J. Nig. Soc. Phys. Sci. 4 (2022) 823

Journal of the
Nigerian Society

of Physical
Sciences

Mathematical Modeling of Waves in a Porous Micropolar Fibre-
reinforced Structure and Liquid Interface

Augustine Igwebuike Anyaa,∗, Uko Ofeb, Aftab khanc

aGST-Mathematics Division, Veritas University Abuja, Bwari-Abuja, Nigeria
bDepartment of Pure and Applied Physics, Veritas University Abuja, Bwari-Abuja, Nigeria

cDepartment of Mathematics, COMSATS University Islamabad, Park Road Chak Shahzad, 44000 Islamabad, Pakistan

Abstract

The present investigation envisages on the Mathematical modeling of waves propagating in a porous micropolar fibre-reinforced structure in a
half-space and liquid interface. The harmonic method of wave analysis is utilized, such that, the reflection and transmission of waves in the media
were modelled and it’s equations of motion analytically derived. It was deduced that incident longitudinal wave in the solid structure yielded four
reflected waves given as; quasi–P wave (qLD), quasi–SV wave, quasi–transverse microrotational (qTM) wave and a wave due to voids and one
transmitted wave known as the quasi-longitudinal transmitted (qLT) wave. The phase velocity in the liquid medium is independent of angle of
propagation as observed. The corresponding amplitude ratios of propagations for both reflected and transmitted waves are analytically derived by
employing Snell’s law. The model would prove to be of relevance in the understanding of modeling of the behavior of propagation phenomena
of waves in micropolar fibre-reinforecd machination systems resulting in solid/liquid interfaces especially in earth sciences and in particular
seismology, amongst others.

DOI:10.46481/jnsps.2022.823

Keywords: Micropolar, Fibre reinforced, Reflection/transmission, Voids/porosity, Liquid interface

Article History :
Received: 20 May 2022
Received in revised form: 17 June 2022
Accepted for publication: 21 June 2022
Published: 20 August 2022

c© 2022 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: T. Latunde

1. Introduction

Researchers in solid mechanics and in particular elastody-
namics have always pursued for ways in trying to decipher the
behaviors of certain disturbances in materials caused either nat-
urally or artificially, especially in disciplines such as; engineer-
ing, structural designing, seismology, material sciences, and
geophysics among others. Although of particular interest in

∗Corresponding author tel. no: +2347038933954
Email address: anyaaugustineigwebuike@gmail.com,

anyaa@veritas.edu.ng (Augustine Igwebuike Anya)

this study is the porous micopolar fibre reinforced materials.
Fibre reinforced materials play a collective host to the strength
of materials used by construction engineers, physicists, material
scientists etc., owing to their high tensile strength, low weight,
and efficacy of fibre-reinforcement cum flexibility of usage [1].
Fibre reinforced media tends to be similar to another impor-
tant type of material known as the Orthotropic material; which
could be considered in the investigation of elastodynamic mod-
els. Some of these materials possess micro-rotation and trans-
lation of local points as given by Erigen [2], theory of microp-
olar elasticity. Thus, in describing the propagation of waves

1



Anya et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 823 2

in such materials, a mathematical model is accompanied along
with certain physical properties and parameters such as; voids
in the material [3], pre-stress, as the case maybe. Voids are
pores in a material which are taken as a volume fraction fields
in equations describing such media. Mathematical models in
terms of equations used for the study of reflection and trans-
mission at a plane half-space of elastic media were initially in-
troduced by Knott [4] and subsequently modified by Jeffreys
[5], Gutenberg [6], etc.

In a similar vein, reflection and transmission coefficients
in fluid-saturated porous media and boundary surfaces, have
received great attention in literature [7-11]. Extensive works
on fiber reinforced media and reflection of waves in an elastic
media with some other physical properties of rotation, gravity,
thermal effects etc., were also conducted [12-18]. Reflections
of waves in a micropolar fibre reinforced medium with other
physical properties of magneto-thermoelastic effects, rotation,
etc., were also carried out [19-20].

