Directional Infragravity Waves Induced by Bichromatic and Bidirectional Waves: Theoretical Approach and Experimental Affirmation

Introduction

Shoaling and Refraction

Theoretical Development of Second-Order Quantities

Second-Order Orbital Wave Motion

Sharma and Dean (1981) were the first authors to derive a second-order solution for describing the velocities induced by a multimodal–multidirectional sea state, computing the subtractive, the additive, and the self-interacting terms for each pair of primary waves. The first- and second-order velocity potential expressions, given by Sharma and Dean (1981 [Eqs. (22) and (30)]), are here rewritten as follows for simplicity and to maintain the same notation used along that paper:

$${\mathrm{\Phi }}^{(1)}={\displaystyle \sum _{i=1}^{\mathrm{\infty }}}{\displaystyle \frac{H_{i}g}{2{\sigma }_i}{\displaystyle \frac{\cosh k_i(h+z)}{\cosh k_{i}h}\sin ({\mathbf{k}}_i\cdot \mathbf{x}-{\sigma }_{i}t+{\epsilon }_i)}}$$

(2)

$$\begin{array}{rl}& \begin{array}{c}{\mathrm{\Phi }}^{(2)}={\displaystyle \frac{1}{4}{\displaystyle \sum _{i=1}^{\mathrm{\infty }}}{\displaystyle \sum _{j=1}^{\mathrm{\infty }}}{\displaystyle \frac{H_{i}g}{2{\sigma }_i}{\displaystyle \frac{H_{j}g}{2{\sigma }_j}{\displaystyle \frac{\cosh [k_{ij}^{-}(h+z)]}{\cosh (k_{ij}^{-}h)}{\displaystyle \frac{D_{ij}^{-}}{{\sigma }_i-{\sigma }_j}\sin ({\varphi }_i-{\varphi }_j)}}}}}\end{array}\\ & +{\displaystyle \frac{1}{4}{\displaystyle \sum _{i=1}^{\mathrm{\infty }}}{\displaystyle \sum _{j=1}^{\mathrm{\infty }}}{\displaystyle \frac{H_{i}g}{2{\sigma }_i}{\displaystyle \frac{H_{j}g}{2{\sigma }_j}{\displaystyle \frac{\cosh [k_{ij}^{+}(h+z)]}{\cosh (k_{ij}^{+}h)}{\displaystyle \frac{D_{ij}^{+}}{{\sigma }_i+{\sigma }_j}\sin ({\varphi }_i+{\varphi }_j)}}}}}\end{array}$$

(3)

where x = (x, y) = horizontal coordinate system; x = cross-shore coordinate positive in the landward direction; y = longshore coordinate positive to the left of the x-axis; z = vertical coordinate positive upwards from the mean sea level; D⁺ and D⁻ are transfer functions shown in Sharma and Dean (1981); and the subscripts “i” and “j” = each one of the waves in the analyzed pair. For each component, the linear dispersion relation holds

$${\sigma }_i^2=g{k}_i\tanh k_{i}h$$

(4)

The negative (−) and positive (+) superscripts represent terms that involve subtraction or addition of each pair of the wave vector numbers (SubI or AddI, respectively): $k_{ij}^{+}=|{\mathbf{k}}_i+{\mathbf{k}}_j|$ and $k_{ij}^{-}=|{\mathbf{k}}_i-{\mathbf{k}}_j|$. The terms that are multiplied by sin(ϕ_i − ϕ_j) in Eq. (3) are responsible for the low frequency motions associated with the IG as well as for the stationary spatial undulations of the mean sea level.

The horizontal velocity field, which results from the superposition of two waves, is obtained as the gradient of the velocity potential, and it is expressed as the sum of the first- and second-order contributions of each individual wave and of the SubI and the AddI contributions as

$$\{\begin{array}{c}u={\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(1)}}{\mathrm{\partial }x}+{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(2)}}{\mathrm{\partial }x}=u_a^{(1)}+u_a^{(2)}+u_b^{(1)}+u_b^{(2)}+u_{a-b}^{(2)}+u_{a+b}^{(2)}}}\\ v={\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(1)}}{\mathrm{\partial }y}+{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(2)}}{\mathrm{\partial }y}=v_a^{(1)}+v_a^{(2)}+v_b^{(1)}+v_b^{(2)}+v_{a-b}^{(2)}+v_{a+b}^{(2)}}}\\ w={\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(1)}}{\mathrm{\partial }z}+{\displaystyle \frac{\mathrm{\partial }{\mathrm{\Phi }}^{(2)}}{\mathrm{\partial }z}=w_a^{(1)}+w_a^{(2)}+w_b^{(1)}+w_b^{(2)}+w_{a-b}^{(2)}+w_{a+b}^{(2)}}}\end{array}$$

(5)

Fig. 5 is an example of the orbital velocity vector at the bottom, along the wave group period, for the superposition of two waves. Higher on the water column, there is an orbital volume, indicating that the momentum transfer associated to the Bi–Bi waves should be better investigated.

