Dynamics of Marine Vehicles

Dynamics of Marine vehicles

By S. Hossein Mousavizedegan Faculty of Marine Technology Amirkabir University of Technology No. 424, Hafez Ave. Tehran, Iran Dedication

This book is dedicated to my family. Preface

This is an introduction to the water wave motion. It is included four chapters. The first chapter is on the basic assumption and the general formulation of the water waves. The second chapter is on the long crested wave theory. The third chapter is about the finite amplitude waves and the effect of the nonlinearities on the wave motion. The final chapter is about the real ocean waves.

March 2010 S. Hossein Mousavizadegan

iii Contents

1 Introduction 1 1.1 Fundamental assumptions on seawater properties and wave motion...... 3 1.2 Boundaryvalueproblemofoceangravitywaves ...... 4

2 Long-Crested, Linear Wave Theory (LWT) 1 2.1 BoundaryvalueproblemforLWT ...... 1 2.2 Analytical solution of the LWT BVP ...... 3 2.3 Dispersion relation ...... 5 2.4 Classiﬁcationofwaterwaves...... 6 2.5 Characteristics of linear plane progressive wave ...... 7 2.5.1 particle motion ...... 8 2.5.2 Pressure distribution ...... 9 2.6 Progressiveobliquewaves ...... 13 2.7 Superpositionofwaves...... 17 2.8 Wavereﬂectionandstandingwave ...... 19 2.9 Wavegroup...... 21 2.10Waveenergy ...... 22 2.10.1 Energypropagation ...... 24 2.10.2 Equationofenergyconservation ...... 25

3 Finite-amplitude waves 28 3.1 Stokes Finite-amplitude waves theory ...... 28 3.1.1 Theﬁrst-orderwavestheory...... 30 3.1.2 Thesecond-orderwavestheory ...... 31 3.2 Trochoidalwavetheory ...... 36 3.3 Wavetransformation...... 37 3.3.1 Waveshoaling ...... 38 3.3.2 Waverefraction...... 39 3.3.3 Wavebreaking ...... 40 3.3.4 Wavediﬀraction ...... 41

4 Real ocean Waves 45 4.1 Introduction...... 45 4.2 Statistical and probabilistic definitions ...... 45 4.2.1 Basic definitions and concept of random process ...... 48 4.3 Irregularwaves ...... 49 4.3.1 Wave height definitions ...... 50 4.3.2 Irregularwaveperiods ...... 52 4.3.3 Probability distribution of a sea state ...... 52 4.4 SpectraldescriptionofOceanwaves ...... 58 4.4.1 Spectraldensityfunction ...... 59 4.4.2 Spectralpropertiesofoceanwaves ...... 65

iv CONTENTS Dynamics of Marine Vehicle

4.4.3 Typicalwaveenergydensityspectrum ...... 66 4.4.4 Directional spectral function ...... 71 4.5 Application of wave energy density spectrum ...... 71 4.5.1 Simulation of wave proﬁle ...... 71 4.5.2 Computation of the average heights characteristics ...... 74 4.5.3 Arbitrarywavespectra...... 74 4.5.4 Waveperiod ...... 75

v Chapter 1

Introduction

Ocean water is permanently subjected to the external forces of nature that dictate the types of induced waves in the ocean. We may distinguish five basic types: sound; capillary; gravity; internal; and planetary waves. Sound waves are due to the water compressibility that is very small. The combination of the turbulent fluctuation of the atmospheric pressure and surface tension induce small waves, of almost regular form, with a high frequency. This type of waves are called capillary waves. These waves are usually unstable and attenuate, due to the surface tension, when the wind calms. Gravity waves, acting on water particles displaced from equilibrium at the ocean surface or at an internal geopotential surface surface in a stratified fluid, induced gravity waves (surface or internal). There are also vary slow and large scale planetary or Rossby waves induced by the variation of the equilibrium potential vorticity, due to changes in depth or latitude. All of the above wave types can occur together, producing more complicated patterns of oscillations. The frequency range associated with external forces is very wide and ocean surface response occupies an extraordinary broad range of wave lengths and periods, from capillary waves, with period less than a second, through wind-induced waves and swell with period of the order of few second, to tidal oscillations with periods of the order of several hours and days. The schematic representation of energy contained in the surface waves is given in Fig. 1.1. The physical mechanisms generating these waves is also listed in Tab. 1.1.

Wave types Physical mechanism periods Capillary waves Surface tension < 10−1 s Wind waves Wind shear, gravity < 15 s Swell Wind waves < 30 s Tsunami Earthquake 10 min 2 h − Tides Gravitational action of the moon and sun, earth rotation 12 24 h − Table 1.1: Waves, physical mechanism, and periods, Masse,l [2]

The gravity waves are of the greatest importance for engineering activity in the sea. The inﬂuence of wind- induced waves on engineering structures is most sensible and hostile. Marine structures must be designed to sustain the forces and motions induced by these waves. A through understanding of the interaction of waves with marine structures has now become a vital factor in the safe and economical design of such structures. The calculation procedures needs to established the structural loading and induced motions generally involve the following steps:

◮ establishing the wave climate in the working area of the marine structure;

◮ estimating the wave conditions for the structure; and

◮ selecting and applying a wave theory to determine the induced motions and hydrodynamic loading

1 Chapter 1 Introduction Dynamics of Marine Vehicle

Figure 1.1: Wave energy spectra, Tichet [5] in which the knowledge of the surface waves is essential. The wind is one the cause of wave generation on the surface of sea. At the initial stage of wave generation, the turbulent fluctuation along with the surface tension cause an almost regular waves which called the capillary wave. As wind velocity increase, waves grow and gravity forces are sufficient to support wave motion. Waves growth is not infinite and it will be reach an limiting steepness that is about 1/7 in deep water. When the waves reach to this limiting value, they break in the form of white caps or spilling or plunging breakers. Knowledge on the mechanism of generation, interaction and decay of ocean waves has been accumulated during the last 80 years. However, modern understanding of the dynamic process involved has been developed only within the last 50 years. The starting points of contemporary wave generation models are the pressure fluctuations and variation in shear stresses at the water surface, associated with the airflow over the waves. In spite of the clear connection between wind and waves, and a long history of theoretical efforts, only in the 1965s and 60s was a basic understanding of the mechanisms of wind-wave generation acquired. In 1956 Ursell, in his review of actual wind-wave generation theories, concluded that all available theories were grossly inadequate to account for observations (Ursell, [6]). The independent and complementary works of Philips [4] and Miles [3] provided the cornerstones on which now rests our theoretical understanding of wind-wave generation. Phillips’ model of wind-wave generation is based on the assumption that the atmospheric turbulent pressure fluctuations are undistributed by the waves and are advected over the sea surface at some velocity U related to the wind speed. Phillips showed that resonance is possible between the advected pressure and those waves which travel at the right speed to keep the forcing. The resonance mechanism accounts for the excitation and initial growth of waves on an undistributed water surface. However, it is too weak to support the continued growth of wind waves. Once waves have appeared on the sea surface, their presence modifies the air flow.

2 Chapter 1 Introduction Dynamics of Marine Vehicle

1.1 Fundamental assumptions on seawater properties and wave motion

We adopt a rectangular coordinate system O xyz so that the origin is at the mean sea surface. The xy plane is horizontal and the z axis is directed opposite− the the gravity force. It is assumed that the seawater is a continuous media. In general,− the equation of motion for the ﬂuid particle depend on the physical properties of the ﬂuid and motion itself. We may assumed that:

1. Seawater is an inviscid fluid. Therefore, there are no shearing stresses. In many oceanic motions, the Reynolds number, the ratio of UL the inertia force to the viscous force, is very large Re = ν . Thus, the viscous influence is often quite negligible over most of the filed of motion. The viscous forces are important in narrow region of the flow, where the local inertia force and viscous forces are comparable. In the ocean, he interfacial layer between the air and the water, as well as the bottom layer are such a regions. The thickness of the 2ν 1/2 surface boundary layer is of the order δ = ω , where ω is a wave frequency, and ν is kinematic coefficient of viscosity (for water is ν =1.2 10−6). For typical ocean wave frequencies, the thickness δ 0.001 m. For the boundary layer near× the natural sea bottom, the eddy viscosity is much higher ≈ (νt 100ν). The thickness of the boundary layer is then 0.1 m, which is still quite small. Therefore, the≈ boundary layer regions are but a very small fraction of a fluid volume, and the influence of the molecular viscosity on the wave motion can be neglected. 2. Seawater is an incompressible fluid. The compressibility of the water is rather small and the Young’s modulus os of order E 3.05 108 Nm−2. The typical speed of seawater is much smaller than the speed of sound and therefore,≈ the× small compressibility of water has no influence on water motion. Accordingly, the continuity equation may be given in the following form.

∂v i =0 i =1, 2, 3 (1.1) ∂xi

3. Seawater salinity and density Seawater is not a pure water. It contains salts, dissolved organic substances, and mineral and organic suspended matter. The relatively constant composition of the main constitutes of sea salt has made it possible to introduce a single parameter deﬁning the salt concentration in seawater. This is known as salinity (S). The salinity of ocean water is close to 35 ppm. In semi-closed seas, where evaporation exceeds precipitation, the salinity may be higher than in the ocean (for example, 42 ppm in Red Sea). On the other, in cold, semi-closed seas, like the Baltic, the salinity is very low due to inﬂowing rivers (7 8 ppm). − The density of seawater is usually derived from the international equation of state of sea water. This equation is valid for salinity S from 0 to 42 ppm, temperature T from 2 to 40oC and of pressure from 0 to 1000 bars. −

ρ = ρ(S,T,p) (1.2)

For example, for S = 0,T = 5oC, p = 0 (atmospheric pressure, the density is ρ = 999.966 kg/m3, while for S = 35 ppm, T = 25oC, p = 0 (atmospheric pressure, the density is ρ = 1023.343 kg/m3. The density if seawater and its distribution in the water column determine the hydrostatic stability of water masses and inﬂuences sound propagation and turbulence. However, the inﬂuence of density on surface waves is negligible, except perhaps for the stage of wave generation under wind action (Massel, [2]).

3 Chapter 1 Introduction Dynamics of Marine Vehicle

4. Motion is irrotational. This means that the individual elementary particles of the fluid do not rotate. The fluid flow is called irrotational if ζ = ∇ v = 0. It implies that for an irrotational fluid flow ×

1 ∂vk ωi = eijk =0 (1.3) 2 ∂xj

As indicated above, in many oceanic motions the influence of the viscous terms is quite negligible. In this event, the Lagrangian theorem indicates that if, at some initial instant, the vorticity vanishes everywhere in the filed of flow, the motion is irrotational. This remain so in the absence of the viscous effects. The consequence of irrotatinality of the flow indicate that the velocity field can be represented as the gradient of a scalar function called as the velocity potential φ.

∂φ vi = i =1, 2, 3 (1.4) ∂xi

In virtue of the continuity equation (??), the velocity potential function is an harmonic function and obeys the Laplace equation.

∂2φ ∂2φ ∂2φ 2φ = + + =0 (1.5) ∇ ∂x2 ∂y2 ∂z2

1.2 Boundary value problem of ocean gravity waves

The water way in a sea region is bounded by:

◮ free surface that is the interface of the air and water. This surface may be deﬁned in form

z = η(x,y,t) (1.6)

that is the elevation of the free surface with respect to the reference frame that is located at the the mean sea surface.

◮ sea bottom, It may be deﬁned as

z = h(x,y,t) (1.7) −

◮ other boundaries, There may be possible some other boundaries in the region that is studied, such as a ship or an oﬀshore structure. We deﬁned them as

B(x,y,z,t)=0 (1.8)

◮ Far filed boundary, we should also consider a far field boundary if the water way is not restricted. It should be in a place that there is no disturbance and is not affected by the presence of wind generated waves.

4 Chapter 1 Introduction Dynamics of Marine Vehicle

Based on the assumption that is described in the last section and the above boundaries of the ﬂuid ﬂow, we may formulate the gravity ocean waves as follows:

2φ =0 on z < η(x,y,t)  ∇   ∂φ P 1 2  ∂t + ρ + 2 φ + gz = c(t) on z < η(x,y,t)  |∇ |      Boundary conditions:    ∂η ∂φ ∂η ∂φ ∂η ∂φ  + + =  ∂t ∂x ∂x ∂y ∂y ∂z     Free Surface 2 2 2 on z = η(x,y,t)   ∂φ 1 ∂φ ∂φ ∂φ   ∂t + gz + 2 ∂x + ∂y + ∂z =0  " #  (1.9)       ∂φ ∂h ∂φ ∂h ∂φ ∂h Fluid Bottom boundary ∂z + ∂t + ∂x ∂x + ∂y ∂y =0 on z = h(x,y,t)  −  n  ∂B  ∂t + ( φ ) B =0 on B(x,y,z,t)=0  ∇ · ∇     Other Boundaries, F = S pn ds on B(x,y,z,t)=0   B      R  M = p(r rG) n ds   SB   − ×      R   ∂φ  Far ﬁeld boundary ∂t =0, φ =0  ∇     Example - 1 Fluid in a U-tube has been forced to oscillate sinusoidally due to an oscillating pressure on one leg of the tube, 1.2. Develop the kinematic boundary condition for the free surface in leg A.

Figure 1.2: Oscillating ﬂow in a U-tube, Dean and Dalrymple [1]

5 Chapter 1 Introduction Dynamics of Marine Vehicle

Solution

The equation of the free surface may be written as follows. F (z,t)= z η(t) = 0 and η(t)= a cos(ωt) − Where a is the amplitude of the free surface and ω is the frequency. The kinematic Boundary condition is: DF (z,t) ∂F (z,t) =0 + V F (z,t)=0 Dt → ∂t · ∇ ∂F (z,t) = aω sin (ωt) ∂t − V = uˆi + vˆj + wkˆ u =0 v =0 V = wkˆ V F (z,t)= w · ∇ w = aω sin (ωt) − Example - 2 A two dimensional bottom of a water may be expressed as follows. z = h(x) − It is illustrated in Fig. 1.3. If the water is considered to be inviscid, show that the kinematic bottom boundary condition states that the ﬂow at the bottom is tangent to the bottom.

