# The Miles' theory of surface wind-waves in finite depth: a predictive physical model in coastal regions

Miguel A Manna

► To cite this version:

Miguel A Manna. The Miles' theory of surface wind-waves in finite depth: a predictive physical model in coastal regions. "Tribute to Ruben Aldrovandi", F. Caruso, J.G. Pereira, A. Santoro (editors) (Editora Livraria da Física, São Paulo, 2024). F. Carusso, J. G Pereira, QA. Santoro (EDITORS) São Paulo Livraria da Física, 2024, pp.153. hal-04752082

HAL Id: hal-04752082

https://hal.science/hal-04752082v1

Submitted on 21 Nov 2024

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# The Miles' theory of surface wind-waves in finite depth: a predictive physical model in coastal regions

M. A. Manna

Université Montpellier, Laboratoire Charles Coulomb UM, CNRS UMR-5221, F-34095, Montpellier, France.

This work is a review of recent developments in the Miles' theory of the growth of surface wind waves. In the linear regime, Miles' theory of wave amplification by wind is extended from deep water to the case of finite depth. A depth-dependent wave growth rate $\gamma$ is derived from the dispersion relation of the wind/water interface. For different values of the dimensionless water depth parameters $\delta = gh/U_1^2$ ($h$ the water depth, $g$ the gravity) and the wave-age parameter $\theta_{fd} = c/U_1$ ($U_1$ a characteristic wind speed, $c$ the wave phase speed) a family of wave growth curves of $\gamma$ in function of $\theta_{fd}$ and $\delta$ is plotted. The model provides a fair agreement with the data and empirical relationships obtained from the Lake George experiment, as well as with the data from the Australian Shallow Water Experiment. Three major results are obtained: (i) For small wave ages $\theta_{fd}$, the wave growth rates are comparable to those of deep water, (ii) for large wave ages $\theta_{fd}$ a finite-depth limited growth is reached with wave growth rates going to zero and (iii) contrary to the deep water case, the wave age at which the growth rate becomes zero is wind-dependent and depth-dependent.

PACS numbers:

## I. INTRODUCTION

### A. wind-generation surface waves in deep water

Surface water waves and their generation by wind is a fascinating problem. The starting point is the Navier-Stokes equations for air and water. This problem is a formidable one for both the physical and the mathematical point of view. Its exactly theoretical solution is impossible so several approximations and assumptions are required in order to obtain particular solutions. Theioneers works are those of Jeffreys [1, 2], Miles [3, 4] and Phillips [5]

The Miles mechanism of wave generation by wind states that waves are produced and amplified through a resonance phenomenon. Resonance appears between the wave-induced pressure gradient on the inviscid airflow and the surface waves. The resonant mechanism happens at a critical height where the airflow speed matches the phase velocity of the surface wave. In Jeffrey's and Miles' theories the viscosity is neglected, the water is considered deep and irrotational and the equations of motions are linearized.

Energy and momentum flow continuously from the air to the water. The wave energy density $\mathcal{E}$ for linear surface waves is given by

$$ \mathcal{E} = 2\rho_w g \eta_0^2, \quad (1) $$

with $\rho_w$ the water density, $g$ the gravity and $\eta_0$ the maximum wave amplitude. However waves may lose energy because of dissipation and in addition energy can be distributed between many models by finite-amplitude four waves interactions. The action balance equation has been commonly adopted to represent the temporal and spatial

dynamics of these processes. In deep water it reads

$$ \frac{\partial}{\partial t} \mathcal{M} + \vec{\nabla} \cdot (\vec{c}_g \mathcal{M}) = S, \quad (2) $$

where $\vec{\nabla}$ is the Laplace operator and $\mathcal{M}$ is the action density defined as

$$ \mathcal{M} = \frac{\mathcal{E}}{\omega}, \quad (3) $$

with $\omega$ the wave frequency and $\vec{c}_g$ the group velocity observed in a frame of reference moving with the wave and the source term $S$ is given by

$$ S = S_{in} + S_{nl} + S_{ds}, \quad (4) $$

with $S_{in}$, $S_{nl}$, $S_{ds}$ representing (respectively) the effects of the wind input, non-linear interactions and dissipation due to white capping.

Modern works [6–8] were almost focused on the computation of the term $S_{in}$ in (4). Later on numerical approaches were developed in order to calculate $S_{nl}$ and $S_{ds}$ [6].