The SubI velocity decays in vertical direction according to $\cosh (k_{ij}^{-}(z+h))$ and is oriented along the direction of the vector ${\mathbf{k}}_{\mathit{i}}-{\mathbf{k}}_{\mathit{j}}$. In case the magnitude of the subtractive wave number is smaller than those of the primary waves, the amplitude decay of the orbital velocity and pressure of the low frequency bound wave will also be weaker. The same effect may hold for the addictive wave number if the angle between primary waves is large (say, 120°). This is quite intriguing, especially in intermediate water depth, because the orbital velocities of the primary waves will decay but, near the bottom, the bound subharmonic (or the bound infragravity) velocity may remain as strong as it is near the free surface. Consequently, the percentual contribution of the infragravity velocity to the total velocity near the bottom may become more significant, in addition to being oriented in a direction different from those of the primary waves. This is an important issue when discussing bottom stresses in the boundary layer and sediment transport.

Mean Water Level

Based on second-order Stokes theory, Dalrymple (1975) analytically derived the following expression for the mean water level as a longshore undulation caused by monochromatic-bidirectional wave trains:

$$\begin{array}{rl}\overline{\eta }& =-{\displaystyle \frac{k}{8\sinh 2kh}}\\ & \times \left\{H_a^2+H_b^2-2H_a{H}_b{\sinh }^{2}kh[1-{\displaystyle \frac{\cos (\mathrm{\Delta }\theta )}{{\tanh }^{2}kh}}]\cos (\mathrm{\Delta }\mathbf{k}\cdot \mathbf{x})\right\}\end{array}$$

(6)

where Δθ = difference between angles of propagation of both incident waves. The SubI bound wave within the group [Figs. 1(c and d)] is determined by time-averaging the Bernoulli equation at the free surface, keeping second-order terms. Later, Dalrymple and Lanan (1976) verified the existence of this stationary long wave under the action of two wave trains with the same frequency but different directions, and confirmed the capacity of generating rip currents and beach cusps.

In nature, this condition may happen as waves are reflected by a structure or by a natural morphological feature and is one of the possible mechanisms for the formation of equally spaced rip currents (Inman et al. 1971). In the surf zone, these spatial undulations may modulate the wave height and wave-induced circulation along the coast, which is usually attributed to edge waves or irregular bottom topography (Dalrymple et al. 2011; Xie 2012).

The inclusion of bidirectional wave groups (Bi–Bi waves) certainly builds one level of complexity, as another degree of freedom develops for the hydrodynamics. Using the velocity potential as defined by Eqs. (2) and (3), the instantaneous free surface is obtained from the Dynamic Free Surface Boundary Condition. Time averaging the resulting equation, analogous to Dalrymple (1975), Eq. (7) is obtained for the mean water level, $\overline{\eta }$, where A⁺ and A⁻ are terms related to the additive and subtractive interaction between the primary waves given by expressions Eqs. (8) and (9), respectively.

$$\begin{array}{rl}& \begin{array}{c}\overline{\eta }=-{\displaystyle \frac{H_a^2{k}_a}{8\sinh (2k_{a}h)}-{\displaystyle \frac{H_b^2{k}_b}{8\sinh (2k_{b}h)}+{\displaystyle \frac{H_a{H}_b}{8g}\left\{{\sigma }_a^2+{\sigma }_b^2-{\displaystyle \frac{{\sigma }_a{\sigma }_b\cos \mathrm{\Delta }\theta }{\tanh (k_{a}h)\tanh (k_{b}h)}+{\sigma }_a{\sigma }_b}\right\}\overline{\cos ({\varphi }_a+{\varphi }_b)}}}}\end{array}\\ & +{\displaystyle \frac{H_a{H}_b}{8g}\left\{{\sigma }_a^2+{\sigma }_b^2-{\displaystyle \frac{{\sigma }_a{\sigma }_b\cos \mathrm{\Delta }\theta }{\tanh (k_{a}h)\tanh (k_{b}h)}-{\sigma }_a{\sigma }_b}\right\}\overline{\cos ({\varphi }_a-{\varphi }_b)}+{\displaystyle \frac{1}{g}({\sigma }_a+{\sigma }_b)\cosh (|{\mathbf{k}}_{\mathit{a}}+{\mathbf{k}}_{\mathit{b}}|h)A_{ab}^{+}\overline{\cos ({\varphi }_a+{\varphi }_b)}}}\\ & +{\displaystyle \frac{1}{g}({\sigma }_a-{\sigma }_b)\cosh (|{\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}}|h)A_{ab}^{-}\overline{\cos ({\varphi }_a-{\varphi }_b)}}\end{array}$$