Figure 1.3: Illustration of bottom boundary condition for the two-dimensional case, Dean and Dalrymple [1]

Solution

The kinematic boundary condition at the bottom of the water is the no-flux boundary condition. It may be written as: DF (x, z) ∂F (x, z) =0 = + V F (x, z)=0 Dt ⇒ ∂t · ∇ Where: F (z,t)= z + h(x)=0 ∂h ∂z V = uî + wk,ˆ and F (x, z)= î + kˆ ∇ ∂x ∂z ∂h (uî + wkˆ) î + kˆ =0 · ∂x ∂h u ∂h u + w =0 = = ∂x ⇒ w −∂x

6 Bibliography

[1] Robert G. Dean, Robert A. Dalrymple, Water wave mechanics for engineers and scientists, World Scientific publishing Co. Pte. Ltd., 2000 [2] Massel, Stanislaw R., Ocean surface waves: Their physics and prediction, World Scientific publishing Co. Pte. Ltd., 1996 [3] Miles, J. W., On the generation of surface waves by shear flows, Jour. Fluid Mech., 3: 185 - 204 [4] Phillips, O. M., On the generation of waves by turbulent wind, Jour. Fluid Mech., 2: 417 - 445 Springer- Verlag, Berlin, pp.445-814, 1960, [5] Tichet, A. H., Phillips, O. M., Hydrodynamics, Open course, MIT, 2005 [6] Ursell, F., Wave generation by wind, In: Batchelor, G. (Editor) Survey in Mechanics, Cambridge Uni- versity Press, 216 - 249

7 Chapter 2

Long-Crested, Linear Wave Theory (LWT)

We formulated the gravity wave in general form in the previous lecture. The gravity wave is a nonlinear boundary value problem (BVM). In addition, the free surface boundary conditions should be applied to a surface z = η(x,y,t) that is initially unknown. We should make some more assumptions to make the problem amenable to an analytical solution. The simplest and most fundamental approach is to seek a linear solution of the problem by taking the wave amplitude A to be very small in compare with the wave length λ. It is also assumed that the waves are two dimensional in the xz plane and the bottom of the water way is a constant ﬂat horizontal surface. The wave theory which results from this additional assumptions is referred to alternatively as small amplitude wave theory, linear wave theory, sinusoidal wave theory, or as Airy theory. We will explain the linear wave theory (LWT), its properties and the associated problem with it in this lecture.

2.1 Boundary value problem for LWT

The most commonly description for wind-generated surface gravity waves is the linear wave theory (LWT). In addition to the basic assumption for gravity waves have been described already, the following assumptions are also taken into account: 1. The waves are two dimensional in xz-plane, (long-crested waves in the y-direction); This assumption reduce a three dimensional problem into a two dimensional problem and we can omit all of the y dependent terms.

A 2. The slopes of the waves are very small, λ << 1; This assumption simpliﬁed the free surface boundary conditions. The nonlinear terms in free surface boundary conditions are negligible in comparison with the remaining linear terms. If we also consider the Taylor series expansion, it can be written that

∂ 1 ∂2 = + η + η2 + (2.1) z=η(x,t) z=0 ∂z z=0 2 ∂z2 z=0 ··· The ﬁrst term on the left hand side of (2.1) is linear and the rest are nonlinear. We keep the linear terms and discard the nonlinear terms. Therefore, the free surface boundary conditions are reduce to:

∂η ∂φ ∂t = ∂z on z =0 (2.2)  ∂φ  ∂t + gη =0  1 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

If we combine theses two conditions, the free surface boundary condition can be given in the following form.

∂η = ∂φ ∂t ∂z ∂2φ ∂φ = 2 + g =0 on z = 0 (2.3)  ∂φ ⇒ ∂t ∂z η = 1  − g ∂t  A 1 As a general rule of thumb for wave amplitude to wavelength ratios of λ < 14 , we can linearize the free surface boundary conditions. 3. The ﬂuid bottom boundary is ﬂat, z = h. − This assumption is also make the bottom boundary condition simple. The boundary value problem (BVP) for the LWT is:

Governing equations  •  2 2  ∂ φ + ∂ φ =0  ∂x2 ∂z2  on z < η(x, t)   ∂φ  p = ρ ∂t ρgz  − −   (2.4)   Boundary conditions: •  2  - Free Surface: ∂ φ + g ∂φ =0 on z =0  ∂t2 ∂z    ∂φ  - Fluid Bottom boundary: ∂z =0 on z = h  −    This boundary value problem is illustrated in Fig. 2.1. The wave terminology is also deﬁned and illustrated

λ crest 2 ∂ φ +g ∂φ =0 A η ∂t2 ∂z SWL

trough

h ∇2φ=0 ∂φ p=−ρ ∂t −ρgz

∂φ z=−h ∂z =0

Figure 2.1: Definition sketch for linear wave theory boundary value problem and the defined terminology in Fig. 2.1. The following notations are used to defined the wave terminology. λ Wave length; • ≡ A Wave amplitude; • ≡ h Water depth; • ≡ 2 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

ω Wave frequency; • ≡ T Wave period; • ≡ k Wave number; • ≡ v or C Phase velocity, or celerity • p ≡ η(x, t) Vertical displacement of water surface at point x and time t; • ≡ 2.2 Analytical solution of the LWT BVP

The Laplace equation is subjected to the free surface and bottom boundary conditions, (2.4). The method of separation of variables may be applied to find the solution. We may assume that: φ(x,z,t)= X(x)Z(z)T (t) (2.5) If we substitute (2.5) in the Laplace equation in (2.4), we obtain X′′ Z′′ = = k2 (2.6) X − Z − where k2 is a separation constant. The resulting two ordinary differential equations are: X′′ + k2X =0 (2.7) Z′′ k2Z =0 (2.8) − The solutions of (2.7) and (2.8) are: X = B cos kx + Dsinkx (2.9) Z = Eekz + Ge−kz (2.10) Thus, the solution can be written in the following form. φ(x,z,t) = (B cos kx + Dsinkx) Eekz + Ge−kz T (t) (2.11) The function T (t) should be a harmonic function from physical point of view. It may be given in the form of cos ωt or sin ωt, where ω is defined as the circular frequency and is given by ω =2π/T =2πf. According to Rahman [3], we may distinguish four independent solution for the Laplace equation. They are:

φ1 = A1Z(z)cos kx cos ωt (2.12)

φ2 = A2Z(z) sin kx sin ωt (2.13)

φ3 = A3Z(z) sin kx cos ωt (2.14)

φ4 = A4Z(z)cos kx sin ωt (2.15) Decomposing the solution in this manner helps in the evaluation of the unknown constants. If we consider the ﬁrst solution (??), the application of the bottom boundary condition gives:

∂φ1 ∂z =0 on z = h − = Eekh = Ge−kh = E = Ge2kh (2.16)  kz −kz ⇒ ⇒  φ1 = A1 Ee + Ge cos kx cos ωt Then ek(z+h) + Ge−k(z+h) φ = 2A Gekh cos kx cos ωt 1 1 2 kh = 2A1Ge cosh k(z + h)cos kx cos ωt. (2.17)

3 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The application of the free surface boundary conditions yields:

1 ∂φ1 η1 = g ∂t on z =0 − 2ω kh = η1 = A1Ge cosh kh cos kx sin ωt (2.18)  kh ⇒ g  φ1 =2A1Ge cosh k(z + h)cos kx cos ωt.

The maximum value of η1 is the wave amplitude A. It will occur when cos kx sin ωt = 1. Therefore, Ag A Gekh = (2.19) 1 2ω cosh kh and subsequently

η1 = A cos kx sin ωt. (2.20)

2π This present a system of standing waves of wavelength k and amplitude of A. The velocity potential φ1 is in the form Ag cosh k(z + h) φ = cos kx cos ωt. (2.21) 1 ω cosh kh

The velocity potential φ1 is a periodic function in x with a wavelength of λ. The wavelength is obtained by 2π λ = k where k is known as the wave number. We may follow the same procedure to ﬁnd the constant of other elementary solution for the velocity potentials given in (2.13), (??) and (??). The ﬁnal solutions are: Ag cosh k(z + h) φ = cos kx cos ωt (2.22) 1 ω cosh kh Ag cosh k(z + h) φ = sin kx sin ωt (2.23) 2 − ω cosh kh Ag cosh k(z + h) φ = sin kx cos ωt (2.24) 3 ω cosh kh Ag cosh k(z + h) φ = cos kx sin ωt (2.25) 4 − ω cosh kh Since the Laplace equation is a linear equation, any linear combination of these elementary solution will also be a solution to the problem. Thus Ag cosh k(z + h) φ = φ + φ = sin (kx ωt) (2.26) 3 4 ω cosh kh − This velocity potential is due to a progressive wave traveling in the positive x direction. The free surface elevation can be obtained from the free surface boundary condition. −

η = 1 ∂φ on z =0 − g ∂t = η = A cos(kx ωt) (2.27)  Ag cosh k(z+h) ⇒ −  φ = sin (kx ωt) ω cosh kh − This solution is periodic in x and t and is called the progressive wave. If an observer moves along with the wave such that his position relative to the wave front remains ﬁxed, then kx ωt = constant. The speed of movement of the observer is: − dx ω λ kx ωt = constant = = = = v (2.28) − ⇒ dt k T p which is known as the wave phase velocity. It may also be denoted by C and called as wave celerity. The progressive wave traveling in the negative x direction can be obtained in the similar manner as we get (??). It can be written that: − Ag cosh k(z + h) φ = φ φ = sin (kx + ωt) (2.29) 3 − 4 ω cosh kh

4 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The associated water elevation is: η = A cos(kx + ωt) (2.30) − Similarly, we can also combined φ1 and φ2 to obtain diﬀerent forms of the progressive wave. Ag cosh k(z + h) φ = φ φ = cos(kx ωt) (2.31) 1 − 2 ω cosh kh − η = A sin (kx ωt) (2.32) − − and Ag cosh k(z + h) φ = φ + φ = cos(kx + ωt) (2.33) 1 2 ω cosh kh η = A sin (kx + ωt). (2.34) Velocity potentials (2.31) and (2.32) are identical to (2.26) and (2.27) except for a phase shift.

2.3 Dispersion relation

Substituting (2.26) in the combined form of the free surface boundary equation, we get the dispersion relation.

∂2φ ∂φ ∂t2 + g ∂z =0 on z =0 2  = ω = gk tanh kh (2.35) Ag cosh k(z+h) ⇒  φ = sin (kx ωt) ω cosh kh − This equation represents the relationship between the wave frequency ω and the wave number k. The same result can be obtained for a progressive wave traveling in the negative x direction. The dispersion relation describes the interaction between the inertia force and gravitational forces.− If we take into account that ω = kvp, we can also obtain a relationship between the phase velocity and the wave number and the depth of the water. g v2 = tanh kh (2.36) p k This relation shows the rate of the propagation of gravity waves as a function of water depth h and wavelength λ. It shows that longer waves are propagating with a higher velocity. The waves of the same length are propagating faster in water with a higher depth. The wavelength can be also obtained from the dispersion relation. gT 2 2πh λ = tanh (2.37) 2π λ The important feature of (2.35) and (2.36) is that the frequency and the phase velocity are functions of wave number and so of wavelength. The gravity waves of diﬀerent wavelengths travel at diﬀerent wave velocities. Such waves are called dispersive, since the waves would disperse as time goes on with various groups of waves such that each group would consists of waves having approximately the same wavelength.

Example - 1 A wave with a period T = 10 s is propagated shore-ward over a uniformly sloping shelf from a depth d = 200 m to a depth d = 3 m. Find the wave phase speed (celerities) vp and lengths λ corresponding to depths d = 200 m and d =3 m.

5 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Solution

It is assumed that the water is deep enough, Therefore: gT 2 9.806 102 λ = = × = 156.067 m 0 2π 2 3.14 × The assumption should be controlled, it means: 2πh 2 π 200 tanh =1 = tanh × × =1 λ ⇒ 156.067 or it can be written that: h 1 200 1 = =1.28 λ ≥ 2 ⇒ 156.067 ≥ 2 thus, it is correct. The wave phase speed is: λ 156.067 v = = = 15.61 m/s p T 10 If it is assumed that the water is also deep in h =3 m, the value of the function 2πh 2 π 3 tanh = tanh × × =0.12 1 λ 156.067 ≤ Therefore, the water is not deep for such a wave. The length of the wave at this depth situation should be calculated. gT 2 2πh λ = tanh 2π λ The solution for λ should be obtained by numerical computation. It may be obtained by using the following MATLAB m-ﬁle. g = 9.806; T = 10 s; h = 3 m; L0 = g*T.^2/2/pi; Lambda = fzero(@(L) L - g*T^2/2/pi*tanh(2*pi*h/L), L0) The computation gives:

λ = 53.145 m and, the wave phase speed is; λ 53.145 v = = =5.31 m/s p T 10 2.4 Classiﬁcation of water waves

The variation of hyperbolic functions are shown in Fig. 2.2. The argument of tanh function in dispersion 2πh − relation are λ . We can write that: h 1 2πh if > then tanh 1 λ 2 λ → and the dispersion relation becomes ω2 = gk (2.38)

6 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

2.5

sinh(x) 2 cosh(x)

1.5 tanh x 1 cosh(t), tanh(t), sinh(t)

0.5

0 0 0.5 1 1.5 2 2.5 3 x

Figure 2.2: Variation of the hyperbolic functions

The wavelength and the wave celerity are: gλ gT 2 v2 = λ = (2.39) p 2π 2π Thus the wave phase speed and wavelength are independent of water depth. If g =9.806 m2/s, then λ =1.56T 2 (2.40) The function tanh x is equal its argument for small values of x, as shown in Fig. 2.3. Therefore, it can be written that: h 1 2πh 2πh if < then tanh λ 20 λ → λ and the dispersion relation becomes ω2 = gk2h (2.41) The phase velocity in such a case is: 2 vp = gh (2.42) that shows no relation to the wave frequency. Therefore, we may distinguish three types of waves with respect to the depth of the water. Shallow-water Intermediate-water Deep-water h/λ < 1/20 1/20 < h/λ < 1/2 h/λ > 1/2

2.5 Characteristics of linear plane progressive wave

A linear progressive wave of frequency ω and wavelength of λ is taken into account. The velocity potential and the water surface elevation may be given as follows. Ag cosh k(z + h) φ = sin (kx ωt) ω cosh kh − η = A cos(kx ωt) (2.43) −