### B. The coastal region

The coastal region is amongst the most dynamic physical system we can observe on the earth's surface. It results from the nonlinear interaction between ocean waves, currents and sediments. Each one of these actors depend on underlying factors and processes. Ocean waves are created by winds, currents depend on bathymetry and tides, sediments (size, mineralogy, quantity, etc) result from geological and climatic factors.
Extreme wave events in coastal regions (or freak waves events) occur in the presence of wind. This issue, in relation to the forecast of extreme waves, is really challenging the engineering and the physics communities.

In references [9, 10] the authors investigated the influence of wind on extreme wave events using the sheltering theory. They have shown that extreme events may be sustained longer by the air-flow separation. This mechanism can only be invoked if the wavecrest is steep enough to produce air-flow separation. Otherwise, for a too low steepness parameter $k\eta_0$ (k wavenumber) separation mechanism becomes irrelevant. Moreover, these works were limited to deep water and hence unable to fully describe winds generating near-shore waves where the wave field is influenced by bottom bathymetry. Consequently they are not adequate to correctly describe or forecast wind influence on extreme wave events in the coastal zone. Therefore an extension of Miles' theory of wind-generated monochromatic waves to the case of finite depth under weak or moderate winds (in terms of adequate finite depth parameters) is needed.

In the finite depth water domain the source term reads

$$S = S_{in} + S_{nl} + S_{ds} + S_{bf} + S_{tri}. \quad (5)$$

The term $S_{in}$ must be recalculated as he is strongly influenced by the finite depth $h$, the term $S_{bf}$ represents bottom friction and $S_{tri}$ is triad nonlinear wave interactions.

Theoretical extensions of wave growth to the finite depth domain were done in [11, 12]. They provided mathematical laws able to qualitatively reproduce features of the fields experiments on growth rate evolution of finite depth wind-waves and supplied a theoretical basis allowing to go beyond the empirical laws. In [12] was studied the wind action on the evolution of a wave-paquet and in [13] the action of a constant vorticity flow on the wave growth $\gamma$. In [14] and [15] was studied the evolution in time of a normal Fourier mode $k$ under the coupled action of weakly nonlinearity, dispersion and anti-dissipation.

This work is based on these last developments and is devoted to give a review of l'etat de l'art in the field. The paper is divided in two sections, Section II and Section III concerning respectively: the linear theoretical laws of growth of surface waves in finite depth and the pioneering experiments on growth of surface wind-waves are presented.

The subsections are organized as follows. In subsection II A we display the model. In subsection II B the linear problem in the water domain is studied. Then in subsection II C the air domain is coupled with the water domain. The linear problem can be solved at the interface and we derive the linear dispersion relation of wave amplification in finite depth. We introduce dimensionless variables and scalings, to obtain an adequate growth rate. The theoretical linear dynamic is discussed. In subsection III A the theoretical laws are compared with both the Young-Verhagen data and plots of empirical relationships from the Lake George experiment and with Donelan's data from the AUSWEX program. In subsection III B are exhibited the critical values of wave ages and wavelengths for which the growth rate goes to zero. In subsection III C the white-capping influence on the phenomenon is discussed. Finally subsection IV draws the conclusions.

## II. WIND GENERATED SURFACE WAVES. THE LINEAR REGIME

### A. The interface problem

We are going to study the stability of an air-water interface. Let the fluid particles be located relatively to a fix rectangular Cartesian frame with origin $O$ and axes $(x, y, z)$, where $Oz$ is the upward vertical direction. We assume translational symmetry along $y$ and we will only consider a sheet of fluid parallel to the $xz$ plane. The plane $z = 0$ characterizes the interface at rest. The perturbed air-water interface will be described by $z = \eta(x, t)$. The water lies between the bottom located at $z = -h$ and the interface $z = \eta(x, t)$. The air occupies the $\eta(x, t) < z < +\infty$ region. We suppose the water as well as the air to be inviscid and incompressible. The unperturbed air flow is a prescribed mean shear flow, only depending on the vertical coordinate $z$. We assume the dynamic to be linear and disregard the air turbulence.

### B. The linearized water dynamics

In the water domain we consider the Euler equations for finite depth with $u(x, z, t)$ and $w(x, z, t)$ being the horizontal and vertical velocities of the fluid. The continuity equation and the linearised equations of motion for $-h \le z \le \eta(x, t)$ are (Lighthill [16])

$$u_x + w_z = 0, \quad \rho_w u_t = -P_x, \quad \rho_w w_t = -P_z - g\rho_w, \quad (6)$$

where $P(x, z, t)$ is the pressure and subscripts in $u, w$ and $P$ denote partial derivatives. The boundary conditions at $z = -h$ and at $z = \eta(x, t)$ are