(7)

$$\begin{array}{r}{A}_{ab}^{+}=-{\displaystyle \frac{H_a{H}_b}{4}{\displaystyle \frac{{\displaystyle \frac{g{k}_a{\sigma }_a}{\sinh (2k_{a}h)}+{\displaystyle \frac{g{k}_b{\sigma }_b}{\sinh (2k_{b}h)}+{\sigma }_a{\sigma }_b({\sigma }_a+{\sigma }_b)\left[{\displaystyle \frac{\cos \mathrm{\Delta }\theta }{\tanh (k_{a}h)\tanh (k_{b}h)}-1}\right]}}}{g|{\mathbf{k}}_{\mathit{a}}+{\mathbf{k}}_{\mathit{b}}|\sinh (|{\mathbf{k}}_{\mathit{a}}+{\mathbf{k}}_{\mathit{b}}|h)-{({\sigma }_a+{\sigma }_b)}^2\cosh (|{\mathbf{k}}_{\mathit{a}}+{\mathbf{k}}_{\mathit{b}}|h)}}}\end{array}$$

(8)

$$\begin{array}{r}{A}_{ab}^{-}=-{\displaystyle \frac{H_a{H}_b}{4}{\displaystyle \frac{{\displaystyle \frac{g{k}_a{\sigma }_a}{\sinh (2k_{a}h)}-{\displaystyle \frac{g{k}_b{\sigma }_b}{\sinh (2k_{b}h)}+{\sigma }_a{\sigma }_b({\sigma }_a-{\sigma }_b)\left[{\displaystyle \frac{\cos \mathrm{\Delta }\theta }{\tanh (k_{a}h)\tanh (k_{b}h)}+1}\right]}}}{g|{\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}}|\sinh (|{\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}}|h)-{({\sigma }_a-{\sigma }_b)}^2\cosh (|{\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}}|h)}}}\end{array}$$

(9)

At this point, it is important to define what is meant by “time averaging,” since there are at least four timescales involved in the problem: the two periods of the primary waves, those of the two interference waves (additive and subtractive), and all combinations of sub- and superharmonics. This is a relevant issue in both analytical and numerical terms because when data is recorded in the field or in the laboratory, an actual averaging procedure must be carried out in order to correctly identify the physical processes predicted by those equations.

If the primary periods are close to each other, the time span of the averaging process might be considered approximately equal to the mean wave period, and the resulting wave pattern is similar to short-crested waves (Wiegel 1964). However, if the primary periods significantly differ, it would be more appropriate to choose the shortest time scale in the problem, that is, the time scale of the sum of the two primary frequencies (σ_a + σ_b) or T_aT_b/|T_a + T_b|. Using this time scale, all terms in Eq. (7) that are multiplied by $\overline{\cos ({\varphi }_a+{\varphi }_b)}$ might theoretically be neglected. If this time scale is chosen, however, numerically time averaging the terms $\overline{\cos ({\varphi }_a-{\varphi }_b)}$ in Eq. (7), or even the primary waves and their second harmonic, will leave a residue. Therefore, the averaging process should be rather understood as a filtering process. Regarding time scales for the SubI wave, it is helpful to compute the periods resulting from the interaction. If the longer period, say T_b, is equal to twice the other period, T_a, the SubI wave will have the same period of the longer period (T_b) and the AddI wave will have period equal to 1/3 of the longer period, eventually being falsely interpreted as a third harmonic of the primary wave.

Comparing Eqs. (6) and (7), there are time varying terms which did not appear in Dalrymple’s original work. Neglecting the AddI, Eq. (6) is retrieved from Eq. (7) if both waves have the same period. Otherwise, the term $\overline{\cos ({\varphi }_a-{\varphi }_b)}$ will become $\cos (\mathrm{\Delta }\mathbf{k}\cdot \mathbf{x}-\mathrm{\Delta }\sigma t+\mathrm{\Delta }\epsilon )$, which represents the SubI/IG wave. Fig. 6 shows the influence of the angle and period differences on the long time scale (Δσ) oscillating mean water level, $\overline{\eta }$, which has been nondimensionalized by H_aH_b/h.

Looking at the resulting equation for the IG wave (or $\overline{\eta }$), its amplitude depends, in a highly nonlinear way, on the difference between main frequencies and primary angles. Taking the shallow water asymptotic approximation, Eq. (7) reduces to

$$\begin{array}{r}\begin{array}{c}\overline{\eta }\approx {\displaystyle \frac{H_a{H}_b}{8h}{D}^{-1}\left\{1+(2-D)\cos \mathrm{\Delta }\theta +(k_{a}h)(k_{b}h)(2+D)+{\displaystyle \frac{h{(\mathrm{\Delta }\sigma )}^2}{g}D}\right\}}\end{array}\end{array}$$

(10a)

$$\begin{array}{r}D=\left[1-{\displaystyle \frac{g|\mathrm{\Delta }\mathbf{k}|}{{(\mathrm{\Delta }\sigma )}^2}\tanh |\mathrm{\Delta }\mathbf{k}|h}\right]\end{array}$$

(10b)

As the primary waves shoal, $\overline{\eta }$ grows at a rate that is the product of the individual growth of the parent waves, but it also depends on the local angle $\mathrm{\Delta }\theta$, on the local relative depth, and on the value of the expression D, which might be interpreted as a mismatch of dispersion. Even though the individual waves may be in shallow water, the magnitude of the difference wave number vector $|\mathbf{\Delta }\mathbf{k}|\mathit{h}$ does not need to be small. In other words, the SubI wave, even though it has a low frequency, may not be a shallow water wave.