7 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

0.4

0.3 x

0.2 tanh

0.1

0 0 0.1 0.2 0.3 0.4 x

Figure 2.3: Variation of the tanh functions in small values −

2.5.1 particle motion The velocity components of the fluid particle in the horizontal and vertical directions are: dx ∂φ Agk cosh k(z + h) cosh k(z + h) u(x,z,t)= = = cos(kx ωt)= Aω cos(kx ωt) dt ∂x ω cosh kh − sinh kh − dz ∂φ Agk sinh k(z + h) sinh k(z + h) w(x,z,t)= = = sin (kx ωt)= Aω sin (kx ωt) (2.44) dt ∂z ω cosh kh − sinh kh − These equations express the velocity components within the wave at any depth z. At a given depth the velocities are seen to be harmonic in x and t. At a given phase angle θ = kx ωt, the hyperbolic function of z causes an exponential decay of the velocity components with distance below− the free surface. The fluid particle velocities at the free surface where z = 0 are: Aω u = u(x, 0,t)= cos(kx ωt) 0 tanh kh − w = w(x, 0,t)= Aω sin (kx ωt) (2.45) 0 − The relative velocities at various depth of the with respect to the fluid particle velocities at the free surface are at different water depth cases as follows.

cosh k(z+h) sinh k(z+h) cosh kh sinh kh

u  kz w  kz =  e =  e (2.46) u0  w0    1 1+ z   h   The vertical particle displacement cannot exceed the wave amplitude A. Therefore, it is assumed that the displacement of any ﬂuid particle from its mean position is small. We can compute the horizontal and

8 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle vertical displacement of the ﬂuid particle from its mean position. It is assumed that x′ = x x¯ = horizontal displacement from mean position p − Agk cosh k(z + h) = udt = cos(kx ωt)dt ω cosh kh − Z Z Agk cosh k(z + h) = sin (kx ωt) (2.47) − ω2 cosh kh − z′ = z z¯ = vertical displacement from mean position p − Agk sinh k(z + h) = wdt = sin (kx ωt)dt ω cosh kh − Z Z Agk sinh k(z + h) = cos(kx ωt) (2.48) ω2 cosh kh − If we assume thatx ¯ =z ¯ = 0, then it can be written that: x2 z2 p + p =1 (2.49) a2 b2 where: cosh k(¯z + h) sinh k(¯z + h) a = A , b = A (2.50) sinh kh sinh kh Equation (2.49) represents an ellipse with a major semi-axis (horizontal) of a and a minor semi-axis (vertical) of b. The particle paths are, therefore, generally elliptic in shape. The speciﬁc form of the particle paths for shallow water and deep water can easily be determine by examining the values of a and b, Rahman [3]. For shallow water waves, it may be readily seen that A Ak(¯z + h) a = , b = . (2.51) kh kh For deep water waves, it can be written that: a = b = Aekz (2.52) The major and minor axes for this case are equal to that each particle describe a circular path. The radii of these circles are given by the formula Aekz, and, therefore, diminish rapidly downwards. Again at the surface, the vertical displacement is equal to the wave amplitude A.

2.5.2 Pressure distribution The pressure ﬁeld within a progressive wave can be obtained from the linearized Bernoulli’s equation. ∂φ p = ρ ρgz (2.53) − ∂t − Using the velocity potential (??), the pressure distribution within the wave is: cosh k(z + h) p = ρgA cos(kx ωt) ρgz cosh kh − − cosh k(z + h) = ρg η z (2.54) cosh kh − In shallow water, the pressure distribution is: p = ρgA cos(kx ωt) ρgz = ρg(η z). (2.55) − − − In deep water, the pressure distribution is as follows. p = ρgAekz cos(kx ωt) ρgz = ρg(ηekz z) (2.56) − − −

9 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Example - 2 The regular wave is progressing in a river or ocean current with a uniform velocity U0. a - Write down the boundary value problem if the fluid in inviscid and the flow is incompressible and irrotational; b - Linearized the Boundary value problem if H 1 ; λ ≤ 7 c - Find the velocity potential of the fluid flow; d - Find the elevation of the free surface, dispersion equation, wave celerity and wave length.

Solution - a

Based on the assumptions that are described, we may formulate the problem as follows for two dimensional case (long crested wave). 2φ =0 on z < η(x, t) ∇ ∂φ + P + 1 φ 2 + gz = c(t) on z < η(x, t)  ∂t ρ 2 |∇ |   Boundary conditions:   ∂η + ∂φ ∂η = ∂φ  ∂t ∂x ∂x ∂z  Free Surface ∂φ 1 ∂φ 2 ∂φ 2 on z = η(x, t)  −→ ( ∂t + gz + 2 ∂x + ∂z = c(t)  h i ∂φ ∂h ∂φ ∂h  Fluid Bottom boundary + + =0 on z = h(x, t)  −→ ∂z ∂t ∂x ∂x −   ∂φ  Far ﬁeld boundary =0, φ = U0ˆi  −→ ∂t ∇   Solution - b

If the amplitude is small in compare with the length of wave, the linearization is justiﬁed and the superposition principles can be applied.

φtotal(x,z,t)= U0x + φwave(x,z,t)

For the sake of brevity, it is considered that φtotal(x,z,t)= φt(x,z,t), φwave(x,z,t)= φ(x,z,t). 2φ =0 on z < η(x, t) Where: φ (x,z,t)= U x + φ(x,z,t) ∇ t t 0 The Bernoulli equation is written: ∂φ P 1 ∂φ P 1 ∂φ 2 ∂φ 2 t + + φ 2 + gz = + + U + + + gz ∂t ρ 2|∇ t| ∂t ρ 2 0 ∂x ∂z " # ∂φ P 1 ∂φ ∂φ 2 ∂φ 2 = + + U 2 +2U + + + gz ∂t ρ 2 0 0 ∂x ∂x ∂z " # ∂φ P 1 ∂φ + + U 2 + U + gz = c(t) on z < η(x, t) ≈ ∂t ρ 2 0 0 ∂x ∂φ ∂φ 1 P = ρ + U + U 2 + gz + ρc(t) on z < η(x, t) − ∂t 0 ∂x 2 0

10 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Boundary conditions: Free Surface ∂η ∂φ ∂η ∂φ + t = t ∂t ∂x ∂x ∂z ∂η ∂φ ∂η ∂φ + U + = ∂t 0 ∂x ∂x ∂z Considering that ∂φ ∂η 0, the kinematic free surface boundary condition (KFSBC) is simplified. ∂x ∂x ≈ ∂η ∂η ∂φ + U = on z =0 ∂t 0 ∂x ∂z The dynamic free surface boundary condition (DFSBC) can also be simplified as follows. ∂φ 1 ∂φ + gη + U 2 +2U = c(t) ∂t 2 0 0 ∂x 1 ∂φ 1 ∂φ η = + U 2 + U + c(t) −g ∂t 2 0 0 ∂x 1 2 When there is no wave on the surface of the water η = 0, therefore, c(t)= 2 U0 then 1 ∂φ ∂φ η = + U on z =0 −g ∂t 0 ∂x Water Bottom boundary: It is assumed that the bottom is flat and therefore: ∂φ =0 on z = h ∂z − The velocity potential is written by the summation of the uniform flow and the velocity potential due to the wave on the free surface of the water.

φt = U0x + φ where the velocity potential φ is obtained by the solution of the following boundary value problem.

2φ =0 Governing equations: ∇ on z < η(x, t) P = ρ ∂φ + U ∂φ + gz  ( − ∂t 0 ∂x  ∂η ∂η ∂φ  KFSBC: ∂t + U0 ∂x = ∂z  Free Surface 1 ∂φ ∂φ on z =0  DFSBC: η = + U0  Boundary conditions:  ( − g ∂t ∂x  ∂φ  Fluid Bottom Boundary ∂z =0 on z = h   ∂φ ˆ −  Far Field Boundary =0, φ = U0i  ∂t ∇     Solution - c

The total velocity potential is:

φt = U0x + φ

11 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The velocity potential due to the wave is obtained by the solution of the Laplace equation. 2φ =0 ∇ The solution for the Laplace equation may be given for a progressive wave after applying the bottom boundary condition as follows. φ = A cosh k(z + h) sin (kx ωt) 1 − Considering DFSBC, it may be written that: 1 η = ( A ω + U A k)cosh kh cos(kx ωt) −g − 1 0 1 − The maximum value of η is the wave amplitude A. It will occur when cos (kx ωt) = 1. Therefore, − 1 A = ( A ω + U A k)cosh kh −g − 1 0 1 Ag Ag A = = 1 ω U k k(v U )cosh kh − 0 p − 0 Ag cosh k(z + h) φ = sin (kx ωt) k(v U ) cosh kh − p − 0 ω Where vp = k is the phase velocity of the wave. Therefore, Ag cosh k(z + h) φ = U x + sin (kx ωt) t 0 k(v U ) cosh kh − p − 0 Solution - d

The surface elevation: • η = A cos(kx ωt) − The dispersion equation: • ∂φ ∂φ P = ρ + U + gz − ∂t 0 ∂x Since the pressure is constant along the free surface, therefore: DP ∂P = + V P =0 Dt ∂t · ∇ ∂ ∂φ ∂φ ∂φ ∂φ ∂φ ∂φ ρ + U + gη + (U + )ˆi + kˆ ρ + U + gz =0 ∂t − ∂t 0 ∂x 0 ∂x ∂z · ∇ − ∂t 0 ∂x If the nonlinear terms are omitted, the combined free surface boundary condition is obtained. ∂2φ ∂2φ ∂2φ ∂φ +2U + U 2 + g =0 on z =0 ∂t2 0 ∂x∂t 0 ∂x2 ∂z The dispersion equation is obtained by inserting the wave velocity potential in the combined free surface boundary condition. Ag ω2 +2U kω U 2k2 + gk tanh kh sin (kx ωt)=0 k(v U ) − 0 − 0 − p − 0 2 2 2 ω 2U0kω + U0 k = gk tanh kh − 2 2 2 ω 2 ω ω 2U0 + U0 2 = gk tanh kh − vp vp U 2 ω2 1 0 = gk tanh kh Dispersion equation − v ⇐ p

12 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The phase velocity: • The dispersion equation may be written in the following form. 2 ω 2 2 (vp U0) = gk tanh kh vp − 2 2 k (vp U0) = gk tanh kh − g (v U )2 = tanh kh p − 0 k g v = U + tanh kh p 0 k r The wave length: • λ g v = = U + tanh kh p T 0 k r 2.6 Progressive oblique waves

If a progressive wave is traveling in a direction that makes an angle θ with the x axis, then the forms of velocity potential and the surface elevation must be modiﬁed. If the wave moving− in r direction as shown in the Fig. 2.4, we can generalized the associated formula as − Ag cosh k(z + h) φ = sin (kr ωt) ω cosh kh − η = A cos(kr ωt) (2.57) − where according to Fig. 2.4, it can be written that:

r vp

θ x

Figure 2.4: The oblique direction of a progressive wave

r = x cos θ + y sin θ (2.58) Thus, the modify form of linear wave equations are as follows. Ag cosh k(z + h) φ = sin (kx cos θ + ky sin θ ωt) ω cosh kh − η = A cos(kx cos θ + ky sin θ ωt) (2.59) − The velocity potential and the surface elevation for various water depth are follows.

Shallow water depth Intermediate water depth Deep water

gA gA cosh k(z+h) gA kz φ= ω sin(kx cos θ+ky sin θ−ωt) φ= ω cosh kh sin(kx cos θ+ky sin θ−ωt) φ= ω e sin(kx cos θ+ky sin θ−ωt)

13 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Example - 3 Consider a wave with a period T =8 s, in a water depth h = 15 m with and a height of H =5.5 m. Find: a - the local horizontal and vertical velocities u and w; and

b - the accelerations ax and az; at an elevation z = 5 m when Θ= (kx ωt)= π . − − 3 Solution - a cosh k(z + h) u = Aω cos(kx ωt) sinh kh − sinh k(z + h) w = Aω sin(kx ωt) sinh kh − It is necessary to ﬁnd, A, ω, k. H 5.5 A = = =2.75 m 2 2 2π 2 3.14 ω = = × =0.7854 1/s T 8 gT 2 2πh 9.806 82 2 π 15 λ = tanh = × tanh × × = λ = 81.767 m 2π λ 2 3.14 × λ ⇒ × 2π 2 3.14 k = = × =0.0768 1/m λ 81.767 cosh k(z + h) cosh [0.0829 ( 5 + 15)] π u = Aω cosΘ = 2.75 0.7854 × − cos =0.993 m/s sinh kh × × sinh (0.0829 15) × 3 × sinh k(z + h) sinh [0.0829 ( 5 + 15)] π w = Aω cosΘ = 2.75 0.7854 × − sin =1.111 m/s sinh kh × × sinh (0.0829 15) × 3 × Solution - b ∂u cosh k(z + h) a = = Aω2 sin(kx ωt) x ∂t sinh kh − ∂w sinh k(z + h) a = = Aω2 cos(kx ωt) z ∂t − sinh kh − cosh [0.0829 ( 5 + 15)] π a =2.75 0.78542 × − sin =1.351 m/s2 x × × sinh (0.0829 15) × 3 × sinh [0.0829 ( 5 + 15)] π w = 2.75 0.78542 × − cos =0.504 m/s2 − × × sinh (0.0829 15) × 3 × Example - 4 Consider a wave in a depth h = 12 m with a height of H =3 m and a period of T = 10 s. The corresponding deep-water wave height is H0 =3.13 m. Find: a - The maximum horizontal and vertical displacement of a water particle from its mean position when z = 0 and z = h. − b - The maximum water particle displacement at an elevation z = 7.5 m when the wave is in inﬁnitely deep water. − c - For the deepwater conditions of above, show that the particle displacements are small relative to the wave height when z = λ0 . − 2

14 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Solution a

A ﬂuid particle is moving in an elliptical path due to the wave motion.

x2 z2 p + p =1 a2 b2 where a is the major semi-axis (horizontal) and b is the minor semi-axis (vertical) of the ellipse.

cosh k(¯z + h) sinh k(¯z + h) a = A , b = A sinh kh sinh kh The parameters a and b are also the maximum horizontal and vertical displacement of a water particle from its mean position. It should be emphasis that A is the amplitude of wave at z = 0. The amplitude of a wave sinh k(¯z+h) at an elevation z in a given depth of h is b = A sinh kh . It is necessary to ﬁnd, k.

gT 2 2πh 9.806 102 2 π 12 λ = tanh = × tanh × × = λ = 99.703 m 2π λ 2 3.14 × λ ⇒ × 2π 2 3.14 k = = × =0.063 1/m λ 99.703 The maximum horizontal and vertical displacement of a water particle from its mean position at z = 0.