$$w(-h) = 0, \quad \eta_t = w(0), \quad P(\eta) = P_a(\eta), \quad (7)$$

where $P_a(x, z, t)$ is the air pressure evaluated at $z = \eta$. The equation $P(\eta) = P_a(\eta)$ is the continuity of the pressure across the air/water interface. We introduce a reduced pressure defined by

$$\mathbf{P}(x, z, t) = P(x, z, t) + \rho_w gz - P_0, \quad (8)$$

where $P_0$ is the atmospheric pressure. In terms of (8) equation (6) and the continuity of the pressure across the air/water interface read

$$u_x + w_z = 0, \quad \rho_w u_t = -\mathbf{P}_x, \quad \rho_w w_t = -\mathbf{P}_z, \quad (9)$$
$$ \mathbf{P}(x, \eta, t) = P_a(x, \eta, t) + \rho_w g \eta - P_0. \quad (10) $$

The linear equations system (7)-(9) can be solved, assuming normal mode solutions as $u = \mathcal{U}(z) \exp(i\theta)$, $w = \mathcal{W}(z) \exp(i\theta)$, $\eta = \eta_0 \exp(i\theta)$, $\mathbf{P} = \mathcal{P}(z) \exp(i\theta)$, with $\theta = k(x - ct)$ where $k$ is the wavenumber, $c$ the phase speed and $\eta_0$ is a constant. We obtain

$$ \mathcal{W}(z) = \frac{-ikc \sinh k(z+h)}{\sinh kh} \eta_0, \quad (11) $$

$$ \mathcal{U}(z) = \frac{kc \cosh k(z+h)}{\sinh kh} \eta_0, \quad (12) $$

$$ \mathcal{P}(z) = \frac{k \rho_w c^2 \cosh k(z+h)}{\sinh kh} \eta_0. \quad (13) $$

The phase speed $c$ is unknown in equations (11)-(13). To determine $c$ we have to consider the boundary conditions (10), not yet used, and the continuity of $P$ at $z = \eta$ which yields

$$ \rho_w \eta_0 \exp(i\theta) \{c^2 k \coth kh - g\} + P_0 = P_a(x, \eta, t). \quad (14) $$

In the single-domain problem $P_a = P_0$ and (14) gives the usual expression for $c$,

$$ c^2 = c_0^2 = \frac{g}{k} \tanh(kh). \quad (15) $$

In the problem under consideration the determination of $c$ needs the use of the air pressure evaluated at $z = \eta$ i.e., $P_a(x, \eta, t)$

### C. The air profile, the linearized air dynamics and the wave growth rate

We consider a steady air flow with a prescribed mean horizontal velocity depending only on the vertical coordinate $z$: $\vec{U} = U(z)\vec{e}_x$. We choose $U(z)$ to be the logarithmic wind profile

$$ U(z) = U_1 \ln(z/z_0), \quad U_1 = \frac{u_*}{\kappa}, \quad \kappa \approx 0.41. \quad (16) $$

This is commonly used to describe the vertical distribution of the horizontal mean wind speed within the lowest portion of the air-side of the marine boundary layer [17]. It can also be justified with scaling arguments and solution matching between the near-surface air layer and the geostrophic air layer [18]. In (16)

* $u_*$ is the friction velocity,

* $\kappa$ the Von Kármán constant,

* $z_0$ the aerodynamic sea surface roughness located just above the interface water/air.

Through this paper, $z_0$ will be regarded as a constant, independent from the sea state. This approximation was first proposed in [19].

The perturbations to the mean flow $U(z)$ and to the pressure $P_a$ are: $u_a(x, z, t)$, $w_a(x, z, t)$ and $P_a(x, z, t)$ where subscript $a$ stands for air. So with $\mathbf{P}_a(x, z, t) = P_a(x, z, t) + \rho_a gz - P_0$, $\rho_a$ the air density, and $U' = dU(z)/dz$ we have the following Euler equations

$$ u_{a,x} + w_{a,z} = 0, \quad (17) $$

$$ \rho_a[u_{a,t} + U(z)u_{a,x} + U'(z)w_a] = -\mathbf{P}_{a,x}, \quad (18) $$

$$ \rho_a[w_{a,t} + U(z)w_{a,x}] = -\mathbf{P}_{a,z}, \quad (19) $$

which must be completed with the appropriate boundary conditions. The first one is the kinematic boundary condition,

$$ \eta_t + U(z_0)\eta_x = w_a(z_0). \quad (20) $$

Using (16) equation (20) can be reduced to

$$ \eta_t = w_a(z_0). \quad (21) $$

Equation (21) describes the influence of the surface perturbation on the vertical perturbed wind speed. We assume $\mathbf{P}_a = \mathcal{P}_a(z) \exp(i\theta)$, $u_a = \mathcal{U}_a(z) \exp(i\theta)$, $w_a = \mathcal{W}_a(z) \exp(i\theta)$ and we add the following boundary conditions on $\mathcal{W}_a$ and $\mathcal{P}_a$,

