![]() Beyond the critical angle, there is no intersection between that line and the slowness curve, and the wave becomes evanescent (Henneke II, 1971 Rokhlin, Bolland and Adler, 1986 Helbig, 1994, p. We get the critical angle when this line is tangent to the slowness curve of the transmission medium. Snell law requires that the end points of all the slowness vectors lie in a common normal line to the interface. ![]() ![]() A geometrical interpretation is that, in the elastic case, critical angles are associated with tangent planes to the slowness surface that are normal to the interface (see Figure 6.6). Moreover, the transmitted wave becomes evanescent.Īccording to Proposition 6.6, at the critical angle and beyond, the Umov–Poynting vector of the transmitted wave is parallel to the interface and the wave becomes evanescent. At the critical angle and beyond, the Umo–-Poynting vector of the transmitted wave is parallel to the interface. Directions of the slowness and Umov-Poynting vectors, corresponding to the critical angle θ C = 36.44° for the elastic case. Actually, the pseudocritical angle does not play any important physical role in the anisotropic case.įigure 6.6. We keep the same interpretation for viscoelastic media. The correct interpretation was given by Henneke II (1971), who defined the critical angle θ C as the angle(s) of incidence beyond which the Umov–Poynting vector of the transmitted wave is parallel to the interface (see also Rokhlin, Bolland and Adler 1986). 9), the critical angle phenomenon is related to the condition s T 3 = 0, but, as we shall see below, this is only valid when the lower medium has p′ 46 = 0 (e.g., transversely isotropic). The pseudocritical angle θ P is defined as the angle of incidence for which the transmitted slowness vector is parallel to the interface. In anisotropic media, two singular angles can be defined depending on the orientation of both the propagation and the Umov–Poynting vectors with respect to the interface. There is then a complete analogy between the reflection–transmission problem for isotropic, lossless acoustic media of equal density and the same problem in electromagnetism, where the media have zero conductivity and their magnetic permeability are similar (perfectly transparent media, see Born and Wolf, 1964, p. On the basis of the acoustic–electromagnetic mathematical analogy ( Carcione and Cavallini, 1995b), the magnetic permeability is equivalent to the material density and the dielectric permittivity is equivalent to the reciprocal of the shear modulus (see Chapter 8). This property can be proved by using Snell law and equation (6.74) (this exercise is left to the reader). In the lossless case and when ρ = ρ′, the reflected and transmitted rays are perpendicular to each other at the Brewster angle, i.e., θ B + θ T = 90°. The condition Im( μ/ μ′) = 0 implies that the Brewster angle exists when Q H = Q′ H, where Q H = Re( μ′)/Im( μ′). Thus, the quality factor ( 4.126) for homogeneous waves in isotropic media is Q = Q H = Re( μ)/Im( μ). In isotropic media, the complex velocity (6.28) is simply v c = μ / ρ. For example, cot θ B is real for Im ( μ / μ′) = 0. The Brewster angle exists only in rare instances.
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