3 cosh(0.063 12) a = × =2.349 m 2 sinh (0.063 12) × 3 b = =1.5 m 2 The maximum horizontal and vertical displacement of a water particle from its mean position at z = d. − 3 1 a = =1.807 m 2 × sinh (0.063 12) × 3 b = =0 2 Solution b

A ﬂuid particle is moving in a circular path due to the wave motion in deep water.

2 2 2 xp + zp = a where a = Aek(¯z is the radius of the circle. The parameter a is also the amplitude of the wave at an elevation z for the free surface of the water. gT 2 9.806 102 λ = = × = 156.073 m 2π 2 3.14 × 2π 2 3.14 k = = × =0.0403 1/m λ 156.073 3.13 a = e[0.0403×(−7.5)] =1.157 m 2 ×

15 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Solution c λ 156.073 z = 0 = = 78.036 m − 2 − 2 − 3.13 a = e[0.0403×(−78.036)] =0.067 m 2 × a 0.067 = =0.0213 0 H 3.13 ≈

Characteristics of a linear plane progressive wave

Shallow water depth Intermediate water depth Deep water

λ λ λ λ h< 20 20 2

φ = gA sin(kx ωt) φ = gA cosh k(z+h) sin(kx ωt) φ = gA ekz sin(kx ωt) ω − ω cosh kh − ω −

η = A cos(kx ωt) η = A cos(kx ωt) η = A cos(kx ωt) − − −

ω2 = k2gh ω2 = gk tanh kh ω2 = gk

u = aω cos(kx ωt) u = Aω cosh k(z+h) cos(kx ωt) u = Aωekz cos(kx ωt) kh − sinh kh − − w = Aω 1+ z sin (kx ωt) w = Aω sinh k(z+h) sin(kx ωt) w = Aωekz sin(kx ωt) kh h − sinh kh − − u =1 u = cosh k(z+h) u = ekz u0 u0 sinh kh uo

w =1+ z w = sinh k(z+h) w = ekz w0 h w0 sinh kh wo

p = ρgA cos(kx ωt) p = ρgA cosh k(z+h) cos(kx ωt) p = ρgAekz cos(kx ωt) d − d cosh kh − d −

cosh k(z+h) kz pd = ρgη Pd = ρg cosh kh η pd = ρge η

p = ρg(η z) p = ρg cosh k(z+h) η z p = ρg ekzη z T − T cosh kh − T − h i x x¯ z z¯ p− p− a2 + b2 =1

A z¯ cosh k(¯z+h) sinh k(¯z+h) kz¯ a = kh and b = A(1 + h ) a = A sinh kh , b = A sinh kh a = b = Ae

16 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Example - 5 Two pressure sensors are mounted according to Fig. 2.5. The amplitudes of dynamic pressure are 20.4 kN/m2 and 25.6 kN/m2 as recorded on sensors 1 and 2, respectively. What are the wave length, water depth and wave amplitude?

Figure 2.5: The position of the sensors in example 6

Solution

cosh k(z + h) p = ρgA cos(kx ωt) d cosh kh − cosh k( h + h) 1 P¯ = ρgA − = ρgA = The dynamic pressure amplitude at the sensor 1 d1 cosh kh cosh kh ⇒ cosh k( h +7.62+ h) cosh7.62k P¯ = ρgA − = ρgA = The dynamic pressure amplitude at the sensor 2 d2 cosh kh cosh kh ⇒ ¯ Pd1 1 2.04 1 ¯ = = = = cosh7.62k =1.255 = k =0.092 Pd2 cosh7.62k ⇒ 2.56 cosh7.62k ⇒ ⇒ 2π =0.092 = λ = 68.42 m λ ⇒

Using the dispersion relationship to ﬁnd the water depth. ω2 = gk tanh kh 2π 2 4π2 4 π2 = gk tanh kh = tanh kh = = × = kh =0.836 T ⇒ gkT 2 9.806 0.092 82 ⇒ × × 0.836 h = =9.086 m 0.092

The amplitude of wave is obtained by using the pressure amplitude at a sensor, say the sensor 1. 1 P¯ cosh kh 2.04 104 cosh0.836 P¯ = ρgA = A = d1 = × × =2.874 m d1 cosh kh ⇒ ρg 992 9.806 × 2.7 Superposition of waves

The boundary value problem associated with the small amplitude plane waves is linear. Therefore, the inﬂuence of a combination of several waves can be obtained by superposing the eﬀects of individual wave components. The velocity potential of a wave system consist of n regular wave are: φ = φ + φ + + φ (2.60) t 1 2 ··· n

17 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle where

Ang cosh kn(z + h) φn = sin (knx ωnt + δn). (2.61) ωn cosh knh ± The minus and plus sign is related to the direction of the propagation of the nth wave. The phase diﬀerence between various waves is denoted by δn and is measured from the origin (kx ωt). The free surface elevation can be obtained from ± 1 ∂φ 1 ∂ ηt = = (φ1 + φ2 + + φn) = η1 + η2 + + ηn. (2.62) −g ∂t z=0 −g ∂t ··· ··· z=0

Therefore, it can be written that:

n η = A cos(k x ω t + δ ) (2.63) t ± i i ± i i i=1 X The other characteristics of wave can also obtained by the superimpose of the characteristics of individual wave components. n cosh k (z + h) u = A ω i cos(k x ω t) (2.64) t i i sinh k h i − i i=1 i X n sinh k (z + h) w = A ω i sin(k x ω t) (2.65) t i i sinh k h i − i i=1 i Xn cosh k (z + h) p = ρg i η z (2.66) t cosh k h i − i=1 i X We may consider such a case that all wave components have the same period and moving in a water of depth h in the same direction. Hence, the circular frequency and the wave numbers are identical for all waves components. Under these special condition, the free surface elevation may be expressed as η = r cos(kx ωt + λ) (2.67) t − where r is such that: n r cos λ = An cos δn i=1 Xn r sin λ = An sin δn. i=1 X Hence 1/2 n 2 n 2 r = A cos δ + A sin δ  n n n n  i=1 ! i=1 ! X X   n A sin δ λ = tan−1 i=1 n n . (2.68) n A cos δ Pi=1 n n For the special caseP in which there are only two wave components that have the same period, we can write that: η = A cos(kx ωt + δ )+ A cos(kx ωt + δ ) t 1 − 1 2 − 2 = cos(kx ωt)(A cos δ + A cos δ ) + sin(kx ωt)(A sin δ + A sin δ ) − 1 1 2 2 − 1 1 2 2 2 2 = cos(kx ωt) A cos δ + sin(kx ωt) A sin δ − i i − i i i=1 i=1 X X = r cos(kx ωt + λ) − 18 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle where

r = A2 + A2 +2A A cos(δ δ ) 1 2 1 2 1 − 2 q A cos δ + A cos δ λ = tan−1 1 1 2 2 . A sin δ + A sin δ 1 1 2 2 In the case that the two waves are in phase, δ1 = δ2, then η = (A + A )r cos(kx ωt + δ ). t 1 2 − 1 If the two components are out of phase, i.e. δ1 = δ2 + π, then η = (A A )r cos(kx ωt + δ ). t 1 − 2 − 1 For δ δ = π/2, then 1 − 2 η = A cos(kx ωt + δ )+ A sin(kx ωt + δ ). t 1 − 1 2 − 1 2.8 Wave reﬂection and standing wave

If there is a barrier at x = b an the way of waves, the waves will be reflected. The reflection coefficient is defined as

amplitude of the reflected wave Ar Kr = = (2.69) amplitude of the incident wave Ai where kr 1. If the value of kr = 1, then the reflection is perfect. Assume that the incident wave is propagating≤ in positive x direction and is reflected by a plane vertical barrier at point x = b. It is assume that the reflection is perfect.− Hence, The velocity potential of the system of waves is Ag cosh k(z + h) φ = [sin (kx ωt) + sin (kx + ωt + δ )] (2.70) ω cosh kh − 2 It is assumed that the barrier is impermeable, the velocity is zero at the barrier. Thus, the boundary condition is ∂φ u = t =0 at x = b (2.71) − ∂x The application of this boundary condition yields cos(kb ωt) cos(kb + ωt + δ )=0 − − 2 Expanding and equating the coefficients of sin ωt and cos ωt, we obtain

cos kb = cos(kb + δ2) = δ2 =2nπ 2kb, n =0, 1, 2, sin kb = sin (kb + δ2) ⇒ − ··· − For two progressive waves moving in opposite directions with the same amplitude, the surface elevation is η = A cos(kx ωt)+ A cos(kx + ωt + δ ) − 2 = A cos(kx ωt)+ A cos(kx + ωt)cos δ + A sin (kx + ωt) sin δ − 2 2 = A cos(kx ωt)+ A cos(kx + ωt)cos(2nπ 2kb)+ A sin (kx + ωt) sin (2nπ 2kb) − − − = A cos(kx ωt)+ A cos(kx + ωt)cos2kb A sin (kx + ωt) sin 2kb − − = A cos kx cos ωt + A sin kx sin ωt + A cos kx cos ωt cos2kb A sin kx sin ωt cos2kb − A sin kx cos ωt sin 2kb A sin kx cos ωt sin 2kb. − − The ﬁnal solution can be written in the form, η =2A cos(kb ωt)cos k(x b) (2.72) − − 19 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The equation (2.69) is the product of two terms, one independent of x and the other independent of t. Thus, there are certain times when η = 0 for all x and there are also certain x of which η = 0 for all times. These later points are called the nodes of the system and are located by the condition 2n +1 cos k(x b)=0 = x = b + π, n =0, 1, 2, (2.73) − ⇒ 2 ··· The condition of stationary nodes deﬁnes standing waves. The slope of the free surface of the incident and reﬂected waves are always equal and opposite at x = b. ∂η t =0 , at x = b for all t (2.74) ∂x For the case that the barrier is at the origin, b = 0, then the standing wave is in the form η =2A cos ωt cos kx (2.75) where is plotted in Fig. 2.6

node node

t =0,T, 2T, ···

0 T 3T

η t = , , 4 4 ···

t = T , 3T , 2 2 ···

0 π π 3π 2π 3π 5π 7π 4π 2k k 2k k k 2k 2k k x

Figure 2.6: Standing waves due to the reﬂection of a progressive wave

The velocity potential for standing waves may be given by inserting the value of δ =2nπ 2kb in (2.70). 2 − 2Ag cosh k(z + h) φ = sin (kb ωt)cos(kx kb) (2.76) ω cosh kh − − The ﬂuid velocity components are due to the presentation of standing waves as follows. ∂φ 2Akg cosh k(z + h) u = = sin (kb ωt) sin (kx kb) (2.77) ∂x − ω cosh kh − − ∂φ 2Akg sinh k(z + h) w = = sin (kb ωt)cos(kx kb) (2.78) ∂z ω cosh kh − − we have seen that the nodes occur where cos k(x b) = 0. Hence, the motion of the ﬂuid particles are horizontal at nodes. The motion of the particles are− vertical where the maximum and minimum variation of the surface take place, Rahman [3].

20 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

2.9 Wave group

Consider two waves of same amplitude, direction and in phase at the origin. The surface elevation can be obtained by summing the eﬀect of each wave. η = A cos(k x ω t)+ A cos(k x ω t) (2.79) t 1 − 1 2 − 2 It can be rewritten as: 1 1 η =2A cos [(k + k )x (ω + ω )t] cos [(k k )x (ω ω )t] (2.80) t 2 1 2 − 1 2 · 2 1 − 2 − 1 − 2 The point of zero amplitude of the wave envelope separate groups of individual wave as shown in the Fig. 2.7. These nodal points are located at the points so that 1 π ω ω (2n + 1)π [(k k )x (ω ω )t]=(2n + 1) = x = 1 − 2 t + . (2.81) 2 1 − 2 − 1 − 2 2 ⇒ node k k k k 1 − 2 1 − 2

vg wave envelope η

λg 2π = − 2 k2 k1

Figure 2.7: Wave pockets and group velocity

Since the position of all nodes is a function of time, they are not ﬁxed and are moving with a speed that is called as the group velocity. At time t = 0, the nodes are located at points x = (2n+1)π , n =0, 1, 2, . k1−k2 The distance between two consecutive nodes is ··· λ 2π λ λ x x = g = = 1 2 (2.82) i+1 − i 2 k k λ λ 1 − 2 2 − 1 The speed of propagation of the nodes, i.e. group velocity, can be given by dx ω ω v = node = 1 − 2 . (2.83) g dt k k 1 − 2 The ω2 and k2 can be written in the form

ω2 = ω1 + δω, k2 = k1 + δk

21 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle then the speed of propagation of the nodes is δω v = . g δk

As ω2 and k2 approach to ω1 and k1, it can be written that dω v = . (2.84) g dk

We now that ω = vpk and thus the group velocity is

dvpk dvp vg = = vp + k dk dk vp 2kh = vg = 1+ (2.85)  2 g ⇒ 2 sinh 2kh  vp = k tanh kh The asymptotic forms of hyperbolic functions are as follows. function large kh small kh

1 kh sinh kh 2 e kh

1 kh cosh kh 2 e 1

tanh kh 1 kh

Therefore, the group velocity at diﬀerent water depth condition are: v 2kh v = p 1+ = Intermediate water depth g 2 sinh 2kh ⇐ v v = p = deep water g 2 ⇐ v = v = shallow water (2.86) g p ⇐ 2.10 Wave energy

The total energy of a harmonic wave is the summation of the potential energy and kinetic energy of the wave. The potential energy of the wave may be computed by ﬁrst ﬁnding the total potential energy of the water in presence of the wave above z = h, P E , minus the potential energy of water above z = h when − 1 − there is no wave on the free surface of the water, P E2. The potential energy of a column of water of height h + η with respect to z = h with the area of dx 1 is: − × ∆(P E ) = g height of the center of mass of the water column the mass of the water column 1 × × h + η = g ρ(h + η)∆x × 2 × h i (h + η)2 = ρg ∆x 2 Thus the average potential energy per unit surface area is ρg t+T x+λ P E = (h + η)2dxdt. 1 2λT Zt Zx Taking into account that η = A cos(kx ωt), we can write that − ρg t+T x+λ 2 ρgh2 ρgA2 P E1 = h + A cos(kx ωt) dxdt = + 2λT t x − 2 4 Z Z h i 22 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

The potential energy without wave on the free surface is: ρg t+T x+λ ρgh2 P E = h2dxdt = 2 2λT 2 Zt Zx Hence, the average potential energy attributed to the presence of wave of the free surface of the water per unit area is ρgA2 P E = P E P E = (2.87) 1 − 2 4 The kinetic energy of the water due to the presence of wave is attributed to the motion of the ﬂuid particles. The components of the water particle velocity are: dx ∂φ Agk cosh k(z + h) cosh k(z + h) u(x,z,t)= = = cos(kx ωt)= Aω cos(kx ωt) dt ∂x ω cosh kh − sinh kh − dz ∂φ Agk sinh k(z + h) sinh k(z + h) w(x,z,t)= = = sin (kx ωt)= Aω sin (kx ωt). (2.88) dt ∂z ω cosh kh − sinh kh − The kinetic energy of a small element of water with the length of δx, height of δz and unit width is: 1 δ(KE) = (u2 + w2)ρδxδz. 2 The average of the kinetic energy of the wave per unit surface area is obtained as follows. ρ t+T x+λ KE = (u2 + w2)ρdxdzdt 2λT Zt Zx Using the velocity components given in (2.88), we obtain ρgA2 KE = (2.89) 4 The total average energy of wave per unit surface area is ρgA2 E = KE + KE = (2.90) 2 Example - 6 An ocean bottom-mounted pressure sensor measures a reversing pressure as a train of swells propagates past the sensor toward the shore. The pressure ﬂuctuations have a 5.5 s period and vary from a maximum of 54.3 kN/m2 to a minimum of 41.2 kN/m2. a - How deep is the pressure sensor (and bottom) below the still water level? b - Determine the wave height, celerity, group celerity and energy as it passes the sensor.

