6.2 Plane wave approximation
6.2.3 Analysis of the dispersion relation
Derivation and discussion
Now, the casesρ→1andd→0ord→hin relation (6.16) present the dispersion relation in a homogeneous fluid over a porous bottom. The dispersion relation in equation (6.16) gets reduced to the dispersion relation for a two-layer fluid over an impermeable bed when G= 0. Corresponding to real values of G, the dispersion relation (6.16) gives rise to two positive real roots, say k1 and k2, restricted by 0 < k1 < k2, which correspond to the propagating modes. This equation also has an infinite number of purely imaginary roots k = kn for n = 3,4,5, . . . , which correspond to the evanescent modes where kn = itn, tn is a real number. These evanescent modes do not contribute to wave propagation.
Furthermore, those real and purely imaginary roots also have negative values which are also roots of this equation and they are wavenumbers of the waves propagating in the negative direction. Since the existence of exactly two positive roots k1 and k2 of (6.16) is ensured, it confirms the existence of two progressive propagating wave modes: (i) Free Surface Mode (SM) and (ii) Interface Mode (IM). Finding the roots of the dispersion relation (6.15) for real G by using the Newton-Raphson Method is a relatively common task. Finding the roots with complex G, on the other hand, is more difficult. The dispersion relation for complex G has complex roots given by kn = ±an±ibn for n = 1,2, . . ., where all an and bn are real. The roots nearer to the real axis correspond to the most prominent progressive waves in surface mode and interfacial mode where the dispersion relation given by equation (6.16) contains a pair of complex roots of opposite signs±k1 and ±k2. It also contains an endless number of complex roots that are located
near the imaginary axis and have opposite signs ±kn, n = 3,4, . . ., and these correspond to the evanescent modes. In the case of the porous-effect parameterG being complex, all roots are complex in nature and they are close to the roots found corresponding to real G.
If we deeply analyse equation (6.16), we obtain a quadratic equation in ω2 as follows:
Aω4 −Bω2+C = 0, (6.20)
where
A =k(ρtanhkdtanhk(h−d) + 1)−G(ρtanhkd+ tanhk(h−d)), B =gk[k(tanhkd+ tanhk(h−d))−G(tanhkdtanhk(h−d) + 1)], C =g2k2(1−ρ) tanhkd(ktanhk(h−d)−G).
From equation (6.20), we obtain two propagating modes
ω1 =± s
B+√
B2−4AC 2A
, ω2 =± s
B−√
B2−4AC 2A
, (6.21) where subscripts1 and2 refer to the propagating mode corresponding to the free surface and interface, respectively. An interesting fact lies here. For realG, wave modes result in the phase velocities in the form
c1 = ω1
k and c2 = ω2
k . (6.22)
Furthermore, the group velocities are given by cg
1 = ∂ω1
∂k and cg
2 = ∂ω2
∂k . (6.23)
Wave energy propagation rate is proportional to the group velocity of the wave train.
Hence, wave energy propagation stops when group velocity is 0 and this phenomenon is referred to as blocking.
For deep water waves for both layers of the fluid, |kd| 1 and |k(h−d)| 1 are valid which yield two ω2 values from (6.21):
ω2 =gk, gk(1−ρ)
(1 +ρ), (6.24)
which correspond to waves in the free surface and interface modes, respectively. Fur- thermore, it is self-evident that in deep water, the angular frequency in both surface and interface modes is unaffected by the porosity of the sea-bed. The phase velocities for
6.2. Plane wave approximation 141
waves in the surface and interface modes are as follows:
c1 = rg
k andc2 = s
g(1−ρ)
k(1 +ρ), (6.25)
which are the same as the wave flow in a two-layer fluid over rigid bottom. As a result, the bottom porosity can be ignored for deep water waves. The phase velocity for deep ocean waves is larger in the free surface mode (c1 > c2). The group velocities are as follows:
cg1 = 1 2
rg
k andcg2 = 1 2
s
g(1−ρ)
k(1 +ρ). (6.26)
In a similar manner, for shallow water waves (kd1and k(h−d)1), equation (6.20) yields
A =k(ρk2d(h−d) + 1)−kG(ρd+ (h−d)), B=gk
k2h−G( k2d(h−d) + 1) , C =g2k3(1−ρ)d
k2(h−d)−G ,
which can be used to find ω2 as well as phase and group velocity.
It is interesting to observe what happens when B2 −4AC = 0. This leads to the free surface and interfacial angular frequency to coincide, resulting in coalescence of both the propagating modes. For shallow water depth, B2−4AC = 0 leads to the algebraic equation
g2k2h
k2(h−Gd(h−d))−G 2 −4k2d(1−ρ)(k2(h−d)−G)
ρd(h−d)k2 +1−G(ρd+ (h−d))}] = 0,
(6.27)
which implies the coincidence of wave modes. However, the important consequence is the sign change ofB2−4AC. The coincidence of wave modes also raises the possibilities of mode swapping. It may be noted that, for ktanhk(h−d)−G= 0, sayk =±kc, there is a situation of no wave propagation. In that case,ω2 = 0 and ω1 satisfies
ω21 = B
A
. (6.28)
Hence, for such wave modes, the system works as a homogeneous fluid system in the presence of the bottom porosity. We can term this as cut-off wave mode kc. Physically, this can be defined as the effect of damping of the wave mode due to bed porosity. For shallow water waves, kc=±p
G/(h−d).
The phase speed of the free surface wave mode and the interfacial wave mode are
0 0.5 1 1.5 2 2.5 3
kh
1 1.5 2 2.5 3 3.5
c 1
Gh=0 Gh=1 Gh=2 Gh=4 Gh=8
kch=0.451 (Gh=1) kch=0.925 (Gh=3) kch=0.643 (Gh=2) kch=1.356 (Gh=4)
(a)
0 0.5 1 1.5 2 2.5 3
kh
0 0.2 0.4 0.6 0.8 1
c2
Gh=0 Gh=1 Gh=2 Gh=4 Gh=8 kch=0.451 (Gh=1) kch=0.643 (Gh=2) kch=0.925 (Gh=3) kch=1.356 (Gh=4)
(b)
Figure 6.3: Phase speeds in free surface and interface mode against non-dimensional length kh by varying Gh corresponding to various values of bed porosity with ρ= 0.75, d/h = 1/2and kc points are expressed in blue box.
computed and examined for various values of real sea-bed porosity in Figures 6.3(a) and 6.3(b). For zero porosity value, smooth change can be observed in both modes of phase velocity, but due to the bottom porosity, major changes can be observed. For moderate porosity values, phase velocity c1 starts from a lower value and it attains its maximum.
However, depending upon the porosity, certain phase shift of kc as well maximum can be observed. Moreover, blocking is not observed with respect to wave modes. Maximum in wave modes may be observed due to the effect of bottom porosity. Larger values of bed porous-effect parameter G result in lower phase values in both modes. In Figure 6.3(b), phase speed in interfacial modes reaches its maximum value immediately after k = kc which shows that the damping reaches its minimum afterk =kc. But zero porosity shows a smooth phase speed reduction. Nature ofc1 changes for differentGh but it starts from the same point and converges after a certain kh. Similar behaviour is observed also for interfacial phase velocity.
Observation of mode swapping
The roots of dispersion relation (6.16) for real G may be found by using an available approach. However, this is not true in the case of complex G. In order to identify such roots, the algorithm of Mendez and Losada [68] is followed so that the roots get transformed from real G to complex G. The benefit of this methodology is that the propagating modes can be clearly differentiated, despite of the fact that they are all complex. Other techniques, such as homotopy perturbation (HP) method of Chang and Liou [18] can also be applied. We here adopt the methodology of Mendez and Losada [68] because of the consistency of the results in both ways, ignoring the computational cost. Furthermore, the direct application of the Newton-Raphson based algorithm through Matlab would definitely provide us the roots but it may be accompanied by a reasonable amount of complication while guessing the suitable initial value. In addition to that, for
6.2. Plane wave approximation 143
D0(k) = 0, with0 denoting differentiation, the Newton-Raphson method does not succeed.
In a similar manner, the eigenfunction expansion also fails for these values of k. In this context, we apply the program Matlab 2019a to implement an algorithm, based on the Newton-Raphson method, to verify the complex roots whereas we find the initial guess for the roots by adopting the method of Mendez and Losada [68].
