω/v. 1-55) where P0 and U0 are the corresponding amplitudes. 1-53) it follows that the ratio of the pressure and velocity amplitudes P0 /U0 equals the characteristic impedance of the medium η= ρ0 = ρ0 v. 1-56) Impedance Intensity. The average acoustic intensity is I = Re 1 ∗ P U0 2 0 = |P0 |2 1 = η |U0 |2 2η 2 [W/m2 ].
1-11) permits us to compute U (x, y, z) at any z if we know U (x, y, 0). 1-11) to obtain ∞ F (kx , ky ) e−j(kx x+ky y) dkx dky . 1-12) is proportional to the two-dimensional (2D) inverse Fourier transform (see Appendix A) of the 2D function F (kx , ky ). 1-11) to determine U (x, y, z). This Fourier approach is useful in understanding 26 Chapter 2 Physical Models wave propagation and in solving inverse problems, as will be shown in subsequent chapters of this book. 1-13) Spherical Wave where r is the distance from the origin, k = ω/c is the wavenumber, and A0 is a constant.
The radius decreases to a minimum value of 2z0 at z = z0 , where the wavefront has the greatest curvature. The radius of curvature subsequently increases as z increases further until R(z) ≈ z for z z0 . The wavefronts are then approximately the same as those of a spherical wave. The pattern of the wavefronts is identical for negative z, except for a change in sign. 1-15), ζ(z), represents a phase retardation independent of ρ. 1-1 Damped Waves A wave traveling through an absorptive or scattering medium is attenuated and its amplitude and intensity decay exponentially, as illustrated in Fig.