CARCIONE_FINAL:Layout 6 ANNALS OF GEOPHYSICS, 57, 1, 2014, G0186; doi:10.4401/ag-6324 G0186 Mathematical analogies in physics. Thin-layer wave theory José M. Carcione1,*, Vivian Grünhut2, Ana Osella2 1 Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Sgonico (Trieste), Italy 2 Universidad de Buenos Aires, Departamento de Física - IFIBA Conicet, Buenos Aires, Argentina ABSTRACT Field theory applies to elastodynamics, electromagnetism, quantum me- chanics, gravitation and other similar fields of physics, where the basic equations describing the phenomenon are based on constitutive relations and balance equations. For instance, in elastodynamics, these are the stress-strain relations and the equations of momentum conservation (Euler-Newton law). In these cases, the same mathematical theory can be used, by establishing appropriate mathematical equivalences (or analo- gies) between material properties and field variables. For instance, the wave equation and the related mathematical developments can be used to describe anelastic and electromagnetic wave propagation, and are exten- sively used in quantum mechanics. In this work, we obtain the mathe- matical analogy for the reflection/refraction (transmission) problem of a thin layer embedded between dissimilar media, considering the presence of anisotropy and attenuation/viscosity in the viscoelastic case, conduc- tivity in the electromagnetic case and a potential barrier in quantum physics (the tunnel effect). The analogy is mainly illustrated with geo- physical examples of propagation of S (shear), P (compressional), TM (transverse-magnetic) and TE (transverse-electric) waves. The tunnel ef- fect is obtained as a special case of viscoelastic waves at normal incidence. 1. Introduction The 19th century was a period when scientists had a global vision of science, making wide use of analo- gies between different field of physics, in particular the analogy between light and elastic waves to study the behaviour of light in matter. For instance, Fresnel’s for- mulae and Maxwell’s equations were obtained from mathematical analogies with shear wave propagation and Hooke’s law, respectively [e.g., Carcione and Cav- allini 1995, Carcione and Robinson 2002]. This practice dates back to the 17th century. Hooke believed light to be a vibratory displacement of a medium (the ether), through which it propagates at finite speed. The role of mathematical analogies has been well illustrated and explained by Tonti in a series of papers [e.g., Tonti 1972, 1976]. Quoting Tonti [1972]: “Many physical theories show formal similarities due to the ex- istence of a common mathematical structure. This structure is independent of the physical contents of the theory and can be found in classical, relativistic and quantum theories; for discrete and continuous systems”. Carcione and Cavallini [1995] found analogies between vector wave files. They show that the 2-D Maxwell equa- tions describing propagation of the transverse-magnetic mode in anisotropic media is mathematically equivalent to the SH wave equation in an anisotropic-viscoelastic solid where attenuation is described with the Maxwell model. Later, Carcione and Robinson [2002] establish the analogy for the reflection-transmission problem at a single interface, showing that contrasts in compressibil- ity yields the reflection coefficient for light polarized perpendicular to the plane of incidence (Fresnel’s sine law - the electric vector perpendicular to the plane of incidence), and density contrasts yields the reflection co- efficient for light polarized in the plane of incidence (Fresnel’s tangent law). These two papers provide the time-domain differentials equations and the analysis for a single interface, while in the present paper, we exploit the analogy for a layer embedded between two dissim- ilar half spaces, showing how the same equation hold for shear, compressional and electromagnetic waves, in addition to the quantum tunnel effect. In 1821, Fresnel obtained the wave surface of an