$$ \lim_{z \to +\infty} (\mathcal{W}'_a + k\mathcal{W}_a) = 0, \lim_{z \to z_0} \mathcal{W}_a = W_0, \quad (22) $$

and

$$ \lim_{z \to +\infty} \mathcal{P}_a = 0, \quad (23) $$

that is, the disturbance plus its derivative vanish at infinity, the vertical component of the wind speed is enforced by the wave movement at the sea surface and the pressure vanishes at infinity. Using equations (17)-(19) and (23) we obtain

$$ w_a(x, z, t) = \mathcal{W}_a \exp(i\theta), \quad (24) $$

$$ u_a(x, z, t) = \frac{i}{k} \mathcal{W}_{a,z} \exp(i\theta), \quad (25) $$

$$ \mathbf{P}_a(x, z, t) = ik\rho_a \exp(i\theta) \int_z^\infty [U-c] \mathcal{W}_a dz'. \quad (26) $$

Eliminating the pressure from the Euler equations we obtain the Rayleigh equation (or inviscid Orr-Sommerfeld equation) [20] $\forall z \setminus z_0 < z < +\infty$

$$ (U-c)(\mathcal{W}_a'' - k^2 \mathcal{W}_a) - U'' \mathcal{W}_a = 0. \quad (27) $$

The Rayleigh equation is singular at the critical height $z_c = z_0 e^{c\kappa/u_*} > z_0 > 0$, where $U(z_c) = c$. This model disregards any kind of turbulence, and so that the critical height is set above any turbulent eddies or other nonlinear phenomena. In equations (24)-(27) neither $\mathcal{W}_a(z)$ nor $c$ are known. In order to find $c$, we have to calculate $P_a(x, \eta, t)$. We obtain

$$ P_a(\eta) - P_0 = -\rho_a g \eta + ik\rho_a e^{(i\theta)} \int_{z_0}^{\infty} [U-c] \mathcal{W}_a dz, \quad (28) $$
where the lower integration bound is taken at the constant roughness height $z_0$ because $W_a(z) = 0$ for $\eta(x,t) \le z < z_0$. Finally, using equation (21) to eliminate the term $ik\rho_a \exp(i\theta)$ in the equation (28), equation (14) yields

$$g(1-s) + c \frac{sk^2}{W_0} I_1 - c^2 \left\{ \frac{sk^2}{W_0} I_2 + k \coth(kh) \right\} = 0, \quad (29)$$

where $s = \rho_a/\rho_w \sim 10^{-3}$ and the integrals $I_1$ and $I_2$ are defined as follow

$$I_1 = \int_{z_0}^{\infty} U W_a dz, \quad I_2 = \int_{z_0}^{\infty} W_a dz. \quad (30)$$

Equation (29) is the dispersion relation of the problem. The parameter $s$ is small and (29) may be approximated as

$$c = c_0 + s c_1 + O(s^2). \quad (31)$$

The explicit form of $c_1$ is calculated in the next section. Therefore, we can find $W_a(z)$ by solving the Rayleigh equation with $c$ replaced by $c_0$, that is to say, of order zero in $s$. The function $W_a(z)$ is complex and consequently $c$ also i.e.,

$$c = \Re(c) + i\Im(c). \quad (32)$$

Its imaginary part $\Im(c)$ gives the growth rate of $\eta(x,t)$ defined by

$$\gamma = k\Im(c), \quad (33)$$

so we have

$$\eta(x,t) = \eta_0 \exp[ik[x-\Re(c)t]] \exp^{\gamma t}. \quad (34)$$

We define three dimensionless parameters $\delta$, $\theta_{dw}$ and $\theta_{fd}$

$$\delta = \frac{gh}{U_1^2}, \quad \theta_{dw} = \frac{1}{U_1} \sqrt{\frac{g}{k}}, \quad \theta_{fd} = \frac{1}{U_1} \sqrt{\frac{g}{k}} T^{1/2}, \quad (35)$$

with $T = \tanh(\frac{\delta}{\theta_{dw}})$. The dimensionless parameter $\delta$, for constant $U_1$, measures the influence of the limit phase velocity $\sqrt{gh}$ on the rate of growth of $\eta$, the parameter $\theta_{dw}$ is a theoretical analogous of the deep water wave age mesuring the relative value of the deep water phase speed in relation to the characteristic wind velocity $U_1$ and $\theta_{fd}$ is a theoretical analogous of the finite depth wave age.