Physically, the Bi–Bi wave causes a progressive long bound wave, which propagates in the direction of the vector that results from the subtraction of the two primary wave number vectors. Similar to the stationary wave associated with a monochromatic-bidirectional sea state (Dalrymple 1975; Dalrymple and Lanan 1976), one may infer that the IG produced by Bi–Bi waves might also be reflected from shore (or other coastal feature) and generate a rhythmic pattern of nearshore circulation. This progressive long wave would also be able to cause the longitudinal displacement of the rip currents. Fowler and Dalrymple (1991) associated this long progressive oscillation to wave groups, a first-order wave–wave interaction characteristic. In contrast, in the present work the second-order interactions are investigated. The migrating rip currents are associated to the SubI/IG wave, and the dynamical processes underlying bichromatic and bidirectional sea states are proposed to be expressed by the radiation stress tensor.

Radiation Stresses

Although the work of Longuet-Higgins and Stewart (1964) had already been published for more than one decade in the 1980s, little attention has been given since then to the radiation stress field associated with two noncollinear waves in the literature.

The general expression for computing the radiation stresses for Bi–Bi waves are given by

$$\begin{array}{r}{S}_{ij}=\overline{{\int }_{-h}^{\eta }\rho {\tilde{u}}_i^{(a)}{\tilde{u}}_j^{(a)}+\rho {\tilde{u}}_i^{(a)}{\tilde{u}}_j^{(b)}+\rho {\tilde{u}}_j^{(a)}{\tilde{u}}_i^{(b)}+\rho {\tilde{u}}_i^{(b)}{\tilde{u}}_j^{(b)}+p{\delta }_{ij}dz}-{\displaystyle \frac{1}{2}\rho g(h+\overline{\eta }{)}^2{\delta }_{ij}}\end{array}$$

(11)

where S_ij = components of the radiation stress tensor; the subscripts i and j = Cartesian components x and y, respectively; ${\tilde{u}}_i^{(a)}$ and ${\tilde{u}}_i^{(b)}$ = components of the horizontal orbital velocities of waves “a” and “b,” respectively; and δ_ij = Kronecker delta. The presence of interaction components (${\tilde{u}}_i^{(a)}{\tilde{u}}_j^{(b)}$ and ${\tilde{u}}_j^{(a)}{\tilde{u}}_i^{(b)}$) and the mean water level itself, given by Eq. (7), reveals that the radiation stress of the Bi–Bi wave is not simply the algebraic sum of the stress components of each individual primary wave.

The complete expressions for the radiation stress tensor, which assumes the general form shown in the following equation, are given in the Appendix:

$$\begin{array}{rl}{S}_{ij}& =S_{ij}^{(a)}+S_{ij}^{(b)}+{\mathrm{\Psi }}^{+}(H_a,H_b,{\sigma }_a,{\sigma }_b,{\mathbf{k}}_a,{\mathbf{k}}_b,h)\overline{\cos ({\varphi }_a+{\varphi }_b)}\\ & +{\mathrm{\Psi }}^{-}(H_a,H_b,{\sigma }_a,{\sigma }_b,{\mathbf{k}}_a,{\mathbf{k}}_b,h)\overline{\cos ({\varphi }_a+{\varphi }_b)}\end{array}$$

(12)

where Ψ⁺ and Ψ⁻ = nonlinear interaction functions that depend on the physical and geometrical properties of waves “a” and “b.” Each component of the total radiation stress tensor is composed by the sum of the respective components of the primary waves tensor added to a nonlinear interaction function, Ψ⁺, for the additive interaction and another function, Ψ⁻, for the subtractive interaction. These additional terms generate a nonuniform and transient stress field whose period is related to the properties of the primary waves. Low frequency components are represented by terms that include the time averaged phase differences, $\overline{\cos ({\varphi }_a-{\varphi }_b)}$, while the high frequency terms are multiplied by $\overline{\cos ({\varphi }_a+{\varphi }_b)}$. These results significantly differ from those originally presented by Longuet-Higgins and Stewart (1964) for unimodal and unidirectional waves, and later expanded to oblique waves by Hsu et al. (2006), Shi and Kirby (2008), and Hsu and Lan (2009) or a simple linear superposition as proposed by Battjes (1972a) for short-crested waves. The expressions presented in the Appendix reduce to the usual form of the radiation stress components for monochromatic–monodirectional waves (Dean and Dalrymple 1991) if their limiting conditions are applied.