op- tically biaxial crystal, assuming that light waves are vi- brations of the ether in which longitudinal vibrations (P waves) do not propagate. An anisotropic elastic medium having orthorhombic symmetry, mathemati- cally analogous to Fresnel’s crystal, exists. This medium was found by Carcione and Helbig [2008], who obtained the differential equations describing the wave motion. In analytical terms, the above mentioned papers deal with a homogeneous medium and a single interface. The next step to exploit the analogy is to study the wave re- Article history Received April 8, 2013; accepted October 14, 2013. Subject classification: Thin layer, Electromagnetism, Viscoelasticity, Reflection and transmission coefficient, Geophysics, Quantum mechanics. sponse of a layer, namely the reflection and transmission of waves by a thin plane layer embedded in a homoge- neous medium. The reflection coefficient for isotropic media is given in Equation (5.18) of Brekhovskikh [1960] and Section 1.6.4. (Equation 57) of Born and Wolf [1964]. These authors report the expressions for the TE case. Here, we mainly consider the TM case. TE waves are equivalent to acoustic (P) waves propagating in liquids, while TM waves are equivalent to shear (SH) waves prop- agating in solids [Carcione and Robinson 2002]. More- over, the equation in Born and Wolf [1964] is given for purely dielectric media but the lossy version can be ob- tained by replacing the real dielectric constant by the complex permittivity. An equivalent viscoelastic model can be obtained in the elastic case, using the Maxwell model. Here, we consider the media to be anisotropic with one of the principal axes parallel to the layering. In geophysics, the problem finds application in re- flection seismology. Widess [1973] and Bakke and Ursin [1998] consider the normal incidence case of a thin layer, while Liu and Schmitt [2003] obtain the reflection coefficient as a function of the incidence angle corre- sponding to P waves. Bradford and Deeds [2006] and Deparis and Garambois [2009] consider the electro- magnetic case applied to amplitude variations with the incidence angle for surface radar problems. In quantum mechanics, the same mathematical approach can be used to analyze the tunnel effect [Anderson 1971]. We choose to present geophysical examples mainly. The analogy is applied to horizontally polarized shear (SH) waves through a fluid-filled fracture and TM waves through thin metal films, which are shown to be similar to the tunnel effect. Moreover, other examples consider P and SH seismic waves reflected from a thin and lossy sandstone bed and TM waves reflected from a dyke of quartz. Finally, we compare the TM and TE cases by using the analogy. 2. Reflection and transmission coefficients of a thin layer. Viscoelasticity We start from the viscoelastic equations and then apply the analogy to obtain the equivalent electromag- netic and quantum expressions. Figure 1 shows the layer embedded between two half spaces with different properties. In the following, we denote particle velocity by v, stress by v, magnetic field by H, electric field by E, density by t, elasticity constant by c, viscosity by h, magnetic permeability by n, dielectric permittivity by e and electrical conductivity by v. Moreover, (x, y, z) in- dicates the spatial variables, ^x a partial derivative with respect to x and a dot above a variable denotes time dif- ferentiation. To distinguish between the stress and con- ductivity components, we use letters and numbers as subindices, respectively, e.g., vxy is a stress component and v11 is a conductivity component. The viscoelastic medium is characterized by the mass density t and elasticity and viscosity matrices (1) respectively [Carcione and Cavallini 1995, Carcione 2007]. It corresponds to a viscoelastic medium de- scribed by the Maxwell mechanical model. The subindices “44” and “66” refer to the Voigt notation of the elasticity and viscosity tensors [e.g., Carcione 2007]. The differential equations describing