Since as a wave “ages” it grows longer and as it grows longer it moves faster so the phase velocity $c$ increases. Consequently $\theta_{fd}$ will be bigger for older waves and smaller for younger waves.

In order to obtain the growth rate, we introduce the following non-dimensional variables and scalings (hats meaning dimensionless quantities)

$$\frac{U}{U_1} = \hat{U}, \quad \frac{W_a}{W_0} = \hat{W}_a, \quad zk = \hat{z}, \quad \frac{c}{U_1} = \hat{c}, \quad \frac{tg}{U_1} = \hat{t}. \quad (36)$$

Using (35) and (36) in equation (29) and retaining only the term of order $s$ we obtain $\hat{c}(\delta, \theta_{dw})$,

$$\hat{c} = \theta_{dw} T^{1/2} - \frac{s}{2} \theta_{dw} T^{1/2} + \frac{s}{2} \{ T \hat{I}_1 \cdot \theta_{dw} T^{3/2} \hat{I}_2 \}. \quad (37)$$

The dimensionless growth rate $\hat{\gamma} = \frac{U_1}{g}\gamma$ is,

$$\hat{\gamma} = \frac{s}{2} \left\{ \frac{T\Im(I_1)}{\theta_{dw}^2} - \frac{T^{3/2}\Im(I_2)}{\theta_{dw}} \right\}. \quad (38)$$

A given set of $(\delta, \theta_{dw})$ fixed the $\gamma$ valued. The unique curve of wave growth rate in deep water is transformed in a family of curves indexed by $\delta = gh/U_1^2$, i.e., a curve for each value of $\delta$.

In Figure 1 is shown a family of six values of $\delta$ against the $\theta_{fd}$ parameter:

* small finite wave age $\theta_{fd}$ corresponds to short surface waves. This stage represents the initial growth of the wave field near the shoreline of a calm lake,

* as the time proceeds the surface waves reaches moderate $\theta_{fd}$ which correspond to mild or moderate wavelengths,

* long waves are found for large $\theta_{fd}$.

Of course, as the wavelengths are increasing, the amplitudes keep on growing. From a physical point of view, this means that Figure 1 is a snapshot of the theoretical dynamical development of the wave which is growing in amplitude and wavelength in time:

* at small $\theta_{fd}$ the growth rate $\gamma$ is equal for all values of $\delta$, the limit being the deep water case i.e.; the Miles' curve,

* as $\theta_{fd}$ increases, the finite-depth effects begin to appear. The growth rate becomes lower than in the deep water limit, for each value of $\delta$. The growth rates are scaled with $\delta$. For a given $\theta_{fd}$, the bigger the $\delta$ the larger the $\hat{\gamma}$. Each $\delta$-curve approaches its own (idealized) theoretical $\theta_{fd}$-limited growth as $\hat{\gamma}$ goes to zero. At this stage the wave reaches a final state of linear progressive wave with zero growth. In others words, for a given $\delta$ the surface wave does not grow old anymore beyond a determined $\theta_{fd}$.

Our results concern the dimensionless growth rate $\hat{\gamma}$ instead of the $\beta$-Miles parameter. We have the following transformation rule between this parameter $\beta$ and dimensionless $\hat{\gamma}$

$$\beta = \frac{2\hat{\gamma}}{s} \theta_{dw}^3 T^{1/2}, \quad (39)$$

where we took $\beta$ as it is usually defined,

$$\Im(c) = c_0 \frac{s}{2} \beta \left( \frac{U_1}{c_0} \right)^2. \quad (40)$$

Its evolution is shown in Figure 2, showing the correct deep water trends, and the new finite depth limits. The effects of depth are critical. $\beta$ is almost constant for small $\theta_{fd}$, as usual, but it goes dramatically to zero when the depth limit is close.
<figure><img src="image_1.png" /><figcaption>FIG. 1: Evolution of the growth rate in semi-logarithmic scale. Every curve but the rightmost one correspond to finite depth. From left to right, they match $\delta = 1, 4, 9, 25, 49, 81$. We can observe that for each depth, there is a $\theta_{fd}$ limited wave growth. The deep water limit, also computed, corresponds to small $\theta_{fd}$ and matches Miles' results.</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>FIG. 2: Evolution of Miles' coefficient $\beta$ for several values of the depth. Each curve is plotted with the same Charnock constant $\alpha_c \approx 0.018$. The finite-depth effect is critical, and high value of $\delta$ correspond to deep water.</figcaption></figure>

### III. YOUNG-DONELAN FIELD EXPERIMENTS ON GROWTH OF SURFACE WIND-WAVES IN FINITE DEPTH

Experiments in Lake George, Australia, described by Young and Verhagen in [21] and [22] provided the first systematic attempts to understand the physics of wave-wind generation in finite depth water. These papers gave a very complete description of the basin geometry and bathymetry, experimental design, used instrumentation, as well as the adopted scaling parameters. The measurements have confirmed the water depth dependence of the asymptotic limits to wave growth.