Furthermore, boundary surfaces of some materials are plane,
grooved or entirely of different shapes in nature. Grooved bound-
ary surface could be visualized as a series of parallel furrows
and ridges whose encounter in mechanical propagation of wave
results to several effects especially across interfaces. Interest-
ingly, some authors had worked on this concept of corrugated
boundaries and other related wave propagation phenomena [21-
29]. Also, certain discussions on materials [30-33] could aid
understanding of material characterizations.

In spite of these contributions, the present study is also of
particular interest, as it seeks to investigate the propagation of
waves in plane boundary of a porous micropolar fibre rein-
forced solid structure and liquid interface. Also the motivation
of the study stems from the fact that classical theories have short
comings in modeling solid structure interactions and its under-
standing of behaviors for any given impact on it; hence mi-
cropolar theory of elasticity by Eringen [2], suggests the basic
assumptions and subsequently, the consideration of 9x9 non-
symmetric material matrix of Micropolar fibre-reinforcement.
In studying this, we employed two concepts of wave propaga-
tion analyses called the normal mode analysis or harmonic anal-
ysis and Snell’s law. Due to the composition of the media, when
P-wave is incident on the solid structure, four waves are re-
flected in the solid structure while only quasi-longitudinal trans-
mitted (qLT) wave is transmitted in the liquid medium. And by
taken continuity conditions at the interface, the reflection and
transmitted wave’s propagation coefficients are derived.

2. Mathematical Formulation of the Problem

The constitutive relations for a micropolar fibre–reinforced
elastic anisotropic solid with voids, considering some existing
works [2-3], [34-35] follow as:

σi j = Ni jmn Emn + S i jmnψmn + ζφδi j, (1)

mi j = N jimn Emn + S mn jiψmn, (2)

we define the deformations and wryness tensors as:

Ei j = u j,i + ε jimφ∗m, ψmn = φ
∗
m,n ,

i = j = m = n = 1, 2, 3.
(3)

The balance laws in the absence of body forces are presented
below as:

σi j,i = ρü j, (4)

mi j,i + ε jmnσmn = ρJφ̈ j
∗, (5)

ξ1(φ,ii) −ωoφ−$φ̇− ζ(ui,i) = ρκφ̈. (6)

where mi j, σi j, φ∗j, u j and φ are the couple stress tensor, stress
tensor, microrotation vector, displacement vector and volume
fraction field respectively. ρ is the density of the solid medium;
ζrepresents the voids parameter; J is the microinertia, Ni jmn,
N jimn are constants of material characterization such that non
symmetric properties of Ni jmn, N jimn and S i jmn are observed.
ε jim is the Levi–Civita tensor and δi j, is the Kronecker–delta
function. The given index after comma denotes partial deriva-
tive with respect to coordinate space and the superscript dot
stipulates partial derivative with respect to time. We tried to
consider the deformation in x1 x3–plane. This is such that the
displacements; u1 , u3 , 0, while u2 = 0. Thus, this im-
plies that the micro-rotation; φ∗ = (0,φ∗2, 0). Repeated indexes
of Einstein summation convention is used. Considering the fact
the tensors are not symmetric in micropolar solid, in the sense
that, a 9x9 matrix of the solid material characterization is uti-
lized, Eqs. (4-6) in components form are given as:

N1u1,11 + (N2 + N3)u3,13 + N6u1,33
+N∗1φ

∗
2,3 + ζφ,1 = ρü1,

(7)

N4u3,11 + N2u1,13 + N5u3,33 + N4φ
∗
2,1 + ζφ,3 = ρü3, (8)

S 1φ
∗
2,11 + S 2φ

∗
2,33 + N

∗
2φ
∗
2 + χ1u1,3 + χ2u3,1 = ρJφ̈

∗
2, (9)

ξ1(φ,ii) −ωoφ−$φ̇− ζ(ui,i) = ρκφ̈, (10)

where

N1 = (λ + 2α + β + 4µL − 2µT ),

N2 = (λ + α), N3 = 2µT ,

N4 = 2µL, N5 = (λ + 2µT ), N6 = (2µL −µT ),

N∗1 = (3µT − 2µL), N
∗
2 = (3µT − 4µL),

χ1 = N4 − (N3/2),χ2 = N4 − N3

3. Analytical Solution of the Problem

We consider a micropalar fibre–reinforced structure occu-
pying the half–space x3 < 0and a liquid medium occupying
the half–spacex3 > 0. Thus, the two media constitutes an in-
terface at the boundary. Hence, an assumption is made for the
displacements in the media as taken below:

u1 = Pe
i{k (x j p j )−ωt}, (11)