the wave mo- tion of SH (shear) waves are (2) The boundary conditions at the interfaces require continuity of [e.g., Carcione 2007]. In the electromagnetic case, con- tinuity of the tangential components of the electric and magnetic fields is required [Born and Wolf 1964] (see below). Let us assume that the incident, reflected and refracted waves are identified by the subscripts and su- perscripts I, R and T. For a single interface, the particle velocities of the incident, reflected and refracted waves are given by and , c c0 0 0 044 66 44 66 h h c cm m and vyz yv i i , i , ,exp exp exp v v R v t s x s z t s x s z t s x s zT y I y y R T x z x z R x z T ~ ~ ~ = = = - - - - - - ^ ^ ^ h h h 6 6 6 @ @ @ , . , v v c v c x xy z yz y z y yz y xy xyx yz44 1 44 1 66 1 66 1 2 2 2 2 v v t h v v h v v + = - =- - = + - - - - o o o CARCIONE ET AL. 2 Figure 1. Plane wave propagating through a layer. The viscoelastic properties are indicated. (3) (4) 3 respectively, where is the slowness vector, ~ is the angular frequency, R and T are the reflection and refraction (transmission) coefficients, t is the time vari- able and i = . The equations obtained below hold for an incident inhomogeneous plane waves (non-uni- form waves in electromagnetism), i.e., waves for which the wavenumber and attenuation vectors do not point in the same direction. In the special case where these two vector coincide, the wave is termed homogeneous (uniform in electromagnetism), and we have where i is the incidence angle and is the complex velocity, with [Carcione and Cavallini 1995, Carcione 2007]. In the general case, the reflection and transmission coefficients (TM case in electromagnetism) are given by where with and with “pv” denoting the principal value [Carcione and Robinson 2002, Carcione 2007]. To obtain the reflection and transmission coeffi- cients of the layer, we follow the procedure indicated in Section 6.4 of Carcione [2007]. At depth z in the layer, the particle-velocity field is a superposition of upgoing and downgoing waves of the form where V− and V+ are upgoing- and downgoing-wave amplitudes. Combining Equations (3) and (12), the nor- mal stress component is Omitting the phase exp[i~(t − sxx)], the particle- velocity/stress vector can be written as Then, the fields at z = 0 and z = h are related by the following equation: where Equation (15) plays the role of a boundary condi- tion. Note that when h = 0, B is the identity matrix. Using Equations (2), (4) and (9), the particle-veloc- ity/stress field at z = 0 and z = h can be expressed as where and R and T are here the reflection and transmission co- efficients of the layer. Substituting Equation (17) into (16), we have If h = 0, { = 0 and we obtain R = (ZI − Z"T)/(ZI + Z"T) as expected. If, moreover, the two media have the same properties, it is Z"T = ZI and the reflection coeffi- cient is nil. Equation (19) coincides with Equation (2) of Bradford and Deeds [2006] in the absence of wave loss. These authors used the convention exp(−i~t), in- stead of exp(i~t) used here, so that the sign of the fre- quency has to be changed in Equation (2). If the upper and lower half spaces have the same properties, we get sin cos v p p66 2 2 44 t i i = + i iandp c p c 1 1 1 1 44 44 1 66 66 1 ~h ~h = + = + -- 44 66 ` `j j , 2 ,r Z Z Z Z Z Z Z I T I T I T I x= + - = +l l l , ,Z p s Z p sI z T z T 44 44= =l l l Snell’s law ,s s s s s sx R x x x R z T I z= = = =-^ h pv ,s p s 1 z T x 44 66 2t t= - l l ll i i i , exp exp exp v z V s z V s z t s x y z T z T x ~ ~ ~ = + + - - - + l l ^ ^ ^ ^ h h h h 6 6@ @ i i i i . exp exp exp z p v Z V s z V s z t s x yz z y T z T z T x 44 2v ~ ~ ~ ~ = = - - - - - + l l l l ^ ^ ^ ^ h h h h 6 6@ @ i i i i . t T exp exp exp exp z v s z s z Z s z Z s z V V z V V y yz z T z T T z T T z T - - / / v ~ ~ ~ ~ = = - = - + - + l l l l l l ^ c ^ ^ ^ ^e c ^ c h m h h h ho m h m ,t B t h0 $=^ ^h h , , , ,t t T Z R Z R h 0 1 1 1 T I = - = + - R R m ^ ^^ ^ ^ h hh h h pv , ,s p s Z p s1z T x T z T 44 66 2 44t t= - = m m m m mm m , i i . cos sin