Conditions at the interfaces and boundaries can be stated in terms of the vertical displacement ∂ψ/∂z, and the pressure given by (3.5). The former is detailed for the homogeneous solution [in spirit of (3.3)] as follows:
∂ψ
∂z =−(ω2a)−1/3 Z ∞
0
{A+Ai0(ζ) +A−[Ai0(ζ)−iBi0(ζ)]}J0(krr)krdkr, (3.22) where the prime denotes differentiation. This is performed similarly for the particular solution. Equations are obtained then considering that between fluid layers the dis-placement and pressure is continuous. As for the wave guide boundaries we refer to the simplest models. On the surface where the water is limited by the atmosphere the pressure is zero, which is called the pressure release boundary; the sea bottom, in turn, can be modelled as rigid boundary, where the vertical displacement is zero. Note that because the conditions hold for all ranges, the integral kernels can be set equal. This yields a system of 2N linear equations for 2N unknown A+’s and A−’s. Algorithms for solving the system of simultaneous equations and subsequently carrying out the inverse Hankel transformation comprise an extensive literature.
Theory and modelling of wave propagation 31
ρ(z) d dz
µ 1 ρ(z)
dΨm dz
¶
+ (k2−krm2 )Ψm = 0, (3.25) in which krm is the separation constant. In addition, the mode shape functions have to satisfy the equations
Ψ(0) = 0 and Ψ0(D) = 0 (3.26)
for the stated boundary conditions.
The modal equation is a classical Sturm-Liouville problem with known properties.
For a nonsingular problem there exist an infinite series ofkr1 > kr2 > . . . eigenvalues, analogous to the eigenfrequencies of a vibrating string, and corresponding Ψm(z) mode shapes. The m’th mode has m roots, and is orthogonal to any other one:
Z D
0
Ψm(z)Ψn(z)
ρ(z) = 0, for m6=n. (3.27)
They are also arbitrary to within a multiplicative constant, and for convenience they are scaled so that
Z D
0
Ψ2m(z)
ρ(z) = 1. (3.28)
For the given problem the modes form a complete set, and thus any function can be represented by a linear combination of them. Hence, the solution is sought in such a form:
p(r, z) = X∞
m=1
Φm(r)Ψm(z). (3.29)
Substituting this into (3.23), and utilising the modal equation (3.25), we find that X∞
m=1
½1 r
d dr
µ
rdΦm(r) dr
¶
Ψm(z) +k2rmΦm(r)Ψm(z)
¾
=−δ(r)δ(z−zs)
2πr . (3.30)
Next, the operator
Z D
0
Ψm(z)(·)
ρ(z) dz (3.31)
is applied. Because of the orthogonality property (3.27) only the m’th term remains from the sum, yielding:
1 r
d dr
·
rdΦm(r) dr
¸
+krn2 Φm(r) =−δ(r)Ψm(zs)
2πrρ(zs) . (3.32)
The solution of this equation can be given in terms of the Hankel function:
Φm(r) = i
4ρ(zs)Ψm(zs)H0(1,2)(krmr). (3.33) For the final solution – due to the radiation condition, i.e. that waves are travelling away from the source but not towards it – the Hankel function of the first kind is chosen, and hence we express the wave field as:
p(r, z) = i 4ρ(zs)
X∞
m=1
Ψm(zs)Ψm(z)H0(1)(krmr). (3.34) An alternative approach starts with the spectral integral representation:
p(r, z) = Z ∞
0
G(z, zs, kr)J0(krr)krdkr
= 1
2 Z ∞
−∞
G(z, zs, kr)H0(1)(krr)krdkr, (3.35) where the general Green’s function satisfies the depth separated equation:
ρ(z)
·G0(z) ρ(z)
¸0
+ [k(z)2−kr2]G(z) = −δ(z−zs)
2π , (3.36)
and generalised boundary conditions:
fT(k2r)G(0) + gT(kr2) ρ(0)
dG(0)
dz = 0, (3.37)
fB(kr2)G(D) + gB(k2r) ρ(D)
dG(D)
dz = 0, (3.38)
superscripts T and B referring to the top and bottom respectively.
This approach gains its utility from a special property of the integral kernel in (3.35). It is straightforward to show that the modal expansion of the Green’s function that satisfies the depth separated equation (3.36) can be written as:
g(z) = 1 2πρ(zs)
X
m
Ψm(zs)Ψm(z)
k2r−krm2 . (3.39)
In this representation it can be seen that the Green’s function has singularities at the modal wave numbers. In addition, certain boundary conditions may introduce further subtleties. For an example let us consider a simple wave guide introduced by Pekeris.
As illustrated in Figure 3.2, the wave guide consists of a water layer of finite depth D on top, and a fluid half space below. The density and the sound speed in the water
Theory and modelling of wave propagation 33
Figure 3.2: Schematic of the Pekeris wave guide. This figure has been reproduced from (Jensenet al. 2000, Fig. 5.5)
and the bottom have different constant values. For such a wave guide it can be shown that the bottom boundaries are described by the equations:
fB(kr2) = 1 and gB(k2r) = ρb/ s
k2r − µω
cb
¶2
. (3.40)
As the horizontal wave number appears in the equation of the boundary condition, and enters through the square root function, it introduces a branch cut in the complex plane of kr for the presently considered wave guide. On the two sides of the branch cut, the vertical wave number is defined by different formulae, which leads to the discontinuity of the integral kernel across the branch cut.
The choice of the branch cut is somewhat arbitrary; a choice called the EJP branch cut is illustrated in Figure 3.3. In this same figure the associated modal wave numbers as poles are also indicated, and the integration path proposed to perform the trans-formation under (3.35). Given that with this choice of the path the integral kernel is analyticl, we can apply the residue theorem (schematically written) as follows:
I
= Z ∞
∞
+ Z
C∞
+ Z
CEJP
= 2πi XM
m=1
res(krm). (3.41)
Because of the decaying property of the Hankel function, as the radius of the semicircle goes to infinity, the associated integral tends to zero. Hence, the wave field is expressed as the residues of a finite number of poles minus the branch cut integral. In fact, considering the boundary value problem of (3.36)-(3.38) and its solution to be found in standard literature, the residues can be put in terms of the mode shape functions as already seen under (3.34). This time, however, there is a finite number of modes only,
Figure 3.3: Location of eigenvalues for the Pekeris wave guide using the EJP branch cut. This figure has been reproduced from (Jensen et al. 2000, Fig. 5.6)
and also the branch cut integral. Although, the latter is generally negligible in the far field.
Numerical computation of the mode shapes requires the discretisation of the depth coordinate, which yields a series of its discrete values: Ψj = Ψ(zj). With this the differential expressions in the modal equation and the boundaries can be replaced by algebraic expressions. This is done by using some differential scheme which involves the linear combination of a few neighbouring Ψj’s. Doing this for all zj, and assembling the resulting linear equations in a matrix form, we are given a standard eigenvalue problem in the form:
(A−k2rI)x= 0, (3.42)
whereIis the identity matrix, andxinvolves all the Ψj’s. The search for the eigenvalue is always a root finding procedure. For realkr’s standard procedures can be used, but there are algorithms available for problems with complex valued solutions as well.