The History Of The Long Wave Theory

Published Date: 02 Nov 2017

2.1 Long wave theory

Waves in shallow water are called long waves or shallow water waves such as tidal waves, tsunamis and other waves with long periods and wave lengths. This study of long waves is of importance in the engineering field regarding the coastal structure design, studying estuaries environments, and so on. Because long wave may cause extreme natural phenomena related to resonance in harbors and mild slope, shoreline change, and natural disasters (Bellotti,2007; Bender et al.,2003).

Long wave is significantly associated with the general wave theory and solution, and small amplitude wave theory as linear wave solution is useful to solve long wave problem physically. In this section, theory of small amplitude wave is presented, after the governing equations of long wave are derived.

The governing equations were derived from the continuity equation with the assumption of irrotational motion, and an incompressible fluid. Then, the continuity equation can be expressed as Eq.(2.1), and it can be represented as a velocity potential (?) which is a gradient of a scalar function (Dean and Dalrymple, 1984).

(2.1)

where,

(2.2)

The continuity equation can be rewritten in Eq.(2.3)

(2.3)

It is clear that the velocity potential should satisfy the continuity equation. This equation is called the Laplace Equation that occurs in many physics and engineering. A solution for the velocity potential is required to obtain the velocity components (u, v, w). It is necessary to determine suitable boundary conditions for the wave problem. Then, the governing equation can be used to solve the wave problem. In general, two-dimensional wave domain is shown in Figure 2.1. The wave motion in the domain is under the Laplace equation. Where x, z is the coordinate system, �� is the surface water elevation, L is wave length, H is wave height, h is water depth. The boundary conditions for bottom, water surface, and lateral in this domain area given as follows.

Figure 2.1 Schematic of fluid domain for two-dimensional wave solution

At the bottom, the boundary condition is assumed to be horizontal direction, a no flow condition applies as Eq.(2.4)

on z = -h (2.4)

At the free surface, two boundary conditions have to be satisfied. The first boundary is the kinematic condition for the displacement of water surface which is given

on z = ��(x, t) (2.5)

The second boundary condition is the dynamic condition for water surface pressure (p��) related to the Bernoulli equation. The boundary condition for free waves expresses as Eq.(2.6)

(2.6)

The lateral boundary condition is a periodicity condition for waves that are periodic in space and time. The periodic lateral boundary conditions apply in both space and time, Eq.(2.7) and Eq.(2.8)

(2.7)

(2.8)

Using the solution for the partial differential equation and the boundary conditions can be obtained the velocity potential. As the solution of partial differential equation, the separation of variables applies in the fluid domain. Then, the velocity potential can be obtained as Eq.(2.9)

(2.9)

where, k : wave number (=2��/L), �� : angular frequency (=2��/T).

Furthermore, the dispersion equation can be derived from the remaining free surface boundary condition. This equation gives the relationship between wave number and angular frequency as seen in Eq.(2.10). The dispersion relationship means that long wave propagates faster than short length wave.

(2.10)

The dispersion relationship for shallow water can be written in the following the hyperbolic functions. If relative depth is small, tanh kh term becomes kh. Then, Eq.(2.10) can be rewritten as Eq.(2.11) and Eq.(2.12) related to wave speed (C) from wave length and wave period.

(2.11)

or

(2.12)

and

(2.13)

The wave speed or celerity in shallow water is determined by the water depth as Eq.(2.13). The celerity equation can be applied easily in many real fields.

Previously, the velocity potential for small amplitude wave was derived. Using Eq.(2.2) and Eq.(2.9), the velocities for a progressive wave are written as follows.

(2.14)

(2.15)

Based on the small amplitude wave theory, as the horizontal pressures are independent of z, the horizontal wave motion is also depth independent, which means the horizontal velocity is not a function of depth. Thus, integrating over depth can apply to the three-dimensional continuity equation and momentum equations. For small amplitude waves, the nonlinear terms can be neglected to simplify the governing equations. Finally the linearized continuity and momentum equations for long wave can also be derived as follows.

(2.16)

(2.17)

(2.18)

where, U and V : depth averaged velocities corresponding u and v. Eqs.(2.16), (2.17), and (2.18), called the shallow water equations, which are widely used to solve the long wave problem in many physics and engineering fields.