Solution a

∂φ P = ρ ρgz − ∂t − gA cosh k(z + h) φ = sin(kx ωt) ω cosh kh − ∂φ cosh k(z + h) = gA cos(kx ωt) ∂t − cosh kh − cosh k(z + h) P = ρgA cos(kx ωt) ρgz cosh kh − − ρgA P = cos(kx ωt)+ ρgh z=−h cosh kh −

cos(kx ωt)=1 = ρgA + ρgh = 54.3 103 − ⇒ cosh kh × cos(kx ωt)= 1 = ρgA + ρgh = 41.2 103 − − ⇒ − cosh kh × 23 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

Summation of these two relationships gives: 95.5 103 2ρgh = 95.5 103 = h = × =4.87 m × ⇒ 2 1000 9.806 × × Solution b

cos(kx ωt)=1 = ρgA + ρgh = 54.3 103 − ⇒ cosh kh × cos(kx ωt)= 1 = ρgA + ρgh = 41.2 103 − − ⇒ − cosh kh × Subtracting these two relationships gives: 2ρgA = 13.3 103 cosh kh × 4π2 4 π2 1 ω2 = gk = k = = × =0.133 ⇒ gT 2 9.806 5.52 m × 2ρgA 13.3 103 = 13/3 103 = H =2A = × cosh (0.133 4.87) = 1.651 m cosh kh × ⇒ 1000 9.806 × × ω2 g 9.806 ω2 = gk = = = v2 = = v =8.59 m/s ⇒ k2 k ⇒ p 0.133 ⇒ p v 8.59 v = p = =4.295 m/s g 2 2 ρgA2 1000 9.806 4.872 E¯ = = × × 10−3 = 23.877 kW 2 2 × 2.10.1 Energy propagation The trajectories of water particles in small-amplitude water waves are closed and therefore there is no transmission of mass as they propagate across a fluid. However, water waves propagate energy. If we consider waves generated by a stone impacting on an initially calm water surface. Some portion of the kinetic energy of the stone is transformed into wave energy. As these waves travel to and perhaps break on the shoreline, it is clear that there has been a propagation of energy away from the generation area. The rate at which the energy is transferred is called the energy flux. It is the rate at which work is being done by the fluid on one side of a vertical section on the fluid on the other side in linear wave theory. We may consider a fixed control volume V to the right of a vertical section S , as shown in Fig. 2.8. The force on an element of the surface with height dz and unit width is dF = pdz where p = ρ ∂φ ρgz. − ∂t −

vp S

Figure 2.8: The control volume and a ﬁxed vertical section in a wave

The instantaneous rate at which work is being done by the pressure force per unit width in the direction of

24 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle wave propagation is η J = p udz (2.91) · Z−h Using (2.54) and (2.88), we can write that η cosh k(z + h) cosh k(z + h) J = ρg η z Aω cos(kx ωt)dz. (2.92) cosh kh − · sinh kh − Z−h The average energy flux is obtained over a wave period as 1 t+T η cosh k(z + h) cosh k(z + h) J = ρg η z Aω cos(kx ωt)dz. (2.93) T cosh kh − · sinh kh − Zt Z−h The final solution for the energy flux after some manipulations according to Dean [2] is 1 ω 1 2kh J = ρgA2 1+ (2.94) 2 k 2 sinh 2kh 1 2kh = E¯ v 1+ | {z } | {zp } 2 sinh 2kh

= E¯ vg J = E¯ v | {z } (2.95) · g It shows that the wave energy is propagating at the speed of group velocity. In other word, we may interpret that the wave group velocity is the speed of advance of wave energy.

2.10.2 Equation of energy conservation If we consider a control volume of V that is limited between the control surfaces of 1 and 2, as given in Fig. 2.9. The ﬂux of energy can be written that

v ∆x p x

J1 J2

Figure 2.9: The Flux of energy

(J J )∆t = ∆E∆x 1 − 2 ∂J ∂2J ∆2x J2 = (J1 + ∆x + + ∂x ∂x2 2 ··· 1 1 keep the linear term, then

∂E ∂J + =0 ∂t ∂x According to (2.95), take into account that J = E¯ v · g 25 Chapter 2 Long-Crested, Linear Wave Theory (LWT) Dynamics of Marine Vehicle

∂E ∂ + v E =0 (2.96) ∂t ∂x g

26 Bibliography

[1] Rahman, M., Water waves, relating modern theory to advanced engineering applications, Oxford Uni- verisit press, 1994 [2] Dean, R. G. and Dalrymple, R. A., Water wave mechanics for engineers and scientists, World Scientiﬁc Publishing Co., 2000

27 Chapter 3

Finite-amplitude waves

The perturbation procedure may be applied to obtained a more close approach to a complete solution for the waves motion. The free surface conditions prevent a complete solution to the waves motion equations. They are linearized by assuming that the contribution of the higher order terms are negligible. However, in many engineering applications, the experimental evidence indicates the nonlinear effects are also important and should be taken into account in computation procedure. More importantly, some of the effect will be missed if we restrict the computation to the linear influence of the wave effects. For example, the drift force is a steady force that act on structures in waves. It can be calculated if we consider the nonlinear effect of the waves on a structure. The Finite-amplitude waves theory, the trochoidal waves and the transformation of waves are consider in this lecture. We also pointed out the nonlinear effect on the wave motions.

3.1 Stokes Finite-amplitude waves theory

The mathematical formulation describing the wave motion are in the following form.

∂2φ ∂2φ ∂x2 + ∂z2 =0  2 2 on z < η(x, t)  ∂φ p 1 ∂φ ∂φ  ∂t + ρ + 2 ∂x + ∂z + gz = c(t)      Boundary conditions:   Free surface;  ∂η ∂φ ∂η ∂φ  + = (3.1)  ∂t ∂x ∂x ∂z   2 2 on z = η(x, t) η = 1 ∂φ + 1 ∂φ + ∂φ  g ∂t 2 ∂x ∂z  − ( " #)     Fluid bottom boundary;  ∂φ  =0 on z = h  ∂z −    It is assumed that the waves are long crested and the fluid bottom boundary is a flat horizontal surface. The free surface boundary conditions may be expressed in a single form according to Sarpkaya and Isaacson [4] as follows. ∂2φ ∂φ ∂ 1 ∂φ ∂ 1 ∂φ ∂ ∂φ 2 ∂φ 2 + g + + + + =0 on z = η(x, t) (3.2) ∂t2 ∂z ∂t 2 ∂x ∂x 2 ∂z ∂z ∂x ∂z " # Stokes (1847, 1880) applied the perturbation method to develop a more generalized formulation and capture the nonlinear effects. It is assumed that the variables describing the flow are expressed as a power

28 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle series of small parameters ε that is called the perturbation parameter. φ = εφ + ε2φ + ε3φ + (3.3) 1 2 3 ··· η = εη + ε2η + ε3η + (3.4) 1 2 3 ··· Substituting (3.3) into Laplace’s equation and the sea bed boundary condition and collecting the terms of order ε,ε2,ε3, , we obtain: ··· 2 2 ∂ φn ∂ φn ∂x2 + ∂z2 =0 for n =1, 2, 3, (3.5) ··· ∂φn =0 on z = h ∂z − The diﬃculties arise in taking into account the free surface boundary condition (3.2) that contains nonlinear terms and should be applied to an unknown surface z = η(x, t). We may apply the Taylor series expansion to express the velocity potential about z = 0. ∂φ η2 ∂2φ φ[x, η(x)] = φ(x, 0) + η + + ∂z 2! ∂z2 ··· z=0 z=0

2 3 = εφ1 + ε φ2 + ε φ3 + ··· z=0

2 3 ∂ 2 3 + εη1 + ε η2 + ε η3 + εφ1 + ε φ2 + ε φ3 + ··· ∂z ··· z=0 2 2 1 2 3 ∂ 2 3 + εη1 + ε η2 + ε η3 + εφ1 + ε φ2 + ε φ3 + + 2! ··· ∂z2 ··· ··· z=0 2 2 ∂φ1 3 ∂φ2 ∂φ1 1 2 ∂ φ1 4 = εφ1(x, 0) + ε φ2 + η1 + ε φ3 + η1 + η2 + η1 + O(ε ) ∂z ∂z ∂z 2 ∂z2 z=0 z=0

(3.6)

Similarly, the derivatives of the velocity potential φ can also be expanded by Taylor series about z = 0 as follows.

∂φ[x, η(x)] ∂φ1(x, 0) 2 ∂φ2 ∂ ∂φ1 = ε + ε + η1 ∂x ∂x ∂x ∂z ∂x z=0 2 3 ∂φ3 ∂ ∂φ2 ∂ ∂φ 1 1 2 ∂ ∂φ1 4 +ε + η1 + η2 + η1 + O(ε ) ∂x ∂z ∂x ∂z ∂x 2 ∂z2 ∂x z=0

∂φ[x, η(x)] ∂φ1(x, 0) 2 ∂φ2 ∂ ∂φ1 = ε + ε + η1 ∂z ∂z ∂z ∂z ∂z z=0 2 3 ∂φ3 ∂ ∂φ2 ∂ ∂φ 1 1 2 ∂ ∂φ1 4 +ε + η1 + η2 + η1 + O(ε ) ∂z ∂z ∂z ∂z ∂z 2 ∂z2 ∂z z=0

2 ∂ φ[x, η(x)] = ε φ1(x, 0) + ε φ2 + η1 φ1 ∇ ∇ ∇ ∂z ∇ z=0 2 3 ∂ ∂ 1 2 ∂ 4 +ε φ3 + η1 φ2 + η2 φ1 + η1 φ1 + O(ε ) (3.7) ∇ ∂z ∇ ∂z ∇ 2 ∂z2 ∇ z=0

29 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

Using (3.6) and (3.7), the free surface boundary condition (3.2) may be written in the following form.

2 2 2 2 2 ∂ φ1 ∂φ1 2 ∂ φ2 ∂φ2 ∂ ∂ φ1 ∂φ1 ∂ ∂φ1 ∂φ1 ε 2 + g + ε  ∂t2 + g ∂z + η1 ∂z ∂t2 + g ∂z + ∂t ∂x + ∂z  + ∂t ∂z         2 ∂ φ3 + g ∂φ3 +  ∂t2 ∂z     2  (3.8)  ∂ ∂φ2 ∂ ∂φ2 ∂ ∂φ1 ∂φ2 ∂φ1 ∂φ2   2 η1 + gη1 +2 + +   ∂t ∂z ∂z ∂z ∂t ∂x ∂x ∂z ∂z    ε3   =0+ O(ε4) on z =0  2 2 2   ∂ η ∂φ1 + 1 η2 ∂ φ1 + g η ∂ ∂φ1 + 1 η2 ∂ ∂φ1 +   ∂t2 2 ∂z 2 1 ∂z2 2 ∂z ∂z 2 1 ∂z2 ∂z  h i  2 2   1 ∂φ1 ∂ ∂φ1 ∂ ∂φ1 ∂φ1   + +   2 ∂x ∂x ∂z ∂z ∂x ∂z     h i       

3.1.1 The first-order waves theory If we only take into account the coefficients of ε and equating them, we obtain the first order theory of wave motion.

2 2 ∂ φ1 ∂ φ1 2 + 2 =0  ∂x ∂z in z < 0  p = ρ ∂φ1 + gz  ∂t  −  Boundary conditions:     Free surface;  2 (3.9)  ∂ φ1 ∂φ1  ∂t2 + g ∂z =0  1 ∂φ1 at z =0 η1 = g ∂t  −    Fluid bottom boundary;  ∂φ1  ∂z =0 at z = h  −    The ﬁrst-order theory was developed in previous lectures. It is referred to as Airy wave theory and the solution of which is given in the previous lectures. These are: Ag cosh k(z + h) φ = εφ = sin (kx ωt) ℓ 1 ω cosh kh − η = εη = A cos(kx ωt) (3.10) ℓ 1 −

30 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

3.1.2 The second-order waves theory If the coeﬃcients of ε2 are taken into account and equating them, the Stokes second-order waves formulations are obtained.