Young in [23, 24] has derived an empirical relation in terms of appropriate dimensionless parameters able to reproduce the experimental data of [21]. Especially the empirical relationship between the fractional energy increase as a function of the inverse wave age, found by

Doneland et al. for deep water in [25], was extended to the finite depth domain. Experimental results and empirical laws have shown that contrary to the deep water case, the wave age at which the growth rate becomes zero is wind-dependent and depth-dependent. So, the point of full development is warped from the deep water case, where it was established in [26]. As a result a growth law against the inverse wave age exists for each value of the wind intensity $U(z)$ and the water depth $h$.

The evolution of the growth rates is such that at small wave ages growth rates are comparable to the deep water limit, at large wave ages the growth rate is lower in shallow water than in deep water, and beyond a limit wave age, the growth rate vanishes.

#### A. Empirical relations versus theoretical laws

Field or laboratory experiments involved measurements of a wind wave train i.e., a superposition of wave Fourier modes. The results are commonly given in terms of the observed phase and group speeds $C_p, C_g$ at the peak frequency $\omega_p$. Consequently qualitative comparison between field (or laboratory) observations and theoretical laws can only be done using the phase and group speeds $c, c_g$ and frequency $\omega$ of only one mode instead of $C_p, C_g$ and $\omega_p$ corresponding to a wave packet.

First of all we are going to show that the theoretical curves for $\hat{\gamma}$ are, mutatis mutandi, in good qualitative agreement with the empirical curves of the dimensionless fractional wave energy increase per radian $\hat{\Gamma}$ as a function of the inverse wave age $U_{10}/C_p$ [23]. In this reference, experimental field data for $\hat{\Gamma}$

$$ \hat{\Gamma} = \frac{C_g}{\omega_p E} \frac{\partial E}{\partial x}, \quad (41) $$

in the finite depth Lake George are adequately represented by the empirical relationship

$$ \hat{\Gamma} = A \left( \frac{U_{10}}{C_p} - 0.83 \right) \tanh^{0.45} \left( \frac{U_{10}}{C_p} - \frac{1.25}{\delta_Y^{0.45}} \right), \quad (42) $$

with A constant, $\delta_Y = gh/U_{10}^2$ the non-dimensional water depth, $U_{10}$ the wind speed measured at a reference height of 10 m, and $C_g$ and $C_p$ the group and phase speeds of the components at the spectral peak frequency $\omega_p$.

In order to make a qualitative comparison between $\hat{\Gamma}$ curves in function of the inverse wave-age $U_{10}/C_p$ and theoretical $\hat{\gamma}$ curves in function of $1/\theta_{fd}$ we need to write the empirical $\hat{\Gamma}$ in terms of theoretical quantities. So, the following changes are necessary:

$$ \text{measured } C_g, C_p, \omega_p \rightarrow \text{theoretical } c_g, c, \omega, \quad (43) $$

$$ \text{and} \quad \frac{U_{10}C_{10}^{1/2}}{\kappa} = u_*/\kappa = U_1, \quad (44) $$

with $C_{10}$ the 10 m drag coefficient as defined in reference [27]. Thus, from the fact that the energy growth rate is
two times the amplitude growth rate, that is

$$ \Gamma = 2\gamma, $$

and using $2c_g = c(1 + 2kh/\sinh(2kh))$, (36), (43), (44) and the expression of $\theta_{fd}$ in (35) we obtain

$$ \hat{\Gamma} = \frac{\theta_{dw}}{T^{1/2}} \hat{\gamma} \left[ 1 + \frac{2\delta}{\theta_{dw}^2 \sinh\left(\frac{2\delta}{\theta_{dw}^2}\right)} \right]. \quad (45) $$

