Guided waves as superposition of body waves
GGuided waves as superposition of body waves
David R. Dalton ∗ , Michael A. Slawinski † , Theodore Stanoev ‡ Abstract
We illustrate properties of guided waves in terms of a superposition of body waves. In particular,we consider the Love and SH waves. Body-wave propagation at postcritical angles—required for a totalreflection—results in the speed of the Love wave being between the speeds of the SH waves in the layerand in the halfspace. A finite wavelength of the SH waves—required for constructive interference—results in a limited number of modes of the Love wave. Each mode exhibits a discrete frequency andpropagation speed; the fundamental mode has the lowest frequency and the highest speed. Let us consider Love wave and the SH waves to examine the concept of a guided wave within a layer as aninterference of body waves therein. In the x x -plane, the nonzero component of the displacement vector ofthe Love wave is (e.g., Slawinski, 2018, Section 6.3) u (cid:96) ( x , x , t ) = C exp ( − ι κ s (cid:96) x ) exp [ ι ( κ x − ω t )]+ C exp ( ι κ s (cid:96) x ) exp [ ι ( κ x − ω t )] , where s (cid:96) := (cid:112) ( v/β (cid:96) ) − v being the speed of the Love wave and β (cid:96) the speed of the SH wave; ω and κ are the temporal and spatial frequencies, related by κ = ω/v . The SH waves travel obliquely inthe x x -plane; different signs in front of x mean that one wave travels upwards and the other downwards.Their wave vectors are k ± := ( κ, , ± κ s (cid:96) ) . Considering their magnitudes, (cid:107) k ± (cid:107) = (cid:112) κ + ( κ s (cid:96) ) , we have (cid:107) k ± (cid:107) = κ (cid:112) s (cid:96) ) = κ vβ (cid:96) ;from which it follows that β (cid:96) v = κ (cid:107) k ± (cid:107) = sin θ , (1)where θ is the angle between k ± and the x -axis. Thus, θ is the angle between the x -axis and a wave-front normal, which means that—exhibiting opposite signs—it is the propagation direction of upward anddownward wavefronts. A necessary condition for the existence of a guided wave is a total reflection on either side of the layer; theenergy must remain within a layer. For the Love waves, this is tantamount to no transmission of the SH ∗ Department of Earth Sciences, Memorial University of Newfoundland, Canada; [email protected] † Department of Earth Sciences, Memorial University of Newfoundland, Canada; [email protected] ‡ Department of Earth Sciences, Memorial University of Newfoundland, Canada; [email protected] a r X i v : . [ phy s i c s . c l a ss - ph ] M a r igure 1: Constructive interference for Love wave : The SH wave reflected twice reproduces itself, and, hence, thenonreflected and twice-reflected SH wavefronts coincide. Herein, (cid:107) AB (cid:107) = a λ , and (cid:107) AB (cid:48) (cid:107) = b λ , where ( a − b ) ∈ N . waves through the surface or the interface. The former is ensured by the assumption of vacuum above thesurface; hence, total reflection occurs for all propagation angles, θ . The latter requires β (cid:96) < β h , where β h is the speed of the SH wave within the halfspace. This inequality results in the existence of a criticalangle, θ c = arcsin( β (cid:96) /β h ) , which is required for a propagation at postcritical angles, θ > θ c .In view of expression (1), the lower limit of v is β (cid:96) , for which sin θ = β (cid:96) /β (cid:96) = 1 ; hence, θ = π/ SH waves that propagate parallel to the x -axis, and can be viewed as the Love wave.The upper limit, v = β h , is a consequence of the critical angle, for which sin θ c = β (cid:96) /β h . If β h → ∞ —which corresponds to a rigid halfspace— θ c → SH waves within the layer can propagate nearlyperpendicularly to the interface and still exhibit a total reflection. This means that v → ∞ , as can be alsoinferred from Figure 2.These limits, β (cid:96) < v < β h , are a consequence of total reflection. Also, the upper limit needs to beintroduced to ensure an exponential amplitude decay in the halfspace (e.g., Slawinski, 2018, Section 6.3.2). Guided waves—as superpositions of body waves—require a constructive interference of body waves. Anecessary