2 2 ∂ φ2 + ∂ φ2 =0  ∂x2 ∂z2  in z < 0  2 2  p = ρ ∂φ2 + 1 ∂φ1 + ∂φ1 + gz  ∂t 2 ∂x ∂z  −    Boundary conditions:   Free surface;     2 2 2 2 (3.11)  ∂ φ2 ∂φ2 ∂ ∂ φ1 ∂φ1 ∂ ∂φ1 ∂φ1  ∂t2 + g ∂z + η1 ∂z ∂t2 + g ∂z + ∂t ∂x + ∂z =0  at z =0  2 2  η = 1 ∂ φ + η ∂φ1 + 1 ∂φ1 + ∂φ1  2 g ∂t 2 1 ∂z 2 ∂x ∂z  −    Fluid bottom boundary;     ∂φ2  =0 at z = h  ∂z −    The free surface boundary condition may be substituted for φ1 and η1 from (3.10). After doing some manipulations, we obtain ∂2φ ∂φ 3A2gkω 2 + g 2 = sin 2(kx ωt). (3.12) ∂t2 ∂z sinh 2kh − The equation (3.12) suggest that the solution for the second-order potential should be in the following form. φ = B cosh2k(z + h) sin 2(kx ωt) (3.13) 2 − Where B is an arbitrary constant. The argument of cosine hyperbolic has been chosen to be double to comply with the second-order theory. If we substitute (3.13) in (3.12), it yields 3 A2ω B = . (3.14) 8 sinh4 kh Hence, the second-order velocity potential is

2 2 3A ω cosh2k(z + h) φq = ε φ2 = sin 2(kx ωt). (3.15) 8 sinh4 kh − If the asymptotic values for the hyperbolic function in deep water and shallow water are taken into account, it can be written that: cosh2k(z + h) 8e2k(z+h) lim = lim =0 kh→∞ sinh4 kh kh→∞ e4kh cosh2k(z + h) 1 lim = kh<π/10 sinh4 kh (kh)4 therefore, the second-order velocity potential in deep and shallow water depths are as follows. φ =0 = Deep water q ⇐ 3A2ω 1 φ = sin 2(kx ωt) = Shallow water (3.16) q 8 (kh)4 − ⇐

31 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

The total velocity potentials up to the second-order approximation are:

2 Ag cosh k(z + h) 3A ω cosh2k(z + h) 3 φ = φℓ + φq = sin (kx ωt)+ sin 2(kx ωt)+ O(ε ) ω cosh kh − 8 sinh4 kh − Ag 3A2ω 1 φ = φ + φ = ekz sin (kx ωt)+ sin 2(kx ωt)+ O(ε3) = Shallow water ℓ q ω − 8 (kh)4 − ⇐ Ag φ = φ + φ = ekz sin (kx ωt)+ O(ε3) = Deep water (3.17) ℓ q ω − ⇐ The second-order free-surface proﬁle according is:

2 2 1 ∂ ∂φ1 1 ∂φ1 ∂φ1 ηq = φ2 + η1 + + −g (∂t ∂z 2 " ∂x ∂z #) z=0

1 3A2ω2 cosh2kh 2 = 4 cos2(kx ωt) A gk tanh kh cos2(kx ωt) −g ( − 4 sinh kh − − − 1 Agk 2 + cos2 (kx ωt) + tanh2 kh sin2 (kx ωt) 2 ω − − ) h i A2k cosh kh A2k = 2+cosh2kh cos2(kx ωt)+ (3.18) 4 sinh3 kh − 2 sinh 2kh The wave elevation due to the second-order effect has two parts: one oscillatory part and one steady part. The steady part shows that the mean free surface in the presence of waves is different than the steel water surface. The total free-surface profile with the second-order approximation is:

2 2 A k cosh kh A k 3 η = ηℓ + ηq = A cos(kx ωt)+ 2+cosh2kh cos2(kx ωt)+ + O(ε )(3.19) − 4 sinh3 kh − 2 sinh 2kh A plot of the free surface elevation is shown in Fig. ??. The maximum values for the free surface elevation, i.e. (crest ηc), are happened where kx ωt = 2nπ, n = 0, 1, 2, . The minimum values free surface elevation, i.e. (trough η ), are occurred where− kx ωt = (2n + 1)π,··· n =0, 1, 2, . t − ··· A2k cosh kh A2k ηc = A + 2+cosh2kh + 4 sinh3 kh 2 sinh 2kh A2k cosh kh A2k ηt = A 2+cosh2kh + − 4 sinh3 kh 2 sinh 2kh Therefore, the free surface elevation up to second-order approximation shows a steeper crest and a flatter trough in compare with the linearized wave theory. This is also observed in Fig. 3.1, i.e. a steeper crest and a flatter trough. The second-order free surface elevation can be obtained for deep and shallow water waves by inserting the asymptotic values of the hyperbolic functions. A2k η = cos2(kx ωt) = Deep water q 2 − ⇐ 3A2k 1 A2 η = cos2(kx ωt)+ = Shallow water (3.20) q 4 (kh)3 − 4h ⇐ Hence, the total free-surface profile up to second-order approximation are in shallow and deep water as follows. 3A2k 1 A2 η = η + η = A cos(kx ωt)+ cos2(kx ωt)+ + O(ε3) = Shallow water ℓ q − 4 (kh)3 − 4h ⇐ A2k η = η + η = A cos(kx ωt)+ cos2(kx ωt)+ O(ε3) = Deep water (3.21) ℓ q − 2 − ⇐

32 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

ηℓ A ηq η

A 2

0 η

A − 2

−A

0 π 3π 5π 7π 2 π 2 2π 2 3π 2 4π kx Figure 3.1: First-order and second-order free surface elevation for waves with kA = 0.1 and kh =1.5

The power series for φ in term of ε is converged if the ratio of the (n + 1)th term divided by the nth term must be less than unity as n goes to inﬁnity. Hence, we must have for the φ series (3.3) εφ 3 kA cosh2kh r = 2 = << 1. (3.22) φ 8 cosh khsinh3kh 1

For deep water waves, when kh >π, the asymptotic values of the hyperbolic functions can be substituted to obtain the relationship for r. εφ 3 kAe2kh r = 2 = = 3(kA)e−2kh (3.23) φ 8 ekhe3kh/8 1

The values of kA is very small and therefore, the values of r should be very small. If we consider the limiting value of kh = π and kA = π/7 (for the wave of maximum steepness), we get π r = 3(kA)e−2kh =3 e−2π =0.0025. 7 It shows that the Stokes perturbation solution is valid for second-order term. In shallow water where kh< π/10, the asymptotic form of (3.22) is εφ 3 kA r = 2 = . (3.24) φ 8 (kh)3 1

The constraint for validation of the Stokes perturbation solution for shallow water waves is

3 kA 8 A 8 r = < 1 = kA < (kh)3 = < (kh)2 (3.25) 8 (kh)3 ⇒ 3 ⇒ h 3 where kh is small. The maximum value for the ratio of A/h is as follows when kh = π/10. A 8 8 A < (kh)2 = π2 = < 0.2632 h 3 300 ⇒ h

33 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

Therefore, the maximum wave amplitude is about 26% of the water depth. The wave amplitude of breaking wave in shallow water is almost 40% of the water depth. Hence, the Stokes expansion up to second-order is not a good approximation for high waves in shallow water. We should consider some other waves theory in shallow water or take into account higher order terms in the Stoke perturbation method. The dispersion relation remains the same as the first-order wave theory. ω2 = gk tanh kh (3.26) The wave phase velocity is obtained with the second-order approximation as follows. g v2 = tanh kh (3.27) p k There is no difference for the second-order and first-order wave phase velocity. By the third-order theory, the celerity takes the following form according to Rahman [3].

2 2 g 2 5+2cosh2kh + 2cosh 2kh vp = tanh kh 1 + (kA) 4 (3.28) k " 8 (sinhkh) # The water particle velocities in the x and z directions are obtained by diﬀerentiation of total velocity potential (3.17) with respect to x and z.− −

∂φ Agk cosh k(z+h) 3A2ωk cosh 2k(z+h) 3 u = ∂x = ω cosh kh cos(kx ωt)+ 4 sinh4 kh cos2(kx ωt)+ O(ε ) Intermediate − − water  ∂φ Agk sinh k(z+h) 3A2ωk sinh 2k(z+h) 3  w = = sin (kx ωt)+ 4 sin 2(kx ωt)+ O(ε ) depth ∂x ω cosh kh − 4 sinh kh − u = ∂φ = Aωekz cos(kx ωt)+ O(ε3)  ∂x − Deep water (3.29)  w = ∂φ = Aωekz sin (kx ωt)+ O(ε3)  ∂x − The components of water particle trajectory beneath a second-order finite-amplitude wave are as follows. cosh k(z + h) kA2 1 3cosh2k(z + h) ξ = A sin (kx ωt)+ 1 sin 2(kx ωt) − sinh kh − 4 sinh2 kh − 2 sinh2 kh − kA2 cosh2k(z + h) + wt (3.30) 2 sinh2 kh sinh k(z + h) 3kA2 sinh 2k(z + h) ζ = A cos(kx ωt)+ cos2(kx ωt) (3.31) sinh kh − 8 sinh4 kh − Where ξ and ζ are the horizontal and vertical displacement of a fluid particle, respectively. The horizontal displacement consist of two oscillatory parts and a non-oscillatory part that is a function of time also. The non-oscillatory part shows that a fluid particle is moved as time is passing. The components of water particle trajectory beneath a second-order finite-amplitude in deep water are: ξ = Aekz sin (kx ωt)+ kA2e2kz wt (3.32) − − ζ = Aekz cos(kx ωt) (3.33) − The vertical motion of a given particle of fluid is strictly periodic but the horizontal motion has a non- periodic term that cause the mass transfer due to the action of the waves motions. This is illustrated in Fig. 3.2. This steady motion is called Stokes drift. The total mean flux is as follows in deep water. 0 1 1 q = ωkA2e2kzdz = ωA2 = kA2v (3.34) 2 2 p Z−∞ The pressure variation is obtained by using the Bernoulli equation as given in (3.11) which is rewritten here. ∂φ 1 ∂φ 2 ∂φ 2 p = ρ q + ℓ + ℓ + gz (3.35) − ∂t 2 ∂x ∂z ( " # ) 34 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

kA2e2kzωT A

3A 4 A 2 A 4

0 ζ A − 4 A − 2 3A − 4 −A

−0.20 0.2 0.4 0.6 0.8 ξ

Figure 3.2: Trajectory a water particle beneath a second-order ﬁnite amplitude wave

Using (3.10) and (3.15) and substituting in (3.34), we obtained the pressure variation in the ﬂuid beneath the second-order ﬁnite amplitude waves. cosh k(z + h) p = ρgz + ρgA cos(kx ωt) − cosh kh − 3kA2 1 cosh2k(z + h) +ρg cos2(kx ωt) 2 sinh 2kh sinh2 kh − kA2 1 kA2 1 ρg cos2(kx ωt) ρg [cosh 2k(z + h) 1] (3.36) − 2 sinh 2kh − − 2 sinh 2kh − In (3.36):

◮ the ﬁrst term is the hydrostatic pressure;

◮ the second term is the hydrodynamic pressure in linearized wave theory;

◮ the third term is the hydrodynamic pressure due to the second-order velocity potential;

◮ the fourth term and fifth terms are the hydrodynamic pressure due to the second-order contribution of the first-order velocity potential. These are due to the water particle velocity components. The fourth term is an oscillatory part and the fifth term is a steady effect. The average energy density is obtained the same as computed for the linearized wave theory. The final computation according to Sarpkaya and Isaacson [4] is: 1 E¯ = ρgA2 + O(ǫ4) (3.37) 2 The energy flux is: 1 1 2kh J¯ = ρgA2v 1+ + O(ǫ4)= Ev¯ (3.38) 2 p 2 sinh 2kh g

35 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

3.2 Trochoidal wave theory

The First theory related to ﬁnite amplitude waves is the trochoidal wave theory that is developed by Gerstner in 1802. It is called the trochoidal wave theory because the free surface and the other constant pressure surfaces are predicted to be trochoidal in shape. These trochoidal waves can be generated by rolling a wheel on a ﬂat surface located above the x axis, as shown in Fig. 3.3. −

0 SWL vp x 1 k

−z

−π −π π π 3π 2π k 2k 0 2k k 2k k

Figure 3.3: Trochoidal wave proﬁles

The radius of the wheel is 1/k since it should be advanced one wavelength per a revolution. The wave amplitude is correspond to the radial position of a point p that is called the generating point. The trochoidal form of wave profile gives an outline which is sharper near the crests and flatter in the trough. These feature become more pronounced as the amplitude is increased. The limiting form have a cusps at the crests as shown in Fig. 3.4. This shows a breaking wave of height 2/k = λ/pi. In practice, experiment confirms that when the h/λ > 1/10, the approximately trochoidal profile of the free surface predicts the shape of a breaking wave.

− 1 2k

− 1 k

− 3 2k

− 2 k

− 5 2k

0 π π 3π 2π 5π 2k k 2k k 2k

Figure 3.4: Trochoidal wave geometry

Since the trochoidal wave is somewhat sharp in the crest and ﬂat in the trough, the lines of orbit center must be somewhat above the corresponding still-water level (SWL) to have equal water volumes, as shown in Fig. 3.3. The amount of displacement is kA2/2. The parametric equations to generate constant pressure

36 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle surface including the free surface are: ωt A x = + ekz0 sin ωt k k 1 A z = z ekz0 cos ωt (3.39) − 0 − k − k

Where z0 is position of the center of generating wheel. It is noted that Gerstner’s solutions are in closed form in contrast to Stokes’ results which are in the form of an inﬁnite series. It should be also indicated that the Gerstner solution is an exact solution of a particular wave motion problem in that no mathematical approximations are made. The equation of trochoidal wave theory can be derived from those of linear (small-amplitude) wave theory. The particle path have the following components: A ξ = α + ekβ sin (kα ωt) k − A ζ = β + ekβ cos(kα ωt) (3.40) k −

Where (α, β) is the original position of the particle at t = 0 and corresponds on (x0,z0) that is the position of the center of the generating wheel. The pressure along a trochoidal surface may be obtained by using the Lagrangian form of the equation of motion. The final solution according to Rahman [3] is 1 p = ρgz + ρv2e2kz0 + Constant. (3.41) − 0 2 p For a particle on the free surface the pressure must be the atmospheric pressure if we neglect the effect of surface tension. It should be indicated that the motion of the fluid in Gerstner’s waves is rotational. According to Sarpkaya and Isaacson [4], the vorticity in the fluid is

2kv e2kz0 ξ = p (3.42) − 1 e2kz0 − The vorticity is infinite at the surface when z0 = 0 and decrease rapidly with increasing depth. It will be zero when z0 . This is unusual behavior which persist in the vorticity distribution of trochoidal wave theory. The→ minus −∞ sign is shown that the vorticity is in the opposite direction of the motion of fluid particles. It follows that the physical realization of such a waves seldom occurs. Except under a few possible circumstances as when waves are progressing against a wind which induces a vorticity within the fluid in the opposite sense to the particle motions, Sarpkaya and Isaacson [4]. For more explanation about the theoretical development of trochoidal waves see Rahman [3] , Sarpkaya and Isaacson [4].