This expression gives the theoretical equivalent of the empirical $\hat{\Gamma}$ in function of $\theta_{dw}$, $\delta$ and $\hat{\gamma}$. The values of $\hat{\gamma}$ for fixed $\delta$'s as a function of $1/\theta_{fd}$ are numerically obtained using $\theta_{fd}$, $T = \tanh(\frac{\delta}{\theta_{dw}^2})$ and (38). Steps (43) and (44) transform $\delta_Y$ and $C_p/U_{10}$ into $\delta$ and $\theta_{fd}$ according to

$$ \delta_Y = \delta \frac{C_{10}}{\kappa^2}, \quad (46) $$

$$ \frac{C_p}{U_{10}} = \theta_{fd} \frac{C_{10}^2}{\kappa}. \quad (47) $$

In [23] the curves of $\Gamma$ versus $U_{10}/C_p$ have been presented for the $\delta_Y$-intervals, rather than for a single value of $\delta_Y$. The intervals were

$$ \delta_Y \in [0.1 - 0.2], \quad \delta_Y \in [0.2 - 0.3], \quad (48) $$

$$ \delta_Y \in [0.3 - 0.4], \quad \delta_Y \in [0.4 - 0.5]. \quad (49) $$

They were determined from the variations in $U_{10}$, the depth $h$ being nearly constant around 2 m. Consequently we substitute the $\delta_Y$-intervals with $\delta$-intervals using (46) and we evaluate the mean value $\bar{\delta}$. For example $\delta_Y \in [0, 1 - 0, 2]$ is transformed into $\bar{\delta} = [13, 17 - 26, 35]$ with $\bar{\delta} = 19, 76$ in Figure 3(a). Figures 3(a), 3(b), 3(c) and 3(d) are displaying a fair concordance of the model with the experimental data and plots of empirical laws for Lake George. The agreement improves as $1/\theta_{fd}$ increases.

## B. The critical wave age and the critical wavelength

In Figure 4 are plotted, against $\delta$, the critical values of the parameter $\theta_{fd}^c$ for which the growth rate $\gamma$ goes to zero. They obey the relation

$$ \theta_{fd}^c = \delta^{\frac{1}{2}}. \quad (50) $$

The above relation, found numerically, is coherent with the parameter formulation of $\theta_{fd}$ in (35). It is indeed a limiting value for $\theta_{fd}$ uniquely determined by the water depth. Young in [24] has shown, from an empirical relationship that $\hat{\Gamma}$, the growth rate, goes to zero as a function of the inverse wave age $U_{10}/C_p$ for

$$ \frac{C_p}{U_{10}} = 0.8 \left( \frac{gh}{U_{10}^2} \right)^{0.45}. \quad (51) $$

Using a $C_{10}$ drag coefficient parametrization such as in [27] we obtain

$$ C_{10} = (0.065U_{10} + 0.8)10^{-3}, \quad (52) $$

and taking an average $U_{10} = 7$ m/s (see [24]), one finds the $U_1$ to $U_{10}$ relationship

$$ U_{10} \approx 28,3 \ u_* \approx 11,6 \ U_1. \quad (53) $$

So, this limiting law reads

$$ \frac{C_p}{U_1} = 1,01 \ \delta^{0,45}, \quad (54) $$

a result in excellent agreement with the theoretical value (50).

With $\theta_{fd}^c$ we can calculate the corresponding critical wave length $\lambda^c$. Using (50) in $\theta_{fd}$ we obtain

$$ \frac{\delta}{\theta_{dw}^2} = \tanh\left(\frac{\delta}{\theta_{dw}^2}\right). \quad (55) $$

Relation (55) means the wave has entered the shallow water region. In such a limit the range of $\delta/\theta_{dw}^2$ is: $0 < \delta/\theta_{dw}^2 < \frac{\pi}{4}$ (see references [28] and [29]). As a result we obtain $\lambda^c = 8h$. For values of $\lambda$ such that $\lambda > \lambda^c$ the phase velocity is in the long wave limit i.e., $c = \sqrt{gh}$. Consequently, if $\lambda > \lambda_c$ the wave feels the bottom, the amplitude does not grow anymore, the resonance wind/phase speed ceases, and the wave reaches its utmost state as a progressive plane wave.

Finally in Figure 4 are also represented data from the Australian Shallow Water Experiment ([25]). A fit is also plotted to show the trend. The raw data consists in the water depth $h$ in metres, the friction velocity $u_*$, the 10 metres wind velocity $U_{10}$ and the ratio of the former with the measured phase speed $c_p$, $U_{10}/c_p$. For example, $u_* = 0.44 \ m.s^{-1}$ and $h = 0.32 \ m$ gives $\delta = 2.7$ and $\theta_{fd} = 1.55$, which gives a small relative error regarding (50). All the points give $(\delta, \theta_{fd})$ coordinates close to the theoretical limit.

## C. The white-capping dissipation influences

This subsection aims at answering the question: why do the $\hat{\Gamma}$ curves seem to be consistent with the empirical fits of reference [23], even though bottom friction dissipation $S_{bf}$ and white-capping dissipation $S_{ds}$ are disregarded ?