cos sin R A A A Z Z Z Z Z Z 1 1 T I T T T T { { { { = + - = + + l m l l mc m , , ,sin coss s v 1 x z i i= R R^ ^h h 1- 2i 2i , exp exp R r r 1 1 2 { { = - - - - ^ ^ h h6 @ ( , )s sx z < i i , B T T cos sin sin cos h Z Z s h 0 T T z T 1 1 $ { { { { { ~ = = - - = - - l l l ^ ^ ^e h h h o THIN-LAYER WAVE THEORY (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) where r is given by Equation (8). Equation (19) is similar to Equation (5.22) of Born and Wolf [1964] if the media are isotropic and lossless. In order to be identical, this equation has to be rewrit- ten as where Equation (21) differs from Equation (5.22) in the fact that r12 and r23 correspond here to TM waves, while in Equation (5.22) these quantities correspond to TE waves. Another difference is the minus sign in the ex- ponent, which arises from the fact that Born and Wolf [1964] use the factor exp(−i~t). In the case in which the media above and below the layer have the same prop- erties, i.e., when r23 = −r12 = −r, Equation (22) be- comes Equation (20). On the other hand, the transmission coefficient can be obtained from Equation (17). It gives which can be written as This expression coincides in form with that of Born and Wolf ([1964], Equation 58). 3. Analogy with a modified acoustic wave equation Carcione and Robinson [2002] introduced a modified acoustic wave equation with a time-depen- dent mass density. Denoting the particle velocities by v and the pressure f ield by p, the equations can be written as where l is the fluid compressibility, tf is the fluid den- sity and c = 0 yields the standard acoustic equations of motion. Equations (26) correspond to a generalized density of the form cI(t) + tf H(t), where H(t) is the Heaviside function and I(t) is the integral operator. In the frequency domain, a complex density governs the wave equation and has the expression tf + c/(i~). This additional density term introduces wave loss as that of the Biot attenuation mechanism in poroelasticity the- ory [e.g., Carcione 2007]. Equations (26) are mathematically analogous to the isotropic-medium SH Equations (2) for the follow- ing correspondence where c44 = c66 = c and h44 = h66 = h. 4. Analogy with electromagnetism 4.1. SH-TM analogy The analogy between SH waves and transverse- magnetic (TM) waves states that the equivalent elec- tromagnetic medium has the following permittivity and conductivity tensors respectively, describing a uniaxial absorbing crystal. The TM Maxwell’s equations are The mathematical analogy is [Carcione and Cavallini 1995, Carcione 2007], where the electromagnetic tensors are redefined here as 2 × 2 matrices for simplicity. 2i i , exp exp T r r1 12 23 12 23 { x x { = + - - ^ ^ h h and . Z Z Z2 T T T 12 23x x x= = +l m l , , , v v p p v v p v v z z x x z z f z fx x x 2 2 2 2 l c t c t + =- - = + - = + o o o v v v p c xy y x z f yz 1 1 , , , , , , v v h t c t l - - - and ,0 0 0 0 0 0 0 0 0 0 0 0 11 11 11 11 3333 e e e v v vf fp p , , . E E H H E E H E E x z z x y z y x x y zx z 11 11 33 33 2 2 2 2 n v e v e - = - = + = + o o o v H E E y yz xy y x z ,v v -f fp p c c0 0 0 044 1 1 66 33 11 , e e - -c cm m 0 0 0 044 1 66 1 11 33 , h h v v- -c cm m ,,t n 1 i 2i , exp exp R r r r r 212 23 12 23 { { = + - + - ^ ^ h h and .r r r Z Z Z Z T T T T 12 23= = + - l m l m ,cos sinT Z i Z Z Z 2 1 I T T T I T 1 { {= + + + - Z Z m l m lc cm m; E CARCIONE ET AL. 4 (21) (22) (23) (24) (25) (26) (27) (28) (29) (32) (31) (30) (33) 5 4.2. SH-TE analogy The TE Maxwell equations are [Carcione 1998, Carcione and Robinson 2002], where we have considered the lossless case (zero conductiv- ity). As can be seen, these equations are similar to those of an isotropic medium, since only one component of the permittivity tensors appears. This fact indicates that TE waves are also equivalent to sound waves. The TM-TE analogy has been given by Carcione [1998] and Carcione and Robinson [2002]. The com- plete mathematical analogy is 5. Analogy with quantum mechanics. Tunnel effect For simplicity, we consider the time-independent