For each individual wave, the radiation stresses may be represented by a Mohr’s circle. However, it is important to certify whether the resulting radiation stresses from the action of both primary waves would also be a plane stress state. Let the total stress tensor be represented according to the following equation as the sum of four terms:

$$\begin{array}{rl}\left(\begin{array}{cc}{S}_{xx}^{ab}& S_{xy}^{ab}\\ S_{yx}^{ab}& S_{yy}^{ab}\end{array}\right)& =\left(\begin{array}{cc}{S}_{xx}^a& S_{xy}^a\\ S_{yx}^a& S_{yy}^a\end{array}\right)+\left(\begin{array}{cc}{S}_{xx}^b& S_{xy}^b\\ S_{yx}^b& S_{yy}^b\end{array}\right)\\ & +\left(\begin{array}{cc}{S}_{xx}^{a+b}& S_{xy}^{a+b}\\ S_{yx}^{a+b}& S_{yy}^{a+b}\end{array}\right)+\left(\begin{array}{cc}{S}_{xx}^{a-b}& S_{xy}^{a-b}\\ S_{yx}^{a-b}& S_{yy}^{a-b}\end{array}\right)\end{array}$$

(13)

Substituting the usual expressions for the radiation stress components for monochromatic waves

$$\begin{array}{rl}& \begin{array}{c}\left(\begin{array}{cc}{S}_{xx}^{ab}& S_{xy}^{ab}\\ S_{yx}^{ab}& S_{yy}^{ab}\end{array}\right)={\displaystyle \underset{(A)}{{\displaystyle \underbrace{\left(\begin{array}{cc}\left({\displaystyle \frac{3}{2}{n}_a-{\displaystyle \frac{1}{2}}}\right)E_a+\left({\displaystyle \frac{3}{2}{n}_b-{\displaystyle \frac{1}{2}}}\right)E_b& 0\\ 0& \left({\displaystyle \frac{3}{2}{n}_a-{\displaystyle \frac{1}{2}}}\right)E_a+\left({\displaystyle \frac{3}{2}{n}_b-{\displaystyle \frac{1}{2}}}\right)E_b\end{array}\right)}}}}\end{array}\\ & +{\displaystyle \underset{(B)}{{\displaystyle \underbrace{{\displaystyle \frac{1}{2}{n}_a{E}_a\left(\begin{array}{cc}\cos 2{\theta }_a& \sin 2{\theta }_a\\ \sin 2{\theta }_a& -\cos 2{\theta }_a\end{array}\right)}}}}}+{\displaystyle \underset{(C)}{{\displaystyle \underbrace{{\displaystyle \frac{1}{2}{n}_b{E}_b\left(\begin{array}{cc}\cos 2{\theta }_b& \sin 2{\theta }_b\\ \sin 2{\theta }_b& -\cos 2{\theta }_b\end{array}\right)}}}}}+{\displaystyle \underset{(D)}{{\displaystyle \underbrace{\left(\begin{array}{cc}{S}_{xx}^{a+b}(t)& S_{xy}^{a+b}(t)\\ S_{yx}^{a+b}(t)& S_{yy}^{a+b}(t)\end{array}\right)}}}}+{\displaystyle \underset{(E)}{{\displaystyle \underbrace{\left(\begin{array}{cc}{S}_{xx}^{a-b}(t)& S_{xy}^{a-b}(t)\\ S_{yx}^{a-b}(t)& S_{yy}^{a-b}(t)\end{array}\right)}}}}\end{array}$$

(14)

Terms (A), (B), and (C) of Eq. (14) represent the sum of two plane stress states, one associated to wave “a” and the other to wave “b” whose sum is itself a plane stress state. This result is illustrated in Fig. 7, which shows the Mohr’s circle for each primary wave and the resulting sum. Term (D) corresponds to the AddI, with high-frequency oscillation. Term (E) corresponds to the SubI, a term that would usually vary in a longer time scale and with a slow undulation in space. This is the term which might be related to the SubI/IG wave, depending on the difference between the periods of the primary waves, and for this reason it should be worth doing a more detailed analysis.

Looking at Eqs. (23)–(25) in the Appendix, the angle (θ_a − θ_b) seems to be very relevant to the orientation and magnitude of the stresses. Examining the time dependent terms in the radiation stresses equation that are simultaneously multiplied by the cosine of the difference of phases

$$\begin{array}{rl}\overline{\cos ({\varphi }_a-{\varphi }_b)}& =\cos (({\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}})\cdot \mathbf{x}-({\sigma }_a-{\sigma }_b)t+({\epsilon }_a-{\epsilon }_b))\\ & =\cos (\mathbf{\Delta }\mathbf{k}\cdot \mathbf{x}-\mathrm{\Delta }\sigma t+\mathrm{\Delta }\epsilon )\end{array}$$

they oscillate very slowly [term (E)]. First, let the constants M₁, M₂, M₃, and M₄ be defined as

$$\begin{array}{r}{M}_1=-{\displaystyle \frac{\rho h{H}_a{H}_b}{8}\left[{({\sigma }_a-{\sigma }_b)}^2+{\sigma }_a{\sigma }_b\left(1-{\displaystyle \frac{\cos (\mathrm{\Delta }\theta )}{\tanh (k_{a}h)\tanh (k_{b}h)}}\right)\right]}\end{array}$$