It is currently admitted that the bottom friction $S_{bf}$ plays a relatively minor role in depth limited growth studies, even though being an important dissipative factor for swell propagating in shallow water [30]. Consequently, we do not consider bottom friction in the following analysis. The white-capping dissipation $S_{ds}$ is an important dissipative mechanism since in finite depth conditions, wind waves show significant wave breaking events. Hence, dissipation due to wave breaking $S_{ds}$ is considered to be the
<figure><img src="image_3.png" /><figcaption>FIG. 3: Growth rate $\hat{\Gamma}$ as a function of inverse wave age $1/\theta_{fd}$ for several values of the parameter $\delta$. White squares correspond to Lake George experiment data, Black squares correspond to the empirical relationship found by Young in [23]. Present results correspond to symbols +, × and *. (a): the dataset covers a range of wind speed corresponding to $\delta_Y = 0.1 - 0.2$, or, using (46) $\delta = 13.17 - 26.35$, and an average value $\langle \delta \rangle = (13.17 + 26.35)/2$ is used. (b): same as (a) with $\delta_Y = 0.2 - 0.3$. (c): same as (a) with $\delta_Y = 0.3 - 0.4$. (d): same as (a) with $\delta_Y = 0.4 - 0.5$.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>FIG. 4: Parameter curves corresponding to zero growth rate. The theoretical limit is given by (50). The AUSWEX data are experimental results from [25] (the sea state is fairly close to the finite depth full development).</figcaption></figure>

dominant dissipative term in finite depth, compared to the deep water case ([22, 30]). Now, what can we observe in plots 3(a), 3(b), 3(c) and 3(d)?

* For young sea regimes ($1/\theta_{fd}$ large), one can observe in figures 3(a), 3(b), 3(c) and 3(d) that the $\langle \delta \rangle$-curves match the experimental $\hat{\Gamma}_Y$ curves in [24] (in black squares). Hence, for large value of $1/\theta_{fd}$ we have $\hat{\Gamma} = \hat{\Gamma}_Y$. Small values of $\theta_{fd}$ correspond to wave propagating in deep water, for any $\langle \delta \rangle$. This is in line with Figure 7.1.c in [24] where $S_{ds} \to 0$ for large frequencies.

* For mature or old sea regimes ($1/\theta_{fd}$ small) both curves $\hat{\Gamma}$ and $\hat{\Gamma}_Y$ go to zero. In developed finite depth seas, wave energy (amplitude) evolution becomes depth limited no matter what the value of $S_{ds}$ is. Hence, finite depth-limitation phenomenon prevails.

* For intermediate regimes the $\hat{\Gamma}$ and $\hat{\Gamma}_Y$ curves are similar in shape. Nevertheless, $\hat{\Gamma} > \hat{\Gamma}_Y$ for all the values of $\langle \delta \rangle$ considered herein. This is in agreement with Figure 7.1.c in [24] where $S_{ds} \lesssim 0$ for intermediate frequencies. Consequently, the wave growth rate $\hat{\gamma}$ ($\hat{\Gamma} = 2\hat{\gamma}$) is overestimated in our model. We can observe in figures 3(a), 3(b), 3(c) and 3(d) that the gap between $\hat{\Gamma}$ and $\hat{\Gamma}_Y$ increases as $\langle \delta \rangle$ decreases, because the white-capping $S_{ds}$ is larger and wider for small kh as we can see in Figure 7.1.c in [24].

## IV. CONCLUSIONS

In this review our aim was exclusively focused on the derivation of a linear Miles' theory for waves propagating on finite depth h. We extended the well known Miles' theory to the finite depth context under breeze to moderate winds conditions. We have linearized the equations
of motion governing the dynamics of the air/water in-
terface problem in finite depth and have studied the lin-
ear instability in time of a normal Fourier mode $k$. The
prediction of exponential growth of wave amplitude (or
energy) is well confirmed by field and laboratory exper-
iments (the wind-to-waves energy transfer rates predic-
tions are smaller than the observations, although their
order of magnitude is the same).

There are many other parameters that influence the
growth of wind-waves in finite depth. For example, wind
speed and wind direction variations with time, geometry
and bathymetry of the lake, bottom friction, surface drift
induced by the air flow, boundary layer turbulence, non-
linear waves interactions, and so on. Taken into account
all these phenomena represents a work that cannot be

handled analytically, even though numerically. Although
our study is a highly idealized, we believe that it may pro-
vide a valuable insight about the effect of depth on the
mechanism of water wave amplification by wind and be
useful in theoretical forecast of wind-wave growth rates
in finite depth.

Acknowledgments

The author thanks the organizers for the invitation to
contribute to the tribute in memory of Professor Ruben
Aldrovandi.

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