Waves in layers

If a substrate has one or more surface layers that are too thin to allow the echoes from different interfaces to be discriminated, then the contrast may be 

The matrix elements are given in Table 10.1. In each layer the angles of the longitudinal and shear waves are given by Snell s law as in (6.87). When (vm/vo) sin do 1 for a particular mode m in a particular layer, then for that mode the angle of refraction dm is complex. This can be seen by defining y = (vm/vo) sin do, and let dm = n/2 + is. Then from sin( /2 + ie) = y, 

The effect of a series of layers may be obtained by matrix multiplication, so that if the matrix describing layer i is denoted a,, and if the product of the matrices for all the layers is written 

Matrix elements a for the ith elastic layer (Brekhovskikh 1980, with unpublished corrections by Tsukahara cf. Brekhovskikh and Godin 1990). The variables are defined p is the density of the layer d is the thickness of the layer ill, Us are the velocities of the longitudinal and shear waves in the layer k, are their wave-vectors with components ai, Os normal to the interfaces between layers and their angles 0i, 05 are related to the angle of the incident wave by Snell s law in the usual way, 

In (10.12) all the parameters except the Mik refer to the substrate. With the normal impedance of the fluid defined as Z0 = pov0/cosd0 ( 6.4.1), the reflectance function has the familiar form 

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Yeh, P., Optical Waves in Layered Media, 2nd edn. Wiley Interscience, New York, 2005... 

For a general detailed description see Optical Waves in Layered Media Yeh, P. Ed. J. Wiley and Sons Inc., New York, 1988. 

L. M. Brekhovskikh, Waves in Layered Media. Academic Press, NY, 1960. 

P. TYGELAND P. HUBRAL —TRANSIENT WAVES IN LAYERED MEDIA A.A. KAUFMAN AND GV. KELLER — INDUCTION LOGGING... 

Yeh P (2005) Optical waves in layered media, 2nd edn. Wiley, New York... 

P. TYGEL AND P. HUBRAL — TRANSIENT WAVES IN LAYERED MEDIA... 

See, for example, J. R. Wait, Electromagnetic Waves in Stratified Media (Pergamon, New York, 1970) L. M. Brekhovskikh, Waves in Layered Media, translated by O. Liberman (Academic, New York, 1980). 

J. L. Rose and A. Pilarski, Surface and plate waves in layered structures. Mater. Eval. 46, 598-605 (1988). 

Yeh, R (1998) Optical Waves in Layered Media, John Wiley Sons, Inc., Hoboken, NJ. 

These equations are valid for TE as well as for TM waves, provided we substitute the appropriate values for g into the formulas. For TE waves one has to use g given by Eq. (5.6.5) and for TM waves by Eq. (5.6.7). By applying these equations to each boundary, 2m equations for 2w + 2 quantities (also for layer 0) are obtained. However, the arriving wave in layer m is assumed to be known and the reflected wave in layer 0 is assumed to be nonexistent therefore, all B and C can be determined, most conveniently by matrix inversion. 

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