(15)

$$M_2=-\rho h({\sigma }_a-{\sigma }_b)\cosh (|{\mathbf{k}}_{\mathit{a}}-{\mathbf{k}}_{\mathit{b}}|h)A_{ab}^{-}$$

(16)

$$\begin{array}{r}{M}_3={\displaystyle \frac{\rho H_a{H}_b{\sigma }_a{\sigma }_b}{16\sinh k_{a}h\sinh k_{b}h}\{{\displaystyle \frac{2g\sinh k_{a}h\sinh k_{b}h}{{\sigma }_a{\sigma }_b}}}\\ -{\displaystyle \frac{\sinh [|{\mathbf{k}}_{\mathit{b}}+{\mathbf{k}}_{\mathit{a}}|h]}{|{\mathbf{k}}_{\mathit{b}}+{\mathbf{k}}_{\mathit{a}}|}+{\displaystyle \frac{\sinh [|{\mathbf{k}}_{\mathit{b}}-{\mathbf{k}}_{\mathit{a}}|h]}{|{\mathbf{k}}_{\mathit{b}}-{\mathbf{k}}_{\mathit{a}}|}}}\}\end{array}$$

(17)

$$\begin{array}{r}{M}_4={\displaystyle \frac{\rho H_a{H}_b{\sigma }_a{\sigma }_b}{16\sinh k_{a}h\sinh k_{b}h}\left\{{\displaystyle \frac{\sinh [|{\mathbf{k}}_{\mathit{b}}+{\mathbf{k}}_{\mathit{a}}|h]}{|{\mathbf{k}}_{\mathit{b}}+{\mathbf{k}}_{\mathit{a}}|}+{\displaystyle \frac{\sinh [|{\mathbf{k}}_{\mathit{b}}-{\mathbf{k}}_{\mathit{a}}|h]}{|{\mathbf{k}}_{\mathit{b}}-{\mathbf{k}}_{\mathit{a}}|}}}\right\}}\end{array}$$

(18)

The stress terms $S_{ij}^{a-b}(t)$ in the Appendix can be rewritten in terms of the preceding constants M_i as

$$\begin{array}{r}{S}_{xx}^{a-b}(t)=[M_1+M_2+M_3+M_4\cos ({\theta }_a+{\theta }_b)]\cos (\mathbf{\Delta }\mathbf{k}\cdot \mathbf{x}-\mathrm{\Delta }\sigma t+\mathrm{\Delta }\epsilon )\end{array}$$

(19)

$$\begin{array}{r}{S}_{yy}^{a-b}(t)=[M_1+M_2+M_3-M_4\cos ({\theta }_a+{\theta }_b)]\cos (\mathbf{\Delta }\mathbf{k}\cdot \mathbf{x}-\mathrm{\Delta }\sigma t+\mathrm{\Delta }\epsilon )\end{array}$$

(20)

$$\begin{array}{r}{S}_{xy}^{a-b}(t)=S_{yx}^{a-b}(t)=[M_4\sin ({\theta }_a+{\theta }_b)]\cos (\mathbf{\Delta }\mathbf{k}\cdot \mathbf{x}-\mathrm{\Delta }\sigma t+\mathrm{\Delta }\epsilon )\end{array}$$

(21)

Hence, the stress coefficients of the tensor $S_{ij}^{a-b}(t)$ can indeed be represented by the Mohr’s circle, in terms of the angle θ_a + θ_b as shown in Fig. 7. A change in orientation of the beach by angle α would represent a change by 2α on the steady part (largest light green circle in Fig. 7) of the radiation stress, a change by α on θ_a, and an equal change on θ_b, adding a change by 2α on θ_a + θ_b.

The final stress state is the product of the (largest) light green circle in Fig. 7 by the oscillating terms given by the cosine function, resulting in the time set of circles shown in Fig. 8. Similarly, for a fixed time, the stress state varies in space, as it will be shown later.

Different cases might be considered regarding the angle Δθ, from small to large angles. For small angles the magnitude of the subtractive wave number is much smaller than the magnitude of the primary wave numbers, while the magnitude of the additive wave number should be larger than those of original waves. As the angle increases, there is a condition when the additive, the subtractive, and the primary wave numbers all have the same magnitude. As the angle difference is increased, the subtractive wave number is much larger than those of the primary waves, while the additive wave number is smaller than those of the primary waves. These conditions affect the expressions of the radiation stresses in shallow water.

Experimental Methodology

Results

Mean Water Level

Eq. (7) allows computing the mean water level for any combination of angle and period of Bi–Bi waves. The curve in Fig. 10 describes the maximum and minimum (dimensional) mean water level elevation under all possible combinations of angle difference between the primary waves. In this example, the same values for height and period of Test T4-B3 were used for the primary waves (periods: 1.3 and 1.7 s; heights: 0.10 and 0.16 m; depth: 0.75 m). This result includes both the steady and the oscillating parts of that expression.