Schrödinger equation where = 1.055 × 10−34 J s is Planck’s constant, m is the mass of the particle, ] is the wave function, E is the total energy of the particle, V is the height of the po- tential barrier, and D is the Laplacian. On the other hand, in the isotropic case, Equation (2) can be written as where p = p44 = p66 and we have assumed an harmonic wave. Equations (36) and (37) are mathematically equiv- alent if ] = vy and outside and inside the barrier, respectively. The tunnel effect occurs when E < V. In this case, the waves within the potential barrier are evanescent with a complex ve- locity given by where we have expressed the energy as E = . The analogy holds with SH waves at normal inci- dence (i = 0). Of interest is the transmission coefficient through the barrier, given by Equation (24), when the media above and below the layer (i.e., the barrier) have the same properties, i.e., when r23 = −r12 = −r, x12 = x and x23 = (Z'T/ZI)x. Then where r and x are given by Equation (8). At normal in- cidence, where the density t is uniform in the whole space and can have any arbitrary value. Alternatively, the analogy can be established by assuming variable density media and uniform stiffness p. In this case, t = sp and t' = s'p, with ZI = ps and Z'T = ps' with p taking any arbitrary value. Equation (40) coincides with Equations (5.11) (E > V) and (5.12) (E < V) of Anderson [1971], considering that this author reports T2 instead of T. The interpretation of the transmission coefficient |T|2 in this case is the probability of a particle to be tunneling through the barrier. An approximation to Equation (40) can be obtained for V > E and { ≫ 1, which is the simplified equation found in many text- books [Feynman et al. 1965]. 6. Results We first consider cases equivalent to the tunnel ef- fect. Particular cases of interest are fractures in elastic media and thin metal films in electromagnetic media. Assuming c'44 = c'66 → ∞ and h'44 = h'66 ≡ h', we obtain a Newtonian fluid from Equation (7), The complex velocity (6) becomes where the third and fourth expressions correspond to SH- and P-waves, respectively. TM SH TE H E E v c E H H c y x z y yz xy y x z 11 44 1 1 33 66 22 , , , , , , , , , , , , e v v n e n n et - - - - ,m E V2 2' ] ]D- = -^ h 0 , ,v s v s py y 2 2~ t D + = = ands mE s m E V2 2 ' '~ ~ = = - l ^ h i ,v s m V2 1 ' ' ~ ~ = =- - - l l^ ^ h h 1 i , i exp exp T r Z Z 2 T I 2 2 { x { = - - -l ^ ^ ^ h h h , and ,Z s Z s s h E s hI T ' t t { ~= = = =l l l l 4 , exp exp exp T V E V E m V E h 1 2 ' . ?{ {- - - = = - - ` ^ ^ ^c j h h h m i .p p p44 66 ~h= = =l l l l ' '~ , , H H E H E H E z x y z x y x z y x z 222 2 2 2 f n n - = = - = o o o i i ,v p t t ~h c l ~ = = = l l l l l l THIN-LAYER WAVE THEORY (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) On the other hand, an anisotropic metal has with where the last expression holds for an isotropic medium, with v'11 = v'33 = v'. Note that in the viscoelastic and electromagnetic cases , while in quantum mechanics (see Equation 39), i.e., there is a r/4 phase difference. Let us first consider SH waves and a fracture thick- ness h = 1 cm. The upper and lower medium are the same and defined by c44 = 10 GPa, c66 = 14 GPa and t = 2000 kg/m3, with zero viscosity. Figure 2 shows the reflection coefficient of SH waves as a function of vis- cosity h' (a) (normal incidence, i.e., i = 0) and incidence angle (b) (for h' = 10 kPa s). The frequency is f = ~/(2r) = 60 Hz. As can be seen, increasing viscosity and inci- dence angle implies decreasing reflection amplitude and increasing transmission. For fluid viscosities com- parable to water, oil or gas, the reflection coefficient is practically 1 for h = 1 cm. To obtain a significant energy transmission the thickness has to be of the order of tens of Angstroms, since the wave has a shear nature whose polarization (particle motion) is parallel to the fracture surface. In real fractures, significant transmission takes place at larger thicknesses, since the surface of the