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Anya et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 823 3

u3 = Qe
i{k (x j p j )−ωt}, (12)

φ∗2 = φ
∗ei{k (x j p j )−ωt}, (13)

φ = φ0e
i{k (x j p j )−ωt}, j = 1, 2. (14)

where P, Q, φ∗ and φ0 are amplitudes of the wave displace-
ments respectively. c = ωk is the phase velocity of the wave, k
is the wave number, and ω is the angular velocity of the wave.
Introducing Eqs. (11-14) into Eqs. (7-10), yields the following
equations below:

{k2 D1 − k2c2ρ}P + {k2(N2 + N3) p1 p3}Q
−ikN∗1 p3φ

∗ − ikζ p1φ0 = 0
(15)

{N2k2 p1 p3}P + {k2 D2 − c2k2ρ}Q
iN4k p1φ∗ − iζk p3φ0 = 0,

(16)

−{ikχ1 p3}P −{ikχ2 p1}Q
+{k2 D3 − N2 −ρJk2c2}φ∗ = 0,

(17)

{i ζk p1}P + {iζk p3}Q
+(ξ1k2 + ω0 −$ikc −ρκk2c2)φ0 = 0,

(18)

where D1 = N1 p21 + N6 p
2
2, D2 = N4 p

2
1 + N5 p

2
2, D3 = S 1 p

2
1 +

S 2 p23,p1 = S inα and p2 = Cosα. For non–trivial solution, Eqs.
(15-18) gives the quartic polynomial below:

γ4 + E1γ
3

+ E2γ
2

+ E3γ + E4 = 0 . (19)

Here γ = k2. This means that the characteristic Eq. (19) with
complex coefficients E1,E2,E3,and E4 (See Appendix) gives four
complex roots. Thus, the four waves propagate with complex
phase velocities: c1, c2, c3 and c4 in the solid medium corre-
sponding to the wave numbers k1, k2, k3and k4 respectively. Hence,
the two dimensional model in the x1 x3- plane of the micropo-
lar fibre–reinforced solid half space, have four waves; quasi–P
wave (qLD), quasi–SV wave, quasi–transverse microrotational
(qTM) wave and wave due to voids travelling in the solid medium.
Following Singh’s work [36], for the liquid medium, consider
N1 = N2 = N5 = λ5,ρ = ρ5,φ∗2 = χ2 = χ1 = ζ = N4 = N3 =
φ = 0 into Eqs. (7-10), we obtain the equation below for a
non–trivial solution:

c∗4ρ2 − c∗2λ5ρ
(
p21 + p

2
3

)
= 0 (20)

The roots of the characteristics Eq. (20) for the liquid medium

can simply be represented asc∗ = ±
√
λ5( p21 + p

2
3)/ρ5. This

shows that one phase velocity; c5,corresponding to the wave
numberk5is taken. Thus, in the liquid medium, one of the roots
of Eq. (20) will be negligible; this entails that only quasi–
Longitudinal transmitted (qLT) wave can propagate. Also ob-
serve that p21 + p

2
3 = 1. Hence, the phase velocity in the liquid

medium will be independent of angle of propagation.

Figure 1. Schematic of the problem showing micropolar fibre–reinforced solid
half-space with voids and liquid interface

4. Reflection/Transmission of Waves and the Geometry of
the Problem

Let us consider the propagation of P-wave (Longitudinal
wave) incident on a micopolar fibre-reinforced half-space in the
x1 x3-plane with voids such that the boundary is plane in nature
and it’s in an interface with a liquid medium. The geometry is
demonstrated in Figure 1 below.

Also any one of the four waves can be chosen as incident
wave. Figure 1 shows that when quasi–P wave(P0) is incident
at micropolar fibre–reinforced anisotropic solid and liquid in-
terface, there exist four reflected waves as quasi–P (P1) or qLD,
quasi–SV (P2) or qTD, quasi–TM (P3) and wave due to voids
(P4)with their angles as α0,α1, α2,α3,α4 respectively. Also
the transmitted wave exists as; quasi–longitudinal transmitted
(qLT); (P5)wave, with angleα5.