At any fixed location, in the presence of the Bi–Bi waves, the mean water level oscillates showing alternately wave setdown and setup because it behaves as a progressive (bound) wave. In contrast, bidirectional–monochromatic waves show a stationary pattern of setups and setdowns at fixed locations (e.g., Dalrymple 1975).

The setup and setdown are more significant at 0° angle difference. Around 45° and 60°, the effects on the mean water level are minimum, and when both waves tend to propagate with opposite directions, the influence of the nonlinear interactions grows.

Fig. 11 shows the directional spectra obtained from the second array of USSs measurements for test T4-B2. It is seen that the BDM method was able to extract the primary waves (σ_a and σ_b), second-order interaction waves (σ_a − σ_b) and also third-order waves (2σ_b − σ_a). Reflections were also captured at frequencies σ_a and σ_a + σ_b. These energies were characterized as reflections, once they do not have the directions predicted by the Stokes theory (θ_a−b = 6.02°).

According to theory, the SubI wave should have H_s = 0.010 m for tests T4-B2, T6-B2, and T4-B4 and 0.004 m for T4-B3. From the directional spectra of each test, the following heights were calculated for the SubI wave inside the wave basin: 0.005 m, 0.013 m, and 0.027 m for the first three tests and 0.017 m for test T4-B3. Table 3 summarizes the main characteristics of the SubI wave modeled by the Stokes II theory and measured by the USS array. Although the wave period and direction have been correctly detected, such small values of wave height within a complex sea state should be difficult to be captured with the desired precision. For test T5, when the primary wave heights were smaller (H_a = 0.05 m, H_b = 0.11 and T_a = 1.6 s, Tb = 2.1 s), the SubI/IG wave energy was not even detected by the USSs array in the cases where Δθ = 30° (Fig. 12).

Tests	Modeled			Measured
	Hs (m)	Dp (°)	Tp (s)	Hs (m)	Dp (°)	Tp (s)
T4-B2	0.01	28.16	5.53	0.005	37.5	5.46
T6-B2	0.01	28.16	5.53	0.013	25.5	5.46
T4-B4	0.01	23.16	5.53	0.027	25.5	5.46
T4-B3	0.004	67.65	5.53	0.017	68.5	5.46

Fig. 13 shows an instantaneous picture of the expected IG waves generated by the Bi–Bi waves inside the simulated basin, for the case where wave “a” is 10° with the x-axis, and wave “b” is aligned with the x-axis (angle equal to 0°). As time evolves, the progressive characteristic of the IG wave becomes evident. The arrows drawn in Fig. 13 indicate the propagation direction of this long bound wave, which is a function of the primary wave numbers.

Second-Order Orbital Wave Motion

The equations developed by Sharma and Dean (1981) enable the calculation of the orbital velocities with its linear and nonlinear interactions at any point in the water column. Fig. 14(a) shows the hodograph representation of the total orbital velocities and its 2D projections on the coordinate planes, at a measuring point 0.35 m below free surface, during test T4-B2. The red lines show the measured velocities by one of the five ADVs, and the black curves represent the computed 3D velocities from Sharma and Dean’s equation [Fig. 14(a)].

Applying the HHT to all five velocity measurements of the array of ADV resulted in 16 IMFs for test T4-B2. For a better visualization, Fig. 15 shows the Hilbert transform of only the IMFs that correspond to the primary waves (IMFs 6 and 7) and to the SubI wave (IMF 9). The white dashed lines represent the theoretical frequencies that are expected for this test. IMFs 6 and 7 oscillate around the theoretical frequencies. This oscillation is due to the mode mixing problem, a common issue of the HHT, which has difficulties in distinguishing between close signal frequencies. However, for this test, the SubI wave has quite a different frequency compared with the other waves (σ_a − σ_b= 0.18 Hz), hence its IMF does not suffer from mode mixing and is almost a straight line on top of the expected frequency (dashed white line).

Fig. 14(b) illustrates the theoretical IG wave hodograph along with the 3D signal of IMF 9 extracted via the HHT procedure. Only five wave cycles (∼28 s) are illustrated. The similarity between the theory and the results from the HHT method is visually clear. A more precise deviation measure between the extracted and the theoretical velocities can be given by the 3D root mean square error (RMSE_3D) defined by

$$RMS{E}_{3D}=\sqrt{RMS{E}_U^2+RMS{E}_V^2+RMS{E}_W^2}$$

(22)

The extracted IG wave for tests T6-B2, T4-B3, T4-B4, and T5-E5 and its RMSE_3D are plotted in Fig. 16.

Within the complex motion generated by the Bi–Bi waves, there is a low frequency modulation with a well-defined amplitude and direction, which does not match the primary motion. It is highlighted that even for test T5-E5, where the SubI/IG wave energy is quite low, the ADV array was capable of extracting this from the velocity measurements, while the USS was not (Fig. 12).

