Guided waves as superposition of body waves

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 total reflection--results in the speed of the Love wave being between the speeds of the SH waves in the layer and 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 and propagation speed; the fundamental mode has the lowest frequency and the highest speed.

Full text

Guided 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 total
reflection—results in the speed of the Love wave being between the speeds of the SH waves in the layer
and 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 and
propagation speed; the fundamental mode has the lowest frequency and the highest speed.
1 Introduction
Let us consider Love wave and the SH waves to examine the concept of a guided wave within a layer as an
interference of body waves therein. In the x1x3-plane, the nonzero component of the displacement vector of
the Love wave is (e.g., Slawinski, 2018, Section 6.3)
u`
2(x1, x3, t) = C1exp (−ι κ s`x3) exp [ι(κ x1−ω t)]
+C2exp (ι κ s`x3) exp [ι(κ x1−ω t)] ,
where s`:= p(v/β`)2−1 , with vbeing the speed of the Love wave and β`the speed of the SH wave; ω
and κare the temporal and spatial frequencies, related by κ=ω/v . The SH waves travel obliquely in
the x1x3-plane; different signs in front of x3mean that one wave travels upwards and the other downwards.
Their wave vectors are k±:= (κ, 0,±κ s`) . Considering their magnitudes,
kk±k=pκ2+ (κ s`)2,
we have
kk±k=κp1+(s`)2=κv
β`
;
from which it follows that β`
v=κ
kk±k= sin θ , (1)
where θis the angle between k±and the x3-axis. Thus, θis the angle between the x3-axis and a wave-
front normal, which means that—exhibiting opposite signs—it is the propagation direction of upward and
downward wavefronts.
2 Total reflection
A necessary condition for the existence of a guided wave is a total reflection on either side of the layer; the
energy 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; dalton.nfld@gmail.com
†Department of Earth Sciences, Memorial University of Newfoundland, Canada; mslawins@mac.com
‡Department of Earth Sciences, Memorial University of Newfoundland, Canada; theodore.stanoev@gmail.com
1
arXiv:1903.05200v1 [physics.class-ph] 11 Mar 2019

Figure 1: Constructive interference for Love wave: The SH wave reflected twice reproduces itself, and, hence, the
nonreflected and twice-reflected SH wavefronts coincide. Herein, kABk=a λ , and kAB0k=b λ, where (a−b)∈N.
waves through the surface or the interface. The former is ensured by the assumption of vacuum above the
surface; hence, total reflection occurs for all propagation angles, θ. The latter requires β`< βh, where
βhis the speed of the SH wave within the halfspace. This inequality results in the existence of a critical
angle, θc= arcsin(β`/βh) , which is required for a propagation at postcritical angles, θ > θc.
In view of expression (1), the lower limit of vis β`, for which sinθ=β`/β`= 1 ; hence, θ=π/2 . It
corresponds to the SH waves that propagate parallel to the x1-axis, and can be viewed as the Love wave.
The upper limit, v=βh, is a consequence of the critical angle, for which sin θc=β`/βh. If βh→ ∞ —
which corresponds to a rigid halfspace— θc→0 ; hence, the SH waves within the layer can propagate nearly
perpendicularly to the interface and still exhibit a total reflection. This means that v→ ∞ , as can be also
inferred from Figure 2.
These limits, β`< v < βh, are a consequence of total reflection. Also, the upper limit needs to be
introduced to ensure an exponential amplitude decay in the halfspace (e.g., Slawinski, 2018, Section 6.3.2).
3 Constructive interference
Guided waves—as superpositions of body waves—require a constructive interference of body waves. A
necessary condition of such an interference is the same phase among the wavefronts of parallel rays. In
Figure 1, this condition means that the difference between kABkand kAB0kmust be equal to a positive-
integer multiple of the wavelength, λ, taking into account the phase shift due to reflection. A reflection at
the surface results in no phase shift (Ud´ıas, 1999, Sections 5.4 and 10.3.1), and the SH-wave postcritical
phase 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 undergoes
a phase change of πradians for any angle. In such a case, the propagation angle is (e.g., Saleh and Teich,
1991, Section 7.1)
θn= arcsin nλ
Z, n = 1,2, . . . , (2)
where λis the wavelength of the SH wave, Zis the layer thickness and nis 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, the
propagation angles, θn, form a set of discrete values; each ncorresponds to a mode of the guided wave. As
illustrated in Figure 2, each mode has its propagation speed, which—in accordance with expression (1)—is
vn=β`
sin θn
,(3)
2

