3. The trapped water waves in a two-layer fluid in an ice-covered ocean of finite depth with a porous bottom.
For the problem mentioned in the first one, the bottom topographical variation can be modeled as two steps of different water depths with finite and infinite regions. The problem can be formulated using matched eigenfunction expansions and then a homogeneous integral equation may be used to derive the horizontal fluid velocity across the line joining the finite and infinite regions. Then by using Fourier expansion method, an infinite system of homogenous linear equations can be obtained, the vanishing of the determinant of which provides trapped mode eigenvalues if they exist. For the second one, the trapped mode can be investigated over a small bottom undulation in particular. Sinusoidal ripples on the sea-bed are of considerable significance due to the ability of an undulating bed to reflect incident wave energy which is important in respect of possible ripple growth if the bed is erodible.
In our thesis we examine the trapped modes in an infinite depth three-layer fluid with layer-wise different densities where we restrict the uppermost layer to be covered by a free surface only. Its extension to an ice-cover or a rigid lid can also be considered. The middle layer considered in this case is of constant density but it can also be extended to the case where the middle layer is linearly stratified. Hence the possible extensions in a three-layer fluid are:
1. Oblique gravity trapped waves in a three-layer fluid bounded above by a rigid lid.
2. Oblique flexural trapped waves in a three-layer fluid of finite depth.
3. The trapped water waves in a three-layer fluid in which density of the middle layer varies linearly with the width of the layer.
We consider the trapped modes arising due to the presence of the impermeable horizontal circular cylinder placed in either layer of a two-layer and a three-layer fluids. The same cylinder may be replaced by a totally submerged horizontal symmetric thin bodies or vertical cliffs placed in either of the layers. Hence the possible extensions in this regard can be:
1. Oblique gravity trapped waves in a finite depth two-layer fluid supported by a symmetric thin body.
2. Trapped waves in a two-layer fluid of finite depth supported by vertical cliffs.
We are fairly confident that these above mentioned problems can be attempted and solved with the knowledge garnered while investigating the problems in this thesis.
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Euler–Bernoulli equation for a thin ice-cover
Let η be the displacement of a point located at the centre of a plane in the direction of the normal to this plane, and we write
κ1 = ∂2η
∂x2, κ2 = ∂2η
∂y2, τ = ∂2η
∂x∂y.
The strain–energy–function takes the form (Love (2007), Pg. 503, Eq. (6)) Ez2
2(1−ν2) h
(κ1+κ2)2−2(1−ν)(κ1κ2−τ2) i
, (A.1)
whereE andν, respectively, denote the effective average Young’s modulus and Poisson’s ratio for ice. The potential energy of bending, estimated per unit area, is obtained by integrating the above expression with respect toz from 0 to h0, with h0 being the thickness of the ice-cover.
The integration results in the following:
D 2
h
(κ1+κ2)2−2(1−ν)(κ1κ2−τ2) i
, (A.2)
whereDis the “flexural rigidity” given by Eh03 12(1−ν2).
The kinetic energy per unit area isT(x, y, t) = 12mη2t, wheremis the mass per unit surface area of the ice-cover. Total potential energy per unit area,V(x, y, t), is the sum of the strain energy due to the curvature of the plate and the potential energy due to the transverse pressure, i.e.,
V(x, y, t) = 1 2D
h
(∇2x,yη)2−2(1−ν)(ηxxηyy−ηxy2 ) i
−qη, (A.3)
whereq is the net external force on the plate per unit area.
Let us consider a rectangular domain R :{(x, y)|0 < x < x0, y0 < y < y1} on the plate andT = (t0, t1) be an interval in time. Then the Lagrangian operator is
L(η) = Z
T
Z
R
(T −V)dx dy dt. (A.4)
Appendix A 106
Hence the Lagrangian is L:= 1
2D h
(∇2x,yη)2−2(1−ν)(ηxxηyy−ηxy2 ) i
−qη−1
2mη2t ≡ L(x, z, t, η, ηt, ηxx, ηyy, ηxy). (A.5) The corresponding Euler–Lagrange equation is
∂L
∂η − ∂
∂x Ã
∂L
∂ηt
! + ∂2
∂x2 Ã
∂L
∂ηxx
! + ∂2
∂y2 Ã
∂L
∂ηyy
! + ∂2
∂x∂y Ã
∂L
∂ηxy
!
= 0. (A.6)
Now,
∂L
∂η = −q; ∂L
∂ηt =−mηt; ∂L
∂ηxx =D[ηxx+ηyy −(1−ν)ηyy];
∂L
∂ηyy = D[ηxx+ηyy−(1−ν)ηxx]; ∂L
∂ηxy = 2D(1−ν)ηxy. Putting these values into Eq. (A.6) gives
D∇4x,yη+mηtt =q, (A.7)
where ∇4x,y = (∂4/∂x4) + 2(∂2/∂x2)(∂2/∂y2) + (∂4/∂y4) is the biharmonic operator in the plane of the ice-cover. In our caseη is the displacement normal to the xy-plane at the upper surface. It is clear that Eq. (A.7) is equivalent to Eq. (3.5).