3.3 Wave transformation

So far We have been concern with the behavior of regular wave train of permanent form over a smooth horizontal seabed without the presence of any obstacle any underlying current. The wave over a variable depth seabed and/or in presence of underlying current or/and an obstacle undergoes changes in height, wavelength and directions of propagation. Processes that can aﬀect a wave as it propagates from a deeper water into a shallower water include: 1. Shoaling; 2. Refraction; 3. Diﬀraction; 4. Dissipation due to friction;

5. Dissipation due to percolation;

37 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

6. Breaking; 7. Additional growth due to the wind; 8. Wave-current interaction; and 9. Wave-wave interactions. The first three effects are propagation effects because they result from convergence or divergence of waves caused by the shape of the bottom topography, which influences the direction of wave travel and causes wave energy to be concentrated or spread out. Diffraction also occurs due to the presence of structures that interrupt wave propagation. The second three effects are sink mechanisms because they remove energy from the wave field through dissipation. The wind is a source mechanism because it represents the addition of wave energy if wind is present. The presence of a large-scale current field can affect wave propagation and dissipation. Wave-wave interactions result from nonlinear coupling of wave components and result in transfer of energy from some waves to others. We will discuss some of them in this section very precisely.

3.3.1 Wave shoaling It is refer to the movement of a wave into a shallow water. This cause that the wave height and wavelength alter that is called as wave shoaling. If the variation of seabed depth is rough then the solution of the associated complete boundary value problem is very complicated. However, there are a host of numerical techniques to solve the complete problem. We may consider that the seabed slope is gentle (but not necessary uniform) in the sense that the seabed slope resulting in a negligible wave reﬂection. The shoaling eﬀect may be estimated under the following assumptions:

◮ the wave motion is two-dimensional;

◮ the wave period remains unchanged;

◮ the average rate of energy remains constant and independent of depth; In other word, the energy is neither supplied (by the wind) nor dissipated (by breaking or by friction/percolation at the seabed).

◮ the selected wave theory applicable to the local wave characteristics at any given depth. Following Sarpkaya and Isaacson [4], it is considered that the deep water is selected as the reference and is denoted by subscript 0. It can be written on the basis of linear theory that: gT gT 2 4π v = , λ = , k = . (3.43) po 2π o 2π o gT 2 According to the constant wave period assumption, we can write: v k = v k = ω = const. p p0 0 v λ k = p = = 0 = tanh kh (3.44)  2 ⇒ vp λ0 k  ω = gk tanh kh = gk0 = const. 0 The ﬂux of energy for the propagation of wave over a variable depth seabed, Fig. 3.6, can be written that ∂E ∂ + v E =0 (3.45) ∂t ∂x g ∂E since there is no energy dissipation or gain then ∂t = 0. Therefore, it can be written that: ∂ 1 1 v E =0 = v E = v E = ρgA2v = ρgA2v ∂x g ⇒ g0 0 g ⇒ 2 0 g0 2 g so that

2 A vg 2 cosh kh = 0 = . (3.46) A v s2kh + sinh 2kh 0 r g 38 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

v ∆x p x

J1 J2

Figure 3.5: The Flux of energy

For shallow water range, it can be written by using the asymptotic vales of hyperbolic function that: v k = v k = ω = const. p p0 0 v λ k = p = = 0 = kh.  2 2 ⇒ vp λ0 k  ω = gk h = gk0 = const. 0 Doing some manipulations, it can be written that: − 1 − 1 v λ h 2 2πh 2 p = =2π = v λ gT 2 λ p0 0 0 − 1 − 1 A 16π2h 4 8πh 4 = = A gT 2 λ 0 0 3.3.2 Wave refraction It is observed in the ocean that when waves approaches a bottom slope obliquely, the speed of the wave front 2 g in shallower water is less than in deeper water in accordance with the dispersion equation, vp = k tanh kh. As a result, the line of wave crest is bent so as to become more closely aligned with bottom contours. This wave phenomenon is known as wave refraction. For more precise explanation about the eﬀect and formulation of the refraction phenomenon see Rahman [3]. It is observed in the ocean that when waves approaches a bottom slope obliquely, the speed of the wave front in shallower water is less than in deeper water in accordance with the dispersion equation, 2 g vp = k tanh kh. As a result, the line of wave crest is bent so as to become more closely aligned with bottom contours. This phenomenon is shown in Fig. 3.6. Circles of radius C∆t, where ∆t is an interval of time, are

Figure 3.6: A geometric construction showing waves approaching shore and the clos- est inshore having a smaller speed, leading to bending of the wave crest, Fenton [3] constructed in Fig. 3.6. It shows where a point on a wave crest can be at a time ∆t later. By drawing an

39 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle envelope to all such circles we have an approximation to the wave crest at the later time, and it is clear that it has apparently turned towards shallower water. The wave is subject to a continuous refraction which tends to align the wave front to the depth contours. This phenomenon is illustrated in Fig. 3.7 where all depth contours evenly spaced and parallel to the shoreline. If a wave crest initially has some angle of approach to

Figure 3.7: Straight shore with all depth contours evenly spaced and parallel to the shoreline, Coastal engineering manual [1] the shore other than 0o, part of the wave at point A will be in shallower water than another part at point B and because the depth at A, hA, is less than the depth at B, hB, the speed of the wave at A will be slower than that at B according to dispersion equation. g g C = tanh kh < tanh kh = C A ω A ω B B The speed differential along the wave crest causes the crest to turn more parallel to shore. The propagation problem becomes one of plotting the direction of wave approach and calculating its height as the wave propagates from deep to shallow water. For the case of monochromatic waves, wave period remains constant, as applied in description of shoaling. In the case of an irregular wave train, the transformation process may affect waves at each frequency differently; consequently, the peak period of the wave field may shift.

3.3.3 Wave breaking Waves approaches the coast increase in steepness as water depth decrease. The shoaling coefficient indicates that the wave height will approach infinity in very shallow water, which is not realistic. At some depth, a wave of given characteristics will become unstable and break, dissipating energy in the form of turbulence. When designing a structure which at times may be inside the surf zone it becomes necessary to be able to predict the location of the breaker line. The surf zone is the region extending from seaward boundary of wave breaking to the limit of wave uprush. Within the surf zone, wave breaking is the dominant hydrodynamic process. Waves break as they reach a limiting steepness which depends on the nature of the bottom and the characteristics of the wave. The limiting steepness is a function of relative depth h/λ and the slope of the bottom m = tan β. There are three main types of wave breaking that are illustrated in Fig. 3.8. Spilling breakers: For very mildly sloping beaches, typically the waves are spilling breakers, and is • characterized by breaking gradually over a long distance such that many waves occur within the surf zone (defined as that region where the waves are breaking, extending from the beach to the seaward limit of the breaking). The rate of energy loss is small, permitting a nearly complete reforming of the waves should they once again progress into deep water, such as when they cross a sand bar. The wave

40 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

remains almost symmetrical, with foam gently spilling down the front face of the breaking wave. By the time the wave reaches the top of the beach, the energy of the wave has been almost completely absorbed, with little or no reﬂection. Plunging breakers: These occur on steeper beaches and are characterized by the crest of the wave curling • over forward and impinging onto part of the wave trough, sometimes trapping air, and plunging with a loud report. Considerable energy is dissipated in this manner by turbulence, and considerable fun is had by surfers before that moment of plunging. There are few measurements of the breaking criteria for such waves. Surging breakers: These waves occur on very steep beaches and are characterized by narrow or non- • existent surf zones, and high reﬂection. Sometimes a fourth type is added collapsing breaker, which is a combination of plunging and surging.

Figure 3.8: Wave breaking types on beaches, Fenton [3]

3.3.4 Wave diffraction When a wave train encounter a large vertical obstacle it has been observed that the wave motion penetrate into the region of geometrical shadow. Wave diffraction is the process by which energy spreads laterally perpendicular to the dominant direction of wave propagation. Consider a long-crested monochromatic wave approaching a semi-infinite barrier (such as a breakwater) in the region where the water depth is constant (i.e. no wave refraction or shoaling) as shown in Fig. 3.9. The portion of the wave that hits the barrier will be reflected and dissipated, with the possible transmission of some wave energy through or over the barrier depending on the cross-section geometry and composition of the barrier. The portion of the wave passing the end of the barrier will have a lateral transfer of wave energy along the wave crest into the lee of the barrier. The diffracted wave crests in the lee of the barrier will form approximately concentric circular arcs with the wave height decreasing exponentially along the crests. The shadow region out to the dashed line will have a wave height that is less than the incident wave height at the end of the barrier, Sorensen [6]. If Hi is the incident wave height at the end of the barrier and Hd is the diffracted wave height at a point of interest in the lee of the barrier, the diffraction coefficient is defined as

Hd Kd = (3.47) Hi

41 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

Figure 3.9: Wave diﬀraction and deﬁnition of terms, Sorensen [6]

where Hd is the diffracted wave height at a point in the lee of the barrier and Hi is the incident wave height at the breakwater tip.The value of the diffraction coefficient Kd depends on the location behind the barrier r defined by r and β, and the incident wave direction θ. It can be write that Kd = f(θ, λ ,β) where λ is the wave length. Since the wave length is a function of the wave period and water depth, the resulting diffraction coefficient for each component of the wave spectrum would depend on the incident direction and period of that component. When waves approach a barrier of finite length and wave diffraction occurs at both ends, a wave crest pattern similar to that shown in Fig. 3.10 will develop. It can be constructed by combining the patterns for semi-infinite barrier diffraction at each end. The wave crests combine along lines like the dashed line to form the higher amplitudes which may be estimated (assuming linear waves) by combining the heights from the two separate patterns.

Figure 3.10: Wave diﬀraction in the lee of a barrier of ﬁnite length, Sorensen [6].

42 Chapter 3 Finite-amplitude waves Dynamics of Marine Vehicle

When waves pass through a gap in a barrier, diffraction occurs in the lee of the barrier on both sides of the gap. As the waves propagate further and passing the gap of the barrier, the zone affected by diffraction grows toward the center line of the gap until the two diffraction zones interact, as shown in Fig. 3.11. The interaction point is depends on the width of the gap, the wider the gap, the further behind the barrier this interaction point occurs. For typical harbor conditions and gap widths greater than about five wavelengths, suggests that the diffraction patterns at each side of the gap opening will be independent of each other. For smaller gap widths, an analysis employing the gap geometry must be used.

Figure 3.11: Wave diﬀraction through a gap, Coastal engineering manual, [1].

The waves approaching a barrier gap will usually not approach in an angle of 90o (in direction normal to the gap). The results for normally incident waves can be used as an approximation for oblique waves by employing a projected imaginary gap width as shown in Fig. 3.12.

Figure 3.12: Oblique wave incident to a barrier gap, Sorensen [6]

43 Bibliography

[1] Coastal manual engineering, Part II, 2008 [2] Dean, R. G. and Dalrymple, R. A., Water wave mechanics for engineers and scientists, World Scientiﬁc Publishing Co., 2000 [3] Fenton, J., Coastal and ocean Engineering, 2013 [4] Rahman, M., Water waves, relating modern theory to advanced engineering applications, Oxford Uni- verisit press, 1994 [5] Sarpkaya, T. and Isaacson, M., Mechanics of wave forces on oﬀshore structures, Van Nostrand Reinhold Company, 1981 [6] Sorensen, R. M., Basic coastal engineering, Springer, 2006

44 Chapter 4

Real ocean Waves

4.1 Introduction

In practice, ocean waves are not regular but random in the sense that the ocean surface is composed of waves moving in different directions and with different amplitudes, frequencies and phases. The wind causes the formation of waves on the free surface of oceans. The variation of the ocean surface is depend on the wind speed. The ocean surface varies from time to time and place to place. The Beaufort number is a means to estimate and report the wind speed and sea state. An individual can derive a Beaufort number and hence a wind speed by observing the ocean surface. Wave traveling out of a distance generating area are called swells while waves generating locally are known as sea. Since the ocean waves are random in nature, they should be describes by their statistical properties. This chapter is started with description of the statistical properties of a random signal x(t) varying continuously with time t. Then, these statistical definition are applied to the random wave field. The random signal may be the free surface elevation η(t) and/or the other wave characteristics and wave effects such as a wave induced force F (t) on a structure.

4.2 Statistical and probabilistic deﬁnitions

The field of statistics deals with the collection, presentation, analysis, and use of data to make decisions, solve problems, and design products and processes. In simple terms, statistics is the science of data. Statistical methods are used to help us describe and understand variability. By variability, it means that successive observations of a system or phenomenon do not produce exactly the same result. Often data are collected over time. In this case, it is usually very helpful to plot the data versus time in a time series plot. Phenomena that might affect the system or process often become more visible in a time-oriented plot and the concept of stability can be better judged. An experiment that can result in different outcomes, even though it is repeated in the same manner every time, is called a random experiment. The set of all possible outcomes of a random experiment is called the sample space of the experiment. The sample space is denoted as S. A sample space is discrete if it consists of a finite or countable infinite set of outcomes. A sample space is continuous if it contains an interval (either finite or infinite) of real numbers. An event is a subset of the sample space of a random experiment. Probability is a number that is assigned to each member of a collection of events from a random experiment that satisfies the following properties: If S is the sample space and E is any event in a random experiment, (1) P (S)=1 (2) 0 P (E) 1 ≤ ≤ (3) For two events E and E with E E = P (E E )= P (E )+ P (E ) 1 2 1 ∩ 2 ⊘ 1 ∪ 2 1 2 Whenever a sample space consists of N possible outcomes that are equally likely, the probability of each 1 outcome is N .