frac- ture is rough and there are many contact points [e.g., Gangi and Carlson 1996]. Next, consider a thin conductive isotropic layer of thickness h = 1 nm embedded in a lossless medium with permittivity components e11 = 8e0 and e33 = 12e0, where e0 = 8.85 10 −12 F/m. The magnetic permeability is the same for the two media, i.e., n = n0 = 4 r 10 −7 H/m. Based on the analogy (30)-(33), Figure 3 shows the reflection coefficient of the electromagnetic wave as a function of conductivity (a) (normal incidence) and incidence angle (b) (for v' = 30000 S/m). The fre- quency is 1 MHz. As can be seen, the film totally re- i and i ,p p44 66 11 33v ~ v ~ = =l l l l i i ,sin cosv 33 2 11 2 n ~ v i v i n v ~ = + = l l l l lc m iv ? iv ? CARCIONE ET AL. 6 Figure 2. Absolute value of the reflection coefficient of SH waves as a function of viscosity (a) (normal incidence, i.e., i = 0) and in- cidence angle (b) (for h' = 10 kPa s). The frequency is f0 = 60 Hz and the fracture thickness is 1 cm. Figure 3. Absolute value of the reflection coefficient of the elec- tromagnetic wave as a function of conductivity (a) (normal inci- dence) and incidence angle (b) (for v' = 30000 S/m). The frequency is f0 = 1 MHz and the layer thickness is h = 1 nm. (45) (46) 7 flects the wave for very high conductivities. This is the case for copper (v' = 6 107 S/m) for instance. Total transmission occurs for very resistive materials. We consider now the tunnel effect. Let us define the reference wavelength As an example, we consider the tunneling of an electron with mass m = 9.1 × 10−31 kg and energy E = 1.6 × 10−19 J (1 eV), giving m = 1.23 nm. Figure 4a and b show the transmission coefficient as a function of h for V = 10E (a), and as a function of V for h = 0.5 nm (b), normalized by m and E, respectively. As expected, the probability of tunneling decreases with increasing h and V. The next example considers a sandstone layer em- bedded in a shale formation, whose properties are taken from Thomsen [1986] (Table 1). The sandstone (Taylor sandstone) has the properties vP = = 3368 m/s, vS = = 1829 m/s, c = (c66 − c44)/(2c44) = 0.255, and t = 2500 Kg/m3, where vP and vS are P- and Swave velocities and c33 is an elasticity constant. The properties of the shale (Mesaverde mudshale) are vP = 4529 m/s, vS = 2703 m/s, c = 0.046, and t = 2520 Kg/m3. We obtain c44 = 18.4 GPa, c66 = 20.1 GPa, c'44 = 8.4 GPa, c'66 = 12.6 GPa, and assume h'44 = h'66 = 0.5 GPa s. This choice of the sandstone viscosity yields the c33 t 2 . mE2 ' m r = THIN-LAYER WAVE THEORY c44 t Medium es (e0) ef (e0) vs (S/m) vf (S/m) e (e0) Ks (GPa) Kf (GPa) z (%) K (GPa) Shale 30 4.23 0.01 0.4 (30.15, −11.31) 20 2.25 15 12.76 Sandstone 5 2 0 10−5 (3.79, −8.6 × 10−4) 39 0.025 35 5.59 Figure 4. Absolute value of the transmission coefficient of an elec- tron through a potential as a function of the width h and height V (properly normalized). In (a) V = 10E and in (b) h = 0.5 nm. Table 1. Properties. A = 3; ~ = 100 MHz. Figure 5. Sandstone layer in mudshale. Absolute value (a) and phase (b) of the SH-wave reflection coefficient as a function of the inci- dence angle and three values of h. quality factors Q44 = ~h'/c'44 = 9 and Q66 = ~h'/c'66 = 6 (see Carcione [2007]: Equation 2.156), which implies high attenuation. In fact, the effective pressure for this sandstone is zero in Table 1 of Thomsen [1986] mean- ing that the medium is over-pressured. We consider a frequency f = 60 Hz, which implies a wavelength of 30 m based on the S-wave velocity of the layer. Figure 5 shows the reflection coefficient of the SH waves, Equa- tion (20): absolute value (a) and phase (b), as a function of the incidence angle and three values of h (1 m, 3 m and 6 m). There is a Brewster angle at approximately 60°, where the reflection coefficients become almost zero and change phase [Carcione 2007]. We may obtain the reflection coefficient of P waves using the analogy (27). Then, we can use Equa- tion (20) by replacing c44 = c66 by t −1 f , t by c −1 33 , c'44 = c'66 by (t'f ) −1, t' by ( c'33) −1, where tf and t'f are the densi- ties of the shale and sandstone, respectively. The vis- cosities are equal to ∞, i.e, there is no attenuation in this case. Figure 6 shows the absolute value (a) and phase (b) as a function of the incidence angle and three values of h (1 m, 3 m and 6 m). In this case, the behaviour of the curves is quite different compared to S-wave case, with no Brewster angle. Next, we compute the reflection coefficient of TM waves from a vein of quartz embedded in a lime- stone.We consider e11 = 9 e0, e33 = 0.8 e11, e'11 = e'33 = 4 e0, n = n0 and zero electrical conductivity for both rocks. The frequency is f = 100 MHz, typical of ground- penetrating radar surveys. The analogy (30)-(33) and Equation (20) are used to obtain the reflection coeffi- cient, shown in Figure 7 for three different thicknesses of the quartz layer. In this case, there is a Brewster angle at 37° approximately. The curves are similar to those of the S wave, since it is known that shear and TM waves are analogous mathematically [e.g., Car- cione and Cavallini 1995]. In the following, we test the SH-TE analogy and compare the TM and TE reflection coefficients. We consider the last example, but in the isotropic case, with n = n0, e11 = e22 = e33 = 9 e0 and e'11 = e'22 = e'33 = 4 e0. The thickness of the layer is h = 20 cm. The TE curve CARCIONE ET AL. 8 Figure 6. Sandstone layer in mudshale. Absolute value (a) and phase (b) of the P-wave reflection coefficient as a function of the incidence angle and three values of h. Figure 7. Vein of quartz in limestone. Absolute value (a) and phase (b) of the TM-wave reflection coefficient as a function of the inci- dence angle and three values of h. 9 is in agreement with Equation (1) of Bradford and Deeds [2006]. It also agrees with Equation (2) of De- paris and Garambois [2009] provided that, in this paper, z = 2dk2 cos ii is replaced by z = 2dk2 cos it, which is the correct expression. The TE and TM curves are very different, mainly at the Brewster angle. Finally, we consider a CO2-saturated sandstone embedded in brine-saturated shale and use a cross- property relation to obtain the bulk modulus as a func- tion of the complex permittivity [Carcione et al. 2007, Mavko et al. 2009]. The gas is in its supercritical state. In the following, K, t, v and e denote bulk modulus, mass density, electrical conductivity and dielectric per- mittivity, respectively, and the subindices “s” and “f” de- note grain and fluid properties. Gassmann equation gives the bulk modulus of the saturated rocks where is the dry-rock modulus and z is the porosity, which can be obtained from the CRIM equation, where e is the rock complex dielectric constant, A is an empirical parameter and “Re” takes the real part. On the other hand, the rock density is given by t = (1 − z)ts + ztf. Assuming ts = 2650 kg/m 3 for both shale and sand- stone, tf = 1000 kg/m 3 for brine and tf = 500 kg/m 3 for CO2, we obtain t = 2400 kg/m 3 and t =1897 kg/m3, respectively. Moreover, we assume the properties given in Table 1, which also shows the calculated Gassmann moduli. We compute the P-wave and TM reflection co- efficients for a wavelength/thickness ratio r = 10, i.e., ~h = 2rc/r, where c is the P-wave velocity or the elec- tromagnetic TM-wave velocity (100 MHz) of the layer. Figure 9 shows the absolute value of the reflection co- efficients versus incidence angle. As can be seen, the re- sults indicate that the layer is more detectable with (high-frequency) electromagnetic methods. 