condition of such an interference is the same phase among the wavefronts of parallel rays. InFigure 1, this condition means that the difference between (cid:107) AB (cid:107) and (cid:107) AB (cid:48) (cid:107) must be equal to a positive-integer multiple of the wavelength, λ , taking into account the phase shift due to reflection. A reflection atthe surface results in no phase shift (Ud´ıas, 1999, Sections 5.4 and 10.3.1), and the SH -wave postcriticalphase shift at the elastic halfspace is presented by Ud´ıas (1999, equation (5.74)).To illustrate the constructive interference—without discussing the phase shift as a function of the inci-dence angle—let us consider an elastic layer above a rigid halfspace, on which a transverse wave undergoesa phase change of π radians for any angle. In such a case, the propagation angle is (e.g., Saleh and Teich,1991, Section 7.1) θ n = arcsin (cid:18) n λZ (cid:19) , n = 1 , , . . . , (2)where λ is the wavelength of the SH wave, Z is the layer thickness and n is a mode of the guided wave; n = 1 is the fundamental mode.Thus—as a consequence of constructive interference—for a given SH wavelength and layer thickness, thepropagation angles, θ n , form a set of discrete values; each n corresponds to a mode of the guided wave. Asillustrated in Figure 2, each mode has its propagation speed, which—in accordance with expression (1)—is v n = β (cid:96) sin θ n , (3)2igure 2: Constructive interference for Love wave : The upgoing and downgoing SH wavefronts at two instants;their speed, (cid:107) βββ (cid:96) (cid:107) , remains constant—regardless of the wavefront orientation—but the Love-wave speed, (cid:107) v n (cid:107) , whosedirection, v n , remains constant, increases as θ n decreases. where, as a consequence of total reflection, θ n ∈ ( θ , π/
2) , where θ > θ c . The specific value of θ dependson Z and λ ; it corresponds to the first postcritical value for which (cid:107) AB (cid:107)−(cid:107) AB (cid:48) (cid:107) = 2 (cid:107) AB (cid:107) cos θ = 2 Z cos θ is a multiple integer of λ .Examining Figure 2, we distinguish the upgoing and downgoing wavefronts, which compose the guidedwave. Its longest permissible wavelength is twice the layer thickness, λ = 2 Z , which corresponds to thefundamental mode; λ = Z , λ = 2 Z/ λ n = 2 Z/n . λ , referred to in the caption of Figure 1 and used in expression (2), corresponds to the SH wave; λ n , where n = 1 , , . . . , corresponds to the guided wave. They are related by the propagation angle, θ n , and by thelayer thickness, Z .The radial frequency of a monochromatic SH wave is constant, ω = 2 π β (cid:96) /λ . The radial frequenciesof the Love wave are distinct for distinct modes, ω n = 2 π v n /λ n . For a given model, β (cid:96) , β h and Z , therelations between ω and ω n , as well as among ω n , where n = 1 , , . . . , are functions of n and θ n ; explicitly, ω n = n π β (cid:96) / ( Z sin θ n ) , and, in general, its behaviour as a function of n cannot be examined analytically.However, in an elastic layer above a rigid halfspace, in accordance with expression (2), ω n = n π β (cid:96) Z sin θ n = π β (cid:96) λ = ω , (4)which is constant for all modes, and depends only on the radial frequency of the SH wave.The constructive interference, illustrated in Figure 1, requires that (cid:107) AB (cid:107) − (cid:107) AB (cid:48) (cid:107) = a λ − b λ = ( a − b ) λ , where—in contrast to a = (cid:107) AB (cid:107) /λ and b = (cid:107) AB (cid:48) (cid:107) /λ — a − b is a positive integer; λ is the SH wavelength.Following trigonometric relations, we write (cid:107) AB (cid:107) − (cid:107) AB (cid:48) (cid:107) = (cid:107) AB (cid:107) − (cid:107) AB (cid:107) cos( π − θ ) = (cid:107) AB (cid:107) (1 − cos( π − θ ))= 2 (cid:107) AB (cid:107) cos θ . Since (cid:107) AB (cid:107) = Z/ cos θ , where θ is the SH -wave propagation angle, it follows that 2 (cid:107) AB (cid:107) cos θ = 2 Z cos θ ,and the constructive interference requires that 2 Z cos θ = ( a − b ) λ , where ( a − b ) ∈ N ; in other words,cos θ = a − b Z λ , θ (cid:62) θ c , to ensure the total reflection, and ( a − b ) λ (cid:54) Z , for θ ∈ R .Using this result and the inverse trigonometric function, we write the first equality of expression (4) as ω n = n π β (cid:96) Z (cid:115) − (cid:18) a n − b n Z (cid:19) λ , (5)which corresponds only to a given value of n and, hence, of θ n , since a n − b n changes with the propagationangle, and needs to be restricted to integer values for each n .Following expression (5), we obtain ω n ω n +1 = nn + 1 (cid:115) − (cid:18) a n +1 − b n +1 Z (cid:19) λ (cid:115) − (cid:18) a n − b n Z (cid:19) λ . (6)Since θ n +1 > θ n , examining Figure 1 and considering given values of λ and Z , we see that—as θ increases— (cid:107) AB (cid:107) − (cid:107) AB (cid:48) (cid:107) decreases. Hence, ( a n − b n ) > ( a n +1 − b n +1 ) , and the root in the numerator is greater thanin the denominator. Consequently, the ratio of roots is greater than unity. However, n/ ( n + 1) < n th mode is higher or lowerthan the frequency of the n + 1 mode. To determine it, we need not only to specify Z and the modelparameters, which result in θ c , but also λ and n , to obtain θ n and θ n +1 , with integer values of (cid:107) AB (cid:107)−(cid:107) AB (cid:48) (cid:107) . To obtain specific values, we let Z = 1000 , β (cid:96) = 2000 , β h = 3000 and λ = 50 , which means that θ c ≈ .
73 , in radians, and ω = 2 π β (cid:96) /λ ≈
251 . For the guided wave, in accordance with Figure 1, we obtain—numerically— θ ≈ .
76 , which corresponds to ( a − b ) = 29 . To include higher modes, using expression (6),we obtain ω /ω ≈ .
52 , ω /ω ≈ .
69 and ω /ω ≈ .
77 , which corresponds to, respectively, ( a n +1 − b n +1 ) =29 − n = 28 , 27 and 26 , and to ω = 17 .
60 , ω = 25 .
55 and ω = 33 .
07 .We might infer that the Love-wave fundamental mode, n = 1 , exhibits the lowest radial frequency—which, following expression (5), is 9 . n . Thehighest allowable mode corresponds to n = 29 , since, for that value, ( a n − b n ) = 1 . For this mode, ω ≈ .
27 ; also, ω /ω ≈ .
966 . Frequencies of distinct modes are shown in the left-hand plot ofFigure 3.Examining expression (6), in view of these results, we conclude that—as n increases—both n/ ( n + 1)and the ratio of roots tend to unity; the former from below, the latter from above. The ratio of successivefrequencies approaches the ratio of successive overtones for a vibrating string, , , , . . . , .Furthermore, using expression (3) and the computed values of θ n , we can obtain the correspondingpropagation speeds of the Love-wave modes. For the fundamental mode, v = β (cid:96) sin θ = 2000sin(0 .
76) = 2903 . , which is the highest speed of this Love wave; it is smaller than β h = 3000 , as required. The lowest speedcorresponds to θ = 1 .
55 , which is nearly π/ SH waves propagate almost parallel to the layer.The speed of the resulting Love wave is v = β (cid:96) / sin θ = 2000 .
63 , which is greater than β (cid:96) = 2000 , asrequired. Speeds of distinct modes are shown in right-hand plot of Figure 3.4igure 3: Frequencies, ω n , and speeds, v n , of Love-wave modes, n = 1 , . . . , Superposition of body waves allows us to examine several properties of guided waves. Body-wave propagationat postcritical angles—required for a total reflection—results in the speed of the Love wave being betweenthe speeds of the SH waves in the layer and in the halfspace. A finite wavelength of the SH waves—requiredfor constructive interference—results in a limited number of modes of the Love wave. Each mode exhibits adiscrete frequency and propagation speed; the first mode has the lowest frequency and the highest speed. Acknowledgments
We wish to acknowledge the graphic support of Elena Patarini. This research was performed in the contextof The Geomechanics Project supported by Husky Energy. Also, this research was partially supported bythe Natural Sciences and Engineering Research Council of Canada, grant 202259.
References
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Fundamentals of photonics . John Wiley & Sons.Slawinski, M. A. (2018).
Waves and rays in seismology: Answers to unasked questions . World Scientific,2nd edition.Ud´ıas, A. (1999).