45 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Beaufort Description Wind speed Wave height Sea conditions number km/h kts m/s m ft 0 calm <1 <1 <0.3 0 0 Flat 1 Light air 1.1 − 5.5 1 − 2 0.3 − 1.5 0 − 0.2 0 − 1 Ripples without crests 2 Light breeze 5.6 − 11 3 − 6 1.6 − 3.4 0.2 − 0.5 1 2 Small wavelets.

3 Gentle breeze 12 − 19 7 − 10 3.4 − 5.4 0.5 − 1 2 − 3.5 Large wavelets. 4 Moderate breeze 20 − 28 11 − 15 5.5 − 7.9 1 − 2 3.5 − 6 Small waves with breaking crests. 5 Fresh breeze 29 − 38 16 − 20 8.0 − 10.7 2 − 3 6 − 9 Moderate waves of some length 6 Strong breeze 39 − 49 21 − 26 10.8 − 13.8 3 − 4 9 − 13 Long waves begin to form. 7 High wind 50 − 61 27 − 33 13.9 − 17.1 4 − 5.5 13 − 19 Sea heaps up. 8 Gale, Fresh gale 62 − 74 34 − 40 17.2 − 20.7 5.5 − 7.5 18 − 25 Moderately high waves 9 Strong gale 75 − 88 41 − 47 20.8 − 24.4 7 − 10 23 − 32 High waves 10 Storm 89 − 102 48 − 55 24.5 − 28.4 9 − 12.5 29 − 41 Very high waves 11 Violent storm 103 − 117 56 − 63 28.5 − 32.6 11.5 − 16 37 − 52 Exceptionally high waves. 12 Hurricane-force ≥118 ≥64 ≥32.7 ≥14 ≥46 Huge waves

Table 4.1: The modern Beaufort scale, From Wikipedia, the free encyclopedia [5]

A probability density function p(x) can be used to describe the probability distribution of a continuous random variable X. If an interval is likely to contain a value for X, its probability is large and it corresponds to large values for p(x). For a continuous random variable X, a probability density function is a function such that: (1) p(x) 0 ≥ ∞ (2) −∞ p(x)dx =1 b (3) RP (a X b)= p(x)dx = area under p(x) from a to b ≤ ≤ a for any a and b R A probability density function provides a simple description of the probabilities associated with a random ∞ b variable. As long as p(x) is non-negative and f(x) = −∞ p(x)dx =1, 0 P (a X b)= a p(x)dx 1 so that the probabilities are properly restricted. A probability density function≤ ≤ is zero≤ for x values that≤ cannot occur and it is assumed to be zero wherever it isR not speciﬁcally deﬁned. If X is a continuousR random variable, for any x1 and x2

P (x X x )= P (x

d ∞ p(u)du = p(x) dx Z−∞ Then, given P (x), P (x) p(x)= (4.2) dx as long as the derivative exists.

46 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Suppose X is a continuous random variable with probability density function p(x). The mean or expected value of X, denoted as µ or E(X), is ∞ µ = E(X)= xp(x)dx (4.3) Z−∞ And more generally, the espected value of any function g(x) of x is given as ∞ E(g(x)) = g(x)p(x)dx (4.4) Z−∞ ∞ provided that the integral −∞ g(x) p(x)dx converge. Adopting this approach we may deﬁne the n th moment of X as | | − ∞ R E(Xn)= xnp(x)dx (4.5) Z−∞ The variance of X, denoted as V (X) or σ2, is ∞ ∞ σ2 = V (X)= (x µ)2p(x)dx = x2p(x)dx µ2 (4.6) − − Z−∞ Z−∞ The standard deviation of X is: σ = √σ2 There are various probability distribution functions depend on the behavior of a physical phenomenon. Two probability distribution function are of particular interest in the study of the random ocean waves are the Gaussian (or normal) distribution and the Rayleigh distribution. These are applied to describe the probability distribution of wave surface elevation η and of wave height H, respectively. The cumulative probability and probability density of the Gaussian distribution are 1 x (x µ )2 P (x)= exp − x dx (4.7) σ √2π − 2σ2 x Z−∞ x P (x) 1 (x µ )2 p(x)= = exp − x (4.8) dx σ √2π − 2σ2 x x where it is shown in Fig. 4.1. The distribution of a random process x(t) with mean zero (µx = 0) and variance unity (σx = 1) is called a standard normal distribution. 1 x x2 P (x)= exp dx (4.9) √2π −∞ − 2 Z P (x) 1 x2 p(x)= = exp . (4.10) dx √2π − 2 The Rayleigh distribution are given as

2 1 exp π x for x 0 − − 4 µx ≥ P (x)=  (4.11)   0 otherwise 2  πx π x  2 exp for x 0 2µx − 4 µx ≥ p(x)=  (4.12)   0 otherwise  It is shown in Fig. 4.2. It should be noted that the Gaussian probability density function is symmetric about the mean value. It can take both the positive and negative values. But, the Rayleigh probability density function can only take positive value and has a value of zero at x = 0. It reaches to a maximum and then goes to zero exponentially as x increases.

47 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Figure 4.1: The Gaussian probability density and cumulative probability distribution

Figure 4.2: The Raleigh probability density and cumulative probability distribution

4.2.1 Basic deﬁnitions and concept of random process

Let us begin with an ensemble of k wave records xk(t) taken under identical macroscopic conditions, i.e.: position on the ocean surface, water depth, mean wind velocity, air-sea temperature. Even under identical conditions, we cannot expect that these wave records will be identical or even closely similar in detail. The family xk(t) represents k realizations of the stochastic process x(t). For a given k, x(t) is a function of time, while when t = t1, xk(t) is a random variable. Stochastic processes may belong to one of three categories: a - Stationary and ergodic; b - Stationary; and c - Non-stationary.

48 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

A random process (or random function) is stationary in the wide range if: E[x(t)]=x ¯ = const. (4.13) R(t ,t )= R(t t )= E[x(t )x(t )] = R(τ), τ = t t 1 2 1 − 2 1 2 1 − 2 in which E[] is the mean of x and K() is an autocorrelation function. Strictly, a random process is stationary if all statistical moments are translationally invarient. These two definitions of stationarity coincide when x is a Gaussian, in which case all statistics of x are completely determines by first and second moments. In general, using the ensemble of wave records xk(t),any function of x, say F , can be developed to find F xk(t). To be more specific, the time t = t1 is selected in the family xk(t). When F is the x value itself, then averaging over k results in the ensemble mean of the process at t = t1, i.e.:

N k=1 xk(t1) E[F xk(t1) ]k = E[xk(t1)] = lim (4.14) { } n→∞ N P The condition N is only conceptual since in practice N is always finite. When F x (t→) ∞ [x (t )]2, then the averaging F x (t ) over k leads to the variance at t = t : { k 1 }∼ k 1 { k 1 } 1 N 2 k=1[xk(t1)] E[F xk(t1) ]k = E [xk(t1)] = lim (4.15) { } { } n→∞ N P Repetition of the above averaging for the different times helps us to obtain the different numerical values for the statistics. However, the repeated observations technique which provides us ensemble of k wave records can be attained in laboratory wave tank, but is inapplicable to observations of wave phenomena in field experiments. To overcome these difficulties, the ergodic theorem in usually invoked. This allows the ensemble averages to be replaced with time average. The ergodic theorem states that: If x(t) is an ergodic stationary function, then the statistics obtained by ensemble averages at a given time t = t∗ are identical to the corresponding statistics computed by the time averaging for any given realization k = k∗. Hence, the ergodic stationary process should satisfy the following equality:

N 2 k=1 F xk(t = t∗) E[F xk(t = t∗) ]k = lim { } = E[F xk=k∗ (t) ]t (4.16) { } n→∞ N { } P 1 T = lim F xk=k∗ (t)dt T →∞ 2T { ( Z−T ) The significant of the ergodic theorem is that it enable us to develop the statistics of the process x(t) using one, sufficiently long realization. However, it is not possible to demonstrate ergodicity for ocean waves since experiments cannot be exactly repeated in the ocean as they can in the laboratory. The mean value, variance and standard deviations describe the spead of values x but do not describe the way that x(t) varies with time. The autocorrelation function Rx(τ) related the value of x at time t to its value at a later time t + τ and so provides an indication of the correlation of the signal with itself for various time lags τ. The autocorrelation function is defined as:

1 T Rx(τ)= E[x(t)x(t + τ)] = lim x(t)x(t + τ)dt (4.17) T →∞ 2T ( Z−T ) 4.3 Irregular waves

It is necessary to have a few simple parameters that in some sense tell us how severe the sea state is and a way to estimate or predict what the statistical characteristics of a wave record might be had it been measured and saved. There are millions of wave records have been observed and a theoretical/empirical basis has evolved to describe the behavior of the statistics of individual records. There are many short- term candidate parameters which may be used to deﬁne statistics of irregular sea states. Two of the most important parameters necessary for adequately quantifying a given sea state are characteristic height, H,

49 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

4 Tc 3

0 Hmax H η, m −1

−2 Tz

−3

Tsi −4 0 10 20 t, s 40 50 60

Figure 4.3: Parameters of an ocean wave and characteristic period, T . Other parameters related to the combined characteristics of H and T , may also be used in the parametric representation of irregular seas. The parameters of an ocean wave are illustrated in Fig. 4.3. In the time-domain analysis of irregular or random seas, wave height and period, wavelength, wave crest, and trough have to be carefully defined for the analysis to be performed. The definitions provided earlier in the regular wave that the crest of a wave is any maximum in the wave record, while the trough can be any minimum. However, these definitions may fail when two crests occur within an intervening trough lying below the mean water line. Also, there is not a unique definition for wave period, since it can be taken as the time interval between either two neighboring wave troughs or two crests. Other more common definitions of wave period are the time interval between successive crossings of the mean water level by the water surface in a downward direction called zero down-crossing period or zero up-crossing period for the period deduced from successive up-crossings. The zero-crossing wave height is the difference in water surface elevation of the highest crest and lowest trough between successive zero-crossings. The definition of wave height depends on the choice of trough occurring before or after the crest. Here, a wave will be identified as an event between two successive zero-upcrossings and wave periods and heights are defined accordingly.

4.3.1 Wave height definitions Characteristic wave height for an irregular sea state may be defined in several ways. These include the mean height, the root-mean-square height, and the mean height of the highest one-third of all waves known as the significant height. Among these, the most commonly used is the significant height, denoted as Hs or H1/3. It was introduced by Sverndrup and Munk in 1947 and has been found to be very similar to the estimated visual height by an experienced observer. The significant wave height may be determine from a wave record in three different ways. The number of (crest-to-trough) waves are counted from a wave record and the highest one-third waves are selected. The average of these waves gives the significant wave height. Thus:

N/3 3 H = H (4.18) 1/3 N i i=1 X where N is the number of individual wave heights, Hi, in a record ranked from highest to lowest. The Tucker (1963) provided a relationship to estimate the signiﬁcant wave height as follows.

H1/3 = √2C1(ηc + ηt) (4.19)

50 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

The notation ηc is the height of the highest crest in a wave record and ηt is and the depth of the lowest trough. The coefficient C1 is calculated by knowing the the number of zero up-crossing cycles Nz. −1 C = (ln N )−1/2 1+0.289 (ln N )−1 0.247 (ln N )−2 (4.20) 1 z z − z h i Similarly, Hp is defined as the average of the first highest pN recorded waves, p = 1/n, where n = 10, , 100. ··· pN 1 H = H (4.21) p pN i i=1 X The mean height H¯ is the average wave height over all records,

1 N H¯ = H = H (4.22) 1 N i i=1 X If there are N waves of heights H1,H2, ,HN , then the root mean square (rms) wave height is calculated by the formula ···

N 1 H = H2 (4.23) rms vN i u i=1 u X t The root mean square wave height is always greater than H1 in real ocean. The largest wave height or the most probable maximum wave height is related to the rms wave height by follows according to Longuet-Higgins (1952). 0.2886 H = √ln N + H (4.24) max √ rms ln N Example - 1

A wave height from an ocean wave record are given in the following table. Find the average height, the signiﬁcant height, the average of one-tenth and one-hundred highest wave height and the root mean square wave height.

Wave height Number of waves of height (1) (1) (2) 5 2 4 25 3 31 2 40 1 4

Solution: Wave number of percentage Cumulative Cumulative One-third One-tenth One-hundred height wave of waves number of number of of highest of highest of highest [ft] height (1) waves wave height waves height waves height waves height (1) (2) (3) (4) (5) (6) (7) 5 2 2% 2 10 10 10 5 4 25 25% 27 110 110 42 3 31 30% 58 203 131 2 40 39% 98 283 1 4 4% 102 287 100%

51 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

287 H¯ = H = =2.81 ft or 1 102 5 2+4 25+3 30+2 39+1 4 H¯ = µ = η p(η )= × × × × × =2.82 ft H i i 100 i X 3 H = 131 3.85 ft 1/3 102 × ≈ 10 H = 42 4.2 ft 1/10 102 × ≈ 100 H = 5 5 ft 1/100 102 × ≈

N 1 2 52 + 25 42 + 31 32 + 40 22 +4 12 H = H2 = × × × × × =2.96 [ft] rms vN i 102 u i=1 r u X t 0.2886 0.2886 H = √ln N + H = √ln 102 + 2.96=6.76 [ft] max √ rms √ ln N ln 102 4.3.2 Irregular wave periods There may be speciﬁed two types of average wave periods of a wave record.

1. Average zero-upcrossing wave period which is denoted by T¯z; and

2. Average crest period which is denoted by T¯c. They are shown in Fig. 4.3. If we denote the total recording length of time by Ts, we may divide it into Nz equal interval of Tsi. The average zero-upcrossing wave period over each interval is given by

¯ Tsi Tzi = (4.25) Nzi where Nzi is the number of zero upcrossing in time interval, i. The average zero-upcrossing wave period of the record is

N 1 z T¯ = T¯ (4.26) z N zi z i=1 X Similarly, the average crest period is given for the interval i as

¯ Tsi Tci = (4.27) Nci where Nci is the number of crest to crest in time interval, i. If the number of intervals is denoted by Nc, the average wave crest period of the record is as follows.

N 1 c T¯ = T¯ (4.28) c N ci c i=1 X 4.3.3 Probability distribution of a sea state The probability density distribution of free surface elevation of sea η, wave height H and wave period T is introduced in this section.

52 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Probability distribution for the surface elevation The sea surface elevation distribution is assumed to be Gaussian. It is based on the assumption that the components phases are arbitrary but uniformly distributed in the interval of (π, