2.2.1.Basic definitions and nomenclature
In the context of the present study, the term ‘wave’ is used to refer to free surface, progressive, gravity waves propagating at the air-fluid interface. In particular, this study
refers to such waves on open bodies of water such as the ocean or lakes, but equally includes such wave features induced under controlled laboratory conditions.
Important basic variables in the description of a wave are: the wave period, T, the wave height, H, or wave amplitude, a (=H/2), and the wavelength, λ. The wave propagates in water of depth, d. These parameters are illustrated using an idealised wave in Figure 2.1. The positive x direction is identified typically as the direction of wave propagation; y is the corresponding normal component in the horizontal plane. In wave theory construction, the vertical component, z, is measured typically from the mean water surface as a negative value; it should be noted that sections of the present study concerned with sediment- and fluid-dynamics processes at the seabed use z measured instead from the sediment/water interface.
Figure 2.1. Schematic illustration of the basic wave parameters.
Waves induce motion beneath the fluid surface; the spatial and temporal displacement of the fluid is a function of the surface displacement and the depth of observation (see Section 2.3.2). Such periodic flows, specifically when observed at the seabed, are referred to herein as ‘oscillatory flows’, howsoever created, e.g. by wave action, or, by mechanical means in a controlled environment.
2.2.2.The spectrum of wave theories
A broad range of wave theories has been developed to describe numerically the motion of water associated with gravity waves. These theories cover the wide spectrum of
hydrodynamic conditions found in nature. Following Sleath (1984), wave theories may be grouped as follows:
(i) Small-amplitude theory. First approximations are commonly called ‘linear’ or ‘Airy’ waves, higher approximations are referred to as ‘Stokes’ waves.
(ii) Shallow-water theory, including ‘cnoidal’ waves and ‘solitary’ waves. (iii) Rotational wave theory, referred to as ‘trochoidal’ theory.
(iv) Numerical solutions, referred to as the ‘Cokelet’ exact solution or the less
computationally intensive but nonexact ‘stream function’ and ‘vocoidal’ theories.
Reviews of practical wave theory in relation to sediment dynamics are presented elsewhere (e.g. Sleath, 1984; Soulsby, 1997; and Dean and Dalrymple, 2002). Further detailed information on the mathematical construction of selected theories may be found in Le Mehaute (1976), Dean and Dalrymple (1990), or Tucker (1991). Aspects of the Airy and second-order Stokes approximations (small-amplitude theory) were chosen for use in the present study.
2.2.3.Validity of wave theories
As summarised by Sleath (1984), the spectrum of wave theories provides approximations of varied complexity and therefore, of varied accuracy. The accuracy of the model is primarily dependant on the particular combination of hydrodynamic parameters being simulated. Although exact numerical solutions for velocity and pressure distributions are available, such solutions may be mathematically complex, making them impractical and/or computationally expensive. Hence, for use in numerical studies, an appropriate wave theory is chosen according to: the hydrodynamic environment being simulated (for mathematical validity); the degree of accuracy required; and the acceptable degree of computational complexity.
After Dean (1970) and Le Mehaute (1976), Sleath (1984) presented a graphical summary, outlining the numerical validity for various wave theories (Figure 2.2). In addition, these and other wave theories are described in more detail by Kirkgoz (1986), Barltrop (1990), Tucker (1991), Soulsby et al. (1993) and Soulsby (1997).
Figure 2.2. Validity limits for various wave theories, in terms of wave height (H), period (T), wavelength (λ) and water depth (d), (from Sleath, 1984). Note: the coloured areas highlight the conditions represented in the present study.
In studies relating to sediment-dynamics, the most relevant region of the validity diagram shown in Figure 2.2, is located to the left of the ‘transitional wave’ to ‘deep-water wave’ limit (d/λ=0.5), i.e. in water sufficiently shallow for oscillatory flow to be ‘felt’ at the seabed, and consequently, to interact with any potentially mobile particles that may be present.
Table 2.1 gives examples of the range of conditions (combinations of d, T and H), best represented by linear or by second-order Stokes theory. These values were calculated directly from Figure 2.2, for the marginal deep water/transitional condition d/gT2=0.0795 (d/λ=0.5) and for a shallow water example at d/gT2=0.0075. A linear interpolation may be used to approximate the limiting values of H, for intermediate values of d.
From Figure 2.2, waves steeper (of greater H) than those given in Table 2.1 for the second- order Stokes solution in deep water, are better described by either: a higher-order Stokes approximation; stream function theory; or, the Cokelet ‘exact’ solution. In water shallower than that given in the shallow water example, cnoidal theory may also become more appropriate, at smaller wave heights.
T d (m) H Lin (m) H Stokes (m) d (m) H Lin (m) H Stokes (m) 2 > 3.12 < 0.04 0.04 - 0.27 0.39 < 0.01 0.01 - 0.08 3 > 7.02 < 0.09 0.09 - 0.62 0.88 < 0.03 0.03 - 0.18 4 > 12.48 < 0.16 0.16 - 1.10 1.57 < 0.05 0.05 - 0.31 5 > 19.50 < 0.25 0.25 - 1.72 2.45 < 0.08 0.08 - 0.49 6 > 28.08 < 0.35 0.35 - 2.47 3.53 < 0.11 0.11 - 0.71 7 > 38.21 < 0.48 0.48 - 3.36 4.81 < 0.15 0.15 - 0.96 8 > 49.91 < 0.63 0.63 - 4.39 6.28 < 0.20 0.20 - 1.26 9 > 63.17 < 0.79 0.79 - 5.56 7.95 < 0.25 0.25 - 1.59 10 > 77.99 < 0.98 0.98 - 6.87 9.81 < 0.31 0.31 - 1.96 11 > 94.37 < 1.19 1.19 - 8.31 11.87 < 0.38 0.38 - 2.37 12 > 112.30 < 1.41 1.41 - 9.89 14.13 < 0.45 0.45 - 2.83 13 > 131.80 < 1.66 1.66 - 11.61 16.58 < 0.53 0.53 - 3.32 14 > 152.86 < 1.92 1.92 - 13.46 19.23 < 0.62 0.62 - 3.85 15 > 175.48 < 2.21 2.21 - 15.45 22.07 < 0.71 0.71 - 4.41
Transitional/Deep water limit 1 Shallower water example 2
Table 2.1. Critical combinations of wave parameters delimiting the validity of linear (Lin) and second-order Stokes (Stokes) theory: (1) for waves in deep water, beginning to ‘feel’ the bed (d/gT2=0.0795); and (2) for shallower water depths (d/gT2=0.0075). Values calculated from Figure 2.2.