5. Boundary Conditions and Results

The following are the boundary conditions taken at the in-
terface of the micropolar fibre reinforced solid with voids and
liquid:

1. Stresses at the common interface are continuous i.e. σα33 =
σl33, σ

α
13 = σ

l
11,atx3 = 0.

2. The conditions due to microrotation and voids takes the
form:

(a) mα32 = 0, observe that m32 = 0 ⇒ φ
∗
2,3 = 0, and

(b) φα
,3 = 0, atx3 = 0, respectively.

3. Normal displacements are continuous at the common in-
terface:
uα3 = u

l
3, at x3 = 0.

3



Anya et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 823 4

We choose the displacement components, micro-rotation vec-
tors and the volume fraction field as:

uα1 = Aα d
α
1 e

iµα, uα3 = F
αAα dα1 e

iµα,
φ∗2 = ikαG

αAα dα1 e
iµα,

φα = HαAα dα1 e
iµα, ul1 = Al d

l
1e

iµl,
ul3 = I

l Al dl1e
iµl.

 , (21)
where µα = kα(x1 pα1 + x3 p

α
3 − cαt), α = 0 correspond to in-

cident wave, α = 1, 2, 3, 4 corresponds to reflected waves in
the solid medium andµ` = k`(x1 p`1 + x3 p

`
3 − c`t), where ` = 5

corresponds to quasi-longitudinal transmitted (qLT) wave in the
inviscid liquid medium. Also, the coupled relations Fα, Gαand
Hαare obtained from Eqs (15-18) for the solid medium, i.e.

Fα = −Fα1 /F
α
2 ,

Gα = ikα(χ2 pα1 F
α + χ1 pα3 )/(k

2
α(D

α
3 −ρJc

2
α) − N2),

Hα = −(kαζi(pα3 F
α + pα1 ))/(k

2
α(ξ1 −ρκc

2
α) + ω0 −$ikc).

where
Fα1 = p

α
3{(D

α
1 −ρc

2
α − N2( p

α
1 )

2)(k2α(D
α
3 −ρJc

2
α) − N2)

+χ1((pα3 )
2 N∗1 − ( p

α
1 )

2 N4)},
Fα2 = p

α
1{(χ2(p

α
3 )

2 N∗1 − (p
α
1 )

2 N4) + ((pα3 )
2(N2 + N3)

−Dα2 + ρc
2
α)(k

2
α(D

α
3 −ρJc

2
α) − N2)},

Dα1 = N1( p
α
1 )

2 + N6(pα3 )
2, Dα2 = N4(p

α
1 )

2 + N5( pα3 )
2,

and Dα3 = S 1( p
α
1 )

2 + S 2( pα3 )
2,

Similarly, the relation I` for the liquid medium assumes the
form:

I` = {λ`(( p`1)
2 − p`1 p

`
3) −ρ`c

2
`
}/λ`((p`3)

2

−p`1 p
`
3) −ρ`c

2
`
, ` = 5.

Using Eq. (21) into the boundary conditions, a system of equa-
tion is obtained after using Snell’s law i.e., the coefficients;ai j of
the system are made possible by using Snell’s law such that:k0 p

(0)
1 =

k1 p11 = k2 p
2
1 = k3 p

3
1 = k4 p

4
1 = k5 p

5
1 = k, and k0c0 = k1c1 =

k2c2 = k3c3= k4c4 = k5c5 = ω.
Thus, the system obtained takes the form:

ai jZ j = bi, i = j = 1, 2, 3, 4, 5. (22)

where

a1r = kr d
r
1{(N2 p

r
1 + N5 p

r
3 F

r )}/k0d
0
1{(N2 p

0
1 + N5 p

0
3 F

0)},

a1` = −k`d
`
1λ`{( p

`
1 + p

`
3 I
`)}/k0d

0
1{(N2 p

0
1 + N5 p

0
3 F

0)},

a2r = kr d
r
1{(N4 F

r pr1 + G
r )}/k0d

0
1{(N4 F

0 p01 + G
0)},

a2` = −k`d
`
1λ`{(I

` p`1)}/k0d
0
1{(N4 F

0 p01 + G
0)},

a3r = k
2
r d

r
1 p

r
3G

r/k20 d
0
1 p

0
3G

0, a3` = 0,

a4r = kr p
r
3 H

r dr1/k0 p
0
3 H

0d01, a4` = 0

a5r = F
r dr1/F

0d01, a5` = −I
`d`1/F

0d01.