7. Conclusions Theories describing wave phenomena - and diffu- sion - in different fields of physics consist in partial dif- ferential equations, which have identical or similar mathematical expressions. Here, we have considered the reflection/transmission problem through an anisotropic and lossy layer. The equations hold for any layer thickness, not necessarily thin, although their uses are relevant for thin layers, i.e., when the wavelength of the signal is much larger than the thickness. We have shown that the same mathematical equations can be used in elastodynamics, electromagnetism and quan- tum mechanics by establishing appropriate analogies Re i i i s s s f f s .z e v ~ e v ~ e e v ~ - - - - -f p 1 ,K K K K K K K K K K 1 m s s f s m m s f z z z z z z = - - + - + - ^ ^ ^ ^ h h h h 1K K 1 1m A sz z= - z z- + -^ ^ ^ ^h h h h THIN-LAYER WAVE THEORY Figure 8. Absolute value (a) and phase (b) of the TE- and TM-wave reflection coefficients as a function of the incidence angle at f = 100 MHz and h = 20 cm. Figure 9. Absolute value of the TM-wave and P-wave reflection co- efficients as a function of the incidence angle, corresponding to a sandstone layer saturated with supercritical gas embedded in brine- saturated shale. (47) (48) (49) between the different physical variables and medium properties. In particular, the analogy has been obtained for P and SH elastic waves, TE and TM electromagnetic waves and wave mechanics in quantum theory. Further research involves the analysis of the pres- ent problem in the space-time domain using numerical simulations. In this case, the same computer code, with appropriate input variables can be used to solve the dif- ferent physical problems. References Anderson, E. (1971). Modern Physics and quantum me- chanics, W.B. Saunders Co. Bakke, N.E., and B. Ursin (1998). Thin-bed AVO effects, Geophysical Prospecting, 46, 571-587. Born, M., and E. Wolf (1964). Principles of Optics, Ox- ford: Pergamon Press. Bradford, J.H., and J.C. Deeds (2006). Ground-pene- trating radar theory and application of thin-bed off- set-dependent reflectivity, Geophysics, 71, K47-K57. Brekhovskikh, L.M. (1960). Waves in layered media, Ac- ademic Press Inc. Carcione, J.M., and F. Cavallini (1995). On the acoustic- electromagnetic analogy, Wave Motion, 21, 149-162. Carcione, J.M. (1998). Radiation patterns for GPR for- ward modeling, Geophysics, 63, 424-430. Carcione, J.M., and E. Robinson (2002). On the acoustic- electromagnetic analogy for the reflection-refrac- tion problem, Studia Geophysica et Geodaetica, 46, 321-345. Carcione, J.M. (2007). Wave Fields in Real Media. The- ory and numerical simulation of wave propagation in anisotropic, anelastic, porous and electromag- netic media, 2nd edition, Elsevier. Carcione, J.M., B. Ursin and J.I. Nordskag (2007). Cross- property relations between electrical conductivity and the seismic velocity of rocks, Geophysics, 72, E193-E204. Carcione, J.M., and K. Helbig (2008). Elastic medium equivalent to Fresnels double refraction crystal, J. Acoust. Soc. Am., 124 (4), 2053-2060. Deparis, J., and S. Garambois (2009). On the use of dis- persive APVO GPR curves for thin-bed properties estimation: Theory and application to fracture char- acterization, Geophysics, 74, J1-J12. Feynman, R.P., R.B. Leighton and M. Sands (1965). The Feynman lectures on physics 13, Addison-Wesley. ISBN 0-7382-0008-5. Gangi, A.F., and R.L. Carlson (1996). An asperity-de- formation model for effective pressure, Tectono- physics, 256, 241-251. Liu, L., and D.R. Schmitt (2003). Amplitude and AVO re- sponses of a single thin bed, Geophysics, 68, 1161-1168. Mavko, G., T. Mukerji and J. Dvorkin (2009). The rock physics handbook: tools for seismic analysis in porous media, Cambridge Univ. Press. Thomsen, L. (1986). Weak elastic anisotropy, Geo- physics, 51, 1954-1966. Tonti, E. (1972). On the mathematical structure of a large class of physical theories, Accademia Nazio- nale dei Lincei, excerpt from ‘Rendiconti della Clas- se di Scienze fisiche, matematiche e naturali’, Series 8, vol. 52 (1). Tonti, E. (1976). The reason for analogies between physical theories, Appl. Math. Modelling, 1, 37-50. Widess, M.B. (1973). How Thin is a Thin Bed?, Geo- physics, 38, 1176-1180. *Corresponding author: José M. Carcione, Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS), Sgonico (Trieste), Italy; email: jcarcione@inogs.it. © 2014 by the Istituto Nazionale di Geofisica e Vulcanologia. All rights reserved. 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