Gh⇓ Err1 Err2 Err3 Err4 Err5 Err6
10 + 5i 0 0 1.22·10−5 1.18·10−5 2.47·10−5 8·10−6 10 + 10i 0 0 2.73·10−5 3.6·10−5 1.14·10−5 3.14·10−5 10 + 15i 0 0 3.51·10−5 5.42·10−5 1.81·10−5 5.54·10−5
Table 6.1: Difference of kh obtained by two different approaches, i.e., Err for different values of Gh with d/h= 0.5, Kh= 0.63 and ρ= 0.9.
In Table 6.1, the difference of the roots computed by the method of Mendez and Losada (k(M L)) and the method of Newton-Raphson (k(N R)) is compared by following the formula below:
Errj =|k(M L)j h−kj(N R)h|.
The result shows no difference up to four places after decimal point.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Kh
0 5 10 15 20 25 30 35 40
k nh
=0.8
=0.85
=0.9
=0.95 k1h
=0.8
=0.85
=0.9
=0.95 k2h
(a)
0 5 10 15 20 25
Gh
0 5 10 15 20 25
k nh
=0.8
=0.85
=0.9
=0.95 k1h
=0.8
=0.85
=0.9
=0.95 k2h
(b)
Figure 6.4: Variation of non-dimensional wave modes in free surface and interface for various values ofρ(a) against non-dimensionalKhwith the valuesd/h= 1/2andGh= 5 and (b) against non-dimensionalGh with the values d/h= 1/2and Kh= 0.19.
The distribution of both the wave modes for non-dimensional frequency and non- dimensional porosity of the sea-bed is examined in Figures 6.4(a) and 6.4(b). No coa- lescence is found between these real propagating modes. Increasing Kh brings changes ink1h for initial values only but no further major change is visible whereas k2h initially starts from the same value and changes with an increase ofKh. Lowerρ results in lower values for both the modes. While changing the porosity, as observed from Figure 6.4(b), free surface mode starts initially from the same point and shoots up with an increase in
porous-effect parameter. Interfacial mode initially shows the difference but after Gh≥5, it exhibits no major difference. For real propagating modes for real G, the illustrations show that there is no coalescence between the modes, and the eigenfunction expansion can be carried forward while assuming distinct characteristics of the modes.
Figures 6.5(a) to 6.5(d) discuss the behaviour of the roots of the dispersion relation
0 0.5 1 1.5 2 2.5 3
Re(kn)h
0 5 10 15 20
Im(k n)h
k1h k2h
k3h k4h
k5h k6h
k7h
(a)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Re(kn)h
0 5 10 15 20
Im(k n)h
k1h k2h
k3h k4h
k5h k6h
k7h
(b)
0 1 2 3 4 5 6 7
Re(kn)h
0 5 10 15 20 25 30 35
Im(k n)h
k1h k2h
k3h k4h
k5h k6h
k7h
(c)
0 0.5 1 1.5 2 2.5 3 3.5
Re(kn)h
0 5 10 15 20
Im(k n)h
k1h k2h
k3h k4h
k5h k6h
k7h
(d)
Figure 6.5: Roots of dispersion relation (6.16) corresponding to varying Im(G) in [0,5]
corresponding to (a) Kh = 0.0943, (b) Kh = 0.2515,(c) Kh = 0.3773, (d) Kh = 3.0182 for d/h= 1.
against the dimensionless depth with the aim of examining whether there is any situation for which the eigenfunction solution fails. The figures depict the complex wavenumbers corresponding to the first seven non-dimensional modes knh, n= 1, . . . ,7, for the values of Re(G) = 1, Im(G) in the range [0,5] and different values of Kh. For Kh = 0.0943, Figure 6.5(a) shows the dimensionless wavenumbers due to complexG. For this case, real parts ofk1 andk2 decrease with higher Im(G) but the real parts of the evanescent modes rise sharply for increasing Im(G). The real part of the evanescent mode k3 crosses the values of k1h and k2h for higher values of Im(G), to be precise, for Im(G) >2.68. This implies that the plane-wave approximation does not have validity for large complex values ofGcorresponding to such values ofKh. Furthermore, no mode swapping appears to take place corresponding to shallow water. If there is any intersection between two curves, then we refer to it as mode swapping. For Figures 6.5(b) and 6.5(c), values ofKh= 0.2515and Kh= 0.3773, respectively, are used andk1h shows an increasing nature corresponding to
6.3. Method of solution 145