The relative importance of the SubI/IG wave contribution to the total orbital velocity grows as the elevation approaches the bottom. For test T4-B2, it was responsible for almost 9% of the maximum velocity module at 35 cm of depth. Using Eq. (5), it is shown that, at the surface, its contribution is approximately 6%, while at the bottom it is more than 10% (Fig. 17) of the total velocity. The contribution of the IG may reach much higher values when looking at each velocity component. In this case, the contribution of the slow motion may rise up to 52% in the y-axis maximum velocity (transverse direction).

Radiation Stresses

The radiation stress expressions, shown in Eqs. (15) to (21) and in the Appendix, depend on the angle difference between the primary waves as shown in Fig. 18 for the 1.3 and 1.7 s primary waves, 0.10 and 0.16 m wave heights and water depth equal to 0.75 m. The diameter of the Mohr circle is bigger when both waves propagate in the same direction. As the angle difference increases, the diameter of the Mohr circle, |S_xx − S_yy|, decreases down to a minimum at 180°, from where it slowly increases again until both waves propagate in the same direction. It is easy to notice that the radiation stress field of Bi–Bi waves strongly depends on the combination of angle and period difference between the primary waves. Therefore, all shore processes will be directly influenced by the angle between primary waves, no matter how subtle this difference may be.

Figs. 19 to 21 show the radiation stress field for each of the three components of the stress tensor (S_xx, S_xy, and S_yy), for the case the difference in angle of propagation is 10°, using the equations shown in the Appendix. The stress components follow a spatial pattern similar to that of the SubI wave elevation, and in fact this pattern changes in time similar to a progressive wave. As the waves shoal up a slope, the steepness of the primary waves become more pronounced, and the magnitude of the radiation stress components also increase. Disregarding the nonlinear terms, as suggested by Battjes (1972a), each stress component would be constant along the entire domain (S_xx = 36.97 N/m; S_xy = 1.23 N/m; and S_yy = 7.90 N/m) and, therefore, there would be no momentum generation outside of the surf zone.

Using Eqs. (7) and (12), the balance between the IG wave and the spatial variation of the radiation stresses can be calculated. Fig. 22 shows the acceleration field generated by the Bi–Bi waves, disregarding bottom and surface friction and turbulence in the momentum equation. Fig. 23 illustrate the time series of horizontal accelerations calculated for the measuring point of one of the ADVs. Modeled (black) and extracted subtractive interaction wave velocities (red) by the HHT procedure are also shown. In this case, the acceleration is oriented 28.16° with respect to the normal to the wavemaker.

Discussion

Conclusion

Appendix. Radiation Stress Tensor

Data Availability Statement

Notation

${\mathbf{c}}_g^{a-b}$

wave group celerity (vector) for the subtractive interaction of waves a and b;

$E_{*}$

wave energy density, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b);

g

acceleration of gravity;

$H_{*}$

wave height, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b, i, j);

h

local water depth;

$k_{ij}^{+}=|{\mathbf{k}}_i+{\mathbf{k}}_j|$

wave number for the additive interference of waves i and j;

$k_{ij}^{-}=|{\mathbf{k}}_i-{\mathbf{k}}_j|$

wave number for the SubI of waves i and j;

${\mathbf{k}}_{*}$

wave number vector, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b, i, j);

$\mathrm{\Delta }\mathbf{k}$

wave number vector for the subtractive interaction of waves a and b;

$|\mathrm{\Delta }\mathbf{k}|$

modulus of the vector;

$L_{*}$

wave length, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b);

$n_{*}$

ratio between group celerity and phase speed, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b);

p

dynamic wave pressure;

$S_{xx}^{*}$, $S_{xy}^{*}$, $S_{yy}^{*}$

components of radiation stress tensor, superscript $(\cdot {)}^{*}$ indicates the wave being considered (a, b, a–b);

$T_{*}$

wave period, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b);

${\tilde{u}}_i^{(*)},{\tilde{u}}_j^{(*)}$

horizontal components of orbital velocity, superscript $(\cdot {)}^{*}$ indicates the wave being considered (a, b, a − b, a + b), subscript i, j corresponds to Cartesian coordinates;

$u_{*}^{(1)},u_{*}^{(2)}$

first- and second-order of x-component of orbital velocity, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b, a − b, a + b);

$v_{*}^{(1)},v_{*}^{(2)}$

first- and second-order of y-component of orbital velocity, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b, a − b, a + b);

$w_{*}^{(1)},w_{*}^{(2)}$

first- and second-order of z-component of orbital velocity, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b, a − b, a + b);

x = (x, y)

horizontal position with (x, y) coordinates;

ɛ

wave arbitrary initial phase;

$\varphi =(\mathbf{k}\cdot \mathbf{x}-\sigma t+\epsilon )$

wave phase;

Φ⁽¹⁾

first-order velocity potential;

Φ⁽²⁾

second-order velocity potential;

ρ

water density;

${\sigma }_{*}$

wave frequency, subscript $(\cdot {)}_{*}$ indicates the wave being considered (a, b); and

Δσ

frequency difference between waves a and b.

References