2.2 Definitions associated with tsunami waves

In general, tsunami water levels such as tsunami height, inundation depth, run-up height are used to indicate the tsunami wave impacts related to the tsunami scale and intensity. Furthermore, these tsunami parameters may be able to use in the early warning system for tsunami disaster. Indeed, the tsunami information is being used as the criteria for tsunami impacts in many fields. However, if wrong its definition is used, it may cause confusion in understating tsunami information. Thus, it is necessary to know the accurate definitions of tsunami wave. In this section, the definitions of tsunami scale levels and water levels will be presented.

According to the tsunami scales and impacts, tsunami event is classified into Level 1 and Level 2. The tsunami return period was reflected in the classification of tsunami level. The characteristics of Level 1 and Level 2 tsunami events are summarized in Table 2.1.

Table 2.1 Classification of tsunami level

Lv. Return period Remarks

Level 1

Tsunami 1 in 100 year return period - Corresponding to the existing structures.

- Protect human lives and properties.

Level 2

Tsunami 1 in 1,000 year return period - Large-scale casualties and property damages.

- Protect human lives at least, including

evacuation planning, recovery planning, etc.

Level 1 tsunami event has relatively low tsunami impacts, and it has one time in 100 year return period. It means that the existing coastal structures can protect human lives and properties from tsunami wave. As the mitigation plan, tsunami disaster is controlled by the structure designing. However, Level 2 tsunami has totally different scale and intensity compared with Level 1, and this tsunami event has a return period of 1,000 year. Level 2 tsunami waves cause large-scale causalities and property damages, and it affects the whole communities. In that case, the mitigation planning includes evacuation and city recovery planning, public education, etc. as the comprehensive measures.

This classification is very useful to assess the tsunami impacts in accordance with the tsunami level, and it is easy to identify the characteristics of each tsunami level. Furthermore, the relationship between significant factors can be obtain from this criteria based on the tsunami level.

Figure 2.2 Definitions of tsunami height, inundation depth, run-up height

With the tsunami scale and intensity level, it is necessary to define the actual quantities of the tsunami such as tsunami height, inundation depth, run-up height. Figure 2.2 shows the schematic of water levels due to the tsunami wave. In general, it is crucial to determine the Still-Water Level (SWL) in estimating the tsunami height and run-up height. Tide level is most commonly used because when the tsunami propagates into coasts and rivers, the wave flows along the tidal motion. Meanwhile, the inundation depth is estimated from the difference between bottom and water surface. Compared with tsunami height and run-up height, the inundation depth data can be applied effectively to assess the effect of tsunami propagation over land.

The presented definitions in accordance with the tsunami wave are equally applicable in this study. Especially, tsunami level classification, Level 1 and Level 2, is important criterion to lead the entire study.

2.3 River mouth morphology

At a river mouth, the two flows from river and ocean always exists, thus the analysis of flow motion is difficult and complex. Moreover, the flow condition is greatly influenced by extreme natural phenomena.

During the natural extreme events for long term and short term, the river mouth has an important role to determine the flow conditions depending on the features of river mouth (Yang et al., 2001; Mao et al, 2004; Tanaka, 2006), further it has been studied through various approaches hydrological, geological and morphological (Pruszak et al., 2005; Cooper, 2001; Tanaka, 2003; Lichter et al., 2011). In particular, the characteristics of river mouth morphology indicate different impacts of natural events. Furthermore, it was found that morphological characteristics are deeply related to the structures and components of river mouth (Mikhailova, 2008) but also of what kind of natural event such as flood and storm events, tsunamis, tides, and so on.

Figure 2.3 shows the general river mouth types which can be seen easily in estuaries as examples of river mouth structure and component. Morphologically, Old-Kitakami River (Figure 2.3-(a)) and Jo River (Figure 2.3-(b)) have the similar river mouth structure that is covered with the coastal structure. Especially, Jo River is located in the port as well as gradually decreasing the river width, and Old-Kitakami has straight jetties at the river mouth, whereas the Naruse-Yoshida Rivers (Figure 2.3-(c)) is shown the narrow river mouth width compared with river channel such as constricted river mouth. Even though Old-Kitakami River and Naruse-Yoshida Rivers are the same jetties type, the differences of structures may affect to the flow characteristics of extreme natural events. In case of Nanakita River (Figure 2.3-(d)), the river mouth is composed of sandy coast with the constriction type same with the structure of Naruse-Yoshida Rivers mouth. Finally, the river mouth can be simply classified simply according to river mouth structures and components. The river mouth classification will be used to assess the effect due to the morphological characteristics (Tanaka et al., 2012).

(a) Old-Kitakami River

(b) Jo River

(c) Naruse-Yoshida Rivers

(d) Nanakita River

Figure 2.3 River mouth morphological features