Coordinate shift
R β Z
φ ψ
Figure B.1: Coordinate shift for Graf’s addition theorem
We are required to transformKn(lrj) cosnθj,j= 1,2, ..., N, where the coordinates (rj, θj) associated with thej-th cylinder are measured from (hj, f) to the coordinates (rp, θp) associated with the p-th cylinder placed at (hp, f) and vice versa. We use Graf’s Addition Theorem for modified Bessel functions (Watson (2008)) given by
Kn(β) cosnψ= X∞
m=−∞
Kn+m(R)Im(Z) cosmφ, (B.1a) Kn(β) sinnψ=
X∞
m=−∞
Kn+m(R)Im(Z) sinmφ, (B.1b)
whereR, β, Z are the sides of the triangle given in Fig. B.1. In the present case, the triangle has sidesrj,rp and |hj−hp|and so upon the following substitutions
β =lrj; ψ= π
2 −θj; Z =lrk; φ=θk−3π
2 ; R=l|hj−hp|, (B.2) and considering the caseshj > hp,hp > hj separately, we have
Kn(lrj) cosnθj = X∞
m=0
³
Cnmjp cosmθp+Djpnmsinmθp
´
Im(lrp), (B.3) Kn(lrj) sinnθj =
X∞
m=0
³
Ajpnmcosmθp+Bnmjp sinmθp
´
Im(lrp), (B.4)
Appendix B 108
where
Ajpnm = εm 2
³
(−1)mKn+m(l|hj−hp|) +Kn−m(l|hj−hp|)
´
sin(n+m)π
2, (B.5) Bnmjp = εm
2
³
(−1)m+1Kn+m(l|hj−hp|) +Kn−m(l|hj−hp|)
´
cos(n+m)π
2, (B.6) Cnmjp = εm
2
³
(−1)mKn+m(l|hj−hp|) +Kn−m(l|hj−hp|)
´
cos(n+m)π
2, (B.7) Dnmjp = −εm
2
³
(−1)m+1Kn+m(l|hj−hp|) +Kn−m(l|hj−hp|)
´
sin(n+m)π
2. (B.8) Using the relations xj =hp−hj +xk andzj =zp in the integral in (6.12), we obtain
(−1)n Z ∞
0
coshnu cos(lxjsinhu)evzj CL(v)du
= (−1)n Z ∞
0
coshnu cos(l|hp−hj|sinhu) cos(lxpsinhu)evzp CL(v)du + (−1)n+1
Z ∞
0
coshnu sin(l|hp−hj|sinhu) sin(lxpsinhu)evzp CL(v)du. (B.9) The well known generating function of modified Bessel functions is utilized (Watson (2008)):
exp h1
2X(T +T−1) i
= X∞
m=0
1 2εm
³
Tm+T−m
´
Im(X), (B.10)
whereε0 = 1, εm = 2, m≥1. Substituting X =−lrp, T = exp[i(θp+iu)] in it and equating real and imaginary parts, the following results are obtained:
evzp cos(lxpsinhu) = evf X∞
m=0
(−1)mεm coshmu Im(lrp) cosmθp, (B.11) evzp sin(lxpsinhu) = evf
X∞
m=0
(−1)m+1εm sinhmu Im(lrp) sinmθp. (B.12) Using the above relations in (B.9), we obtain
(−1)n Z ∞
0
coshnucos(lxjsinhu)evzjCL(v)du= X∞
m=0
h
αjpnmcosmθp+βnmjp sinmθp i
Im(lrp), (B.13) with
αjpnm = (−1)m+nεm Z ∞
0
coshnu coshmu cos(l|hp−hj|sinhu)evf CL(v)du, (B.14) βnmjp = (−1)m+nεm
Z ∞
0
coshnu sinhmu sin(l|hp−hj|sinhu)evf CL(v)du. (B.15) Similarly, for the integral in (6.14), we get
(−1)n+1 Z ∞
0
sinhnu sin(lxjsinhu)evzj CL(v)du= X∞
m=0
h
ajpnmcosmθp+bjpnmsinmθp i
Im(lrp), (B.16)
with
ajpnm = (−1)m+n+1εm Z ∞
0
sinhnu coshmu sin(l|hp−hj|sinhu)evf CL(v)du, (B.17) bjpnm = (−1)m+nεm
Z ∞
0
sinhnu sinhmu cos(l|hp−hj|sinhu)evf CL(v)du. (B.18) These are the results used in Chapter 6 for shifting local coordinates from any arbitrary cylinderj to a fixed cylinder p.