Hence, Z j is the reflection and transmission coefficients, and the
amplitude ratios of the reflected and transmitted waves areZi =
|Ai/A0| , bi = −1,i = j = 1, 2, 3, 4, 5, r = 1, 2, 3, 4, and` = 5.
Also observe that k0=k1, c0=c1and the speed of the waves;
c0, c1, c2, c3, c4, and c5 depends upon the material parameters.

The components of propagation and unit displacement vectors
are as follows:

p(0)1 = S inα0, p
(0)
3 = Cosα0, d

(0)
1 = S inα0,

d(0)3 = Cosα0,
p(1)1 = S inα1, p

(1)
3 = −Cosα1, d

(1)
1 = S inα1,

d(1)3 = −Cosα1,
p(2)1 = S inα2, p

(2)
3 = −Cosα2, d

(2)
1 = Cosα2,

d(2)3 = S inα2,
p(3)1 = S inα3, p

(3)
3 = −Cosα3, d

(3)
1 = Cosα3,

d(3)3 = S inα3,
p(4)1 = S inα3, p

(4)
3 = −Cosα4, d

(4)
1 = Cosα4,

d(4)3 = S inα4,
p(5)1 = S inα5, , p

(5)
3 = Cosα5, d

(5)
1 = S inα5,

d(5)3 = Cosα5.

(23)

6. Conclusion

This article dealt wholly on the formulation and the analyt-
ical solution of waves in a micropolar fibre-reinforced medium
having pores and interfaced with liquid medium. Four reflected
waves namely; quasi–P or qLD, quasi–SV or qTD, quasi–TM
and wave due to voids traveling in the solid medium were found
while quasi–longitudinal transmitted (qLT) was found traveling
in the liquid medium due to the negligible viscosity of the liq-
uid. The analytical derivation of the amplitude ratios of both
reflected and transmitted waves respectively, were derived and
presented. Also, we observed that the phase velocity in the liq-
uid medium is independent of the angle of propagation and the
reverse is the case in the solid medium such that the phase ve-
locity of propagation were found to be dependent on the angle
of propagation. Thus, it is worthy of note to state that the sig-
nificance of the model should prove useful to new researchers,
scientists, material scientists working to ascertain models that
could be imperative in predicting some seismological analyses
in some solid/liquid structures in terms of propagating phenom-
ena. Future works to this study-“mathematical modeling of
waves in a porous micropolar fibre reinforced structure and liq-
uid interface”, could incorporate rotating viscoelasticity of the
solid/liquid media, grooved boundary conditions and non-local
effects to the model.

Acknowledgments

The authors are very grateful to the editors and anonymous
referees for their careful reading of this manuscript and helpful
suggestions.

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Appendix

E1 = (D1(D3
(
i2ζ2 p23 −ω

2ρξ1
)
− i2 N4 p21ξ1χ2

−D2(
(
Jω2ρ + N2

)
ξ1 + D3

(
iω$ + ω2κρ−ω0

)
))

+D2
(
D3

(
i2ζ2 p1 p3 −ω2ρξ1

)
− i2 p23ξ1χ1 (N1)

∗
)

+p1 p23(−D3(i
2ζ2 N3 p1 − N22 p1 (ω (i$ + ωκρ) −ω0)

+N2(i2ζ2 p3 + p1(i2ζ2 + N3 (−ω (i$ + ωκρ) + ω0))))
+p1ξ1(N32 + N

2
2

(
Jω2ρ + N3

)
+ i2 N3 N4χ1+

N2


Jω2ρN3

+i2
(

N4χ1+
χ2 (N1)∗

) )))/D3 (D1 D2 − N2 (N2 + N3) p21 p23) ξ1,
E2 = (−i2ω2ζ2ρD3 p1 p3 − i2ω2ζ2ρD3 p23 + i