Figure 2: Constructive interference for Love wave: The upgoing and downgoing SH wavefronts at two instants;
their speed, kβ
β
β`k, remains constant—regardless of the wavefront orientation—but the Love-wave speed, kvnk, whose
direction, vn, remains constant, increases as θndecreases.
where, as a consequence of total reflection, θn∈(θ1, π/2) , where θ1> θc. The specific value of θ1depends
on Zand λ; it corresponds to the first postcritical value for which kABk−kAB0k= 2 kABkcos2θ= 2 Zcos θ
is a multiple integer of λ.
Examining Figure 2, we distinguish the upgoing and downgoing wavefronts, which compose the guided
wave. Its longest permissible wavelength is twice the layer thickness, λ1= 2Z, which corresponds to the
fundamental mode; λ2=Z,λ3= 2Z/3 , and, in general, λn= 2Z/n .
4 Frequencies of body and guided waves
λ, referred to in the caption of Figure 1 and used in expression (2), corresponds to the SH wave; λn, where
n= 1,2, . . . , corresponds to the guided wave. They are related by the propagation angle, θn, and by the
layer thickness, Z.
The radial frequency of a monochromatic SH wave is constant, ω= 2π β`/λ . The radial frequencies
of the Love wave are distinct for distinct modes, ωn= 2π vn/λn. For a given model, β`,βhand Z, the
relations between ωand ωn, as well as among ωn, where n= 1,2, . . . , are functions of nand θn; explicitly,
ωn=n π β`/(Zsin θn) , and, in general, its behaviour as a function of ncannot be examined analytically.
However, in an elastic layer above a rigid halfspace, in accordance with expression (2),
ωn=n π β`
Zsin θn
=πβ`
λ=ω
2,(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
kABk−kAB0k=a λ −b λ = (a−b)λ ,
where—in contrast to a=kABk/λ and b=kAB0k/λ—a−bis a positive integer; λis the SH wavelength.
Following trigonometric relations, we write
kABk−kAB0k=kABk−kABkcos(π−2θ) = kABk(1 −cos(π−2θ))
= 2 kABkcos2θ .
Since kABk=Z/ cos θ, where θis the S H-wave propagation angle, it follows that 2kABkcos2θ= 2Zcos θ,
and the constructive interference requires that 2Zcos θ= (a−b)λ, where (a−b)∈N; in other words,
cos θ=a−b
2Zλ ,
3

where θ>θc, to ensure the total reflection, and (a−b)λ62Z, for θ∈R.
Using this result and the inverse trigonometric function, we write the first equality of expression (4) as
ωn=n π β`
Zs1−an−bn
2Z2
λ2
,(5)
which corresponds only to a given value of nand, hence, of θn, since an−bnchanges with the propagation
angle, and needs to be restricted to integer values for each n.
Following expression (5), we obtain
ωn
ωn+1
=n
n+ 1
s1−an+1 −bn+1
2Z2
λ2
s1−an−bn
2Z2
λ2
.(6)
Since θn+1 > θn, examining Figure 1 and considering given values of λand Z, we see that—as θincreases—
kABk − kAB0kdecreases. Hence, (an−bn)>(an+1 −bn+1 ) , and the root in the numerator is greater than
in the denominator. Consequently, the ratio of roots is greater than unity. However, n/(n+ 1) <1 .
We cannot, in general, determine analytically if the radial frequency of the nth mode is higher or lower
than the frequency of the n+ 1 mode. To determine it, we need not only to specify Zand the model
parameters, which result in θc, but also λand n, to obtain θnand θn+1 , with integer values of kABk−kAB0k.
5 Numerical example
To obtain specific values, we let Z= 1000 , β`= 2000 , βh= 3000 and λ= 50 , which means that θc≈
0.73 , in radians, and ω= 2π β`/λ ≈251 . For the guided wave, in accordance with Figure 1, we obtain—
numerically— θ1≈0.76 , which corresponds to (a1−b1) = 29 . To include higher modes, using expression (6),
we obtain ω1/ω2≈0.52 , ω2/ω3≈0.69 and ω3/ω4≈0.77 , which corresponds to, respectively, (an+1−bn+1 ) =
29 −n= 28, 27 and 26 , and to ω2= 17.60 , ω3= 25.55 and ω4= 33.07 .
We might infer that the Love-wave fundamental mode, n= 1 , exhibits the lowest radial frequency—
which, following expression (5), is 9.123—and that the frequency increases monotonically with n. The
highest allowable mode corresponds to n= 29 , since, for that value, (an−bn) = 1. For this mode,
ω29 ≈182.27 ; also, ω28 /ω29 ≈0.966 . Frequencies of distinct modes are shown in the left-hand plot of
Figure 3.
Examining expression (6), in view of these results, we conclude that—as nincreases—both n/(n+ 1)
and the ratio of roots tend to unity; the former from below, the latter from above. The ratio of successive
frequencies approaches the ratio of successive overtones for a vibrating string, 1
2,2
3,3
4, . . . , 28
29 .
Furthermore, using expression (3) and the computed values of θn, we can obtain the corresponding
propagation speeds of the Love-wave modes. For the fundamental mode,
v1=β`
sin θ1
=2000
sin(0.76) = 2903.09 ,
which is the highest speed of this Love wave; it is smaller than βh= 3000 , as required. The lowest speed
corresponds to θ29 = 1.55 , which is nearly π/2 ; hence, the SH waves propagate almost parallel to the layer.
The speed of the resulting Love wave is v29 =β`/sin θ29 = 2000.63 , which is greater than β`= 2000 , as
required. Speeds of distinct modes are shown in right-hand plot of Figure 3.
4

Figure 3: Frequencies, ωn, and speeds, vn, of Love-wave modes, n= 1, . . . , 29
6 Conclusions
Superposition of body waves allows us to examine several properties of guided waves. Body-wave propagation
at postcritical angles—required for a total reflection—results in the speed of the Love wave being between
the speeds of the SH waves in the layer and 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 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 context
of The Geomechanics Project supported by Husky Energy. Also, this research was partially supported by
the Natural Sciences and Engineering Research Council of Canada, grant 202259.
References
Saleh, B. E. A. and Teich, M. C. (1991). 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). Principles of Seismology. Cambridge University Press.
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