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

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