2 Jω2ζ2

ρN2 p21 p
2
3 + i

2ζ2 N22 p
2
1 p

2
3 − iJω

3$ρN22 p
2
1 p

2
3

−Jω4κρ2 N22 p
2
1 p

2
3 − iω$N

3
2 p

2
1 p

2
3 −ω

2κρN32 p
2
1 p

2
3

+i2 Jω2ζ2ρN3 p21 p
2
3 + i

2ζ2 N2 N3 p21 p
2
3

−iJω3$ρN2 N3 p21 p
2
3 − Jω

4κρ2 N2 N3 p21 p
2
3

−iω$N22 N3 p
2
1 p

2
3 −ω

2κρN22 N3 p
2
1 p

2
3

+i2 Jω2ζ2ρN2 p1 p33 + i
2ζ2 N22 p1 p

3
3 + ω

4ρ2 D3ξ1
−i3ω$N2 N4 p21 p

2
3χ1 − i

2ω2κρN2 N4 p21 p
2
3χ1

−i3ω$N3 N4 p21 p
2
3χ1 − i

2ω2κρN3 N4 p21 p
2
3χ1

+i4ζ2 N4 p1 p33χ1 − i
4ζ2 N4 p31 p3χ2 + i

2ω2ρN4 p21ξ1χ2
+Jω2ρN22 p

2
1 p

2
3ω0 + N

3
2 p

2
1 p

2
3ω0 + Jω

2ρN2 N3 p21 p
2
3ω0

+N22 N3 p
2
1 p

2
3ω0 + i

2 N2 N4 p21 p
2
3χ1ω0

+i2 N3 N4 p21 p
2
3χ1ω0 + D1(−i

2 Jω2ζ2ρp23 − i
2ζ2 N2 p23

+Jω4ρ2ξ1 + ω2ρN2ξ1 + i3ω$N4 p21χ2
+i2ω2κρN4 p21χ2 + ω

2ρD3
(
iω$ + ω2κρ−ω0

)
+D2

(
Jω2ρ + N2

)
(ω (i$ + ωκρ) −ω0)

−i2 N4 p21χ2ω0) − i
4ζ2 p43χ1 (N1)

∗ + i2ω2ρp23ξ1χ1 (N1)
∗

5



Anya et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 823 6

+i4ζ2 p21 p
2
3χ2 (N1)

∗
− i3ω$N2 p21 p

2
3χ2

(N1)∗ − i2ω2κρN2 p21 p
2
3χ2 (N1)

∗ + i2 N2 p21 p
2
3χ2ω0 (N1)

∗

+D2(−i2ζ2
(
Jω2ρ + N2

)
p1 p3

+Jω4ρ2ξ1 + ω2ρN2ξ1 + ω2ρD3(ω (i$ + ωκρ) −ω0)
+i3ω$p23χ1 (N1)

∗ + i2ω2κρp23χ1

(N1)∗ − i2 p23χ1ω0 (N1)
∗))/D3

(
D1 D2 − N2

(
N2+
N3

)
p21 p

2
3

)
ξ1,

E3 = −ω2ρ(iω3$ρD3 + ω4κρ2 D3 − i2 Jω2ζ2ρp1 p3−
i2ζ2 N2 p1 p3 − i2 Jω2ζ2ρp23 − i

2ζ2 N2 p23
+Jω4ρ2ξ1 + ω2ρN2ξ1 + i3ω$N4 p21χ2

+i2ω2κρN4 p21χ2 + D2
(
Jω2ρ + N2

) (
iω$ + ω2κρ−ω0

)
+D1

(
Jω2ρ + N2

)
(ω (i$ + ωκρ) −ω0) −ω2ρD3ω0

−i2 N4 p21χ2ω0 + i
3ω$p23χ1 (N1)

∗ + i2ω2κρp23χ1 (N1)
∗

−i2 p23χ1ω0 (N1)
∗)/D3

(
D1 D2−
N2 (N2 + N3) p21 p

2
3

)
ξ1,

E4 = ω4ρ2
(
Jω2ρ + N2

)(
ω

(
i$+
ωκρ

)
−ω0

)
/D3

(
D1 D2−
N2 (N2 + N3) p21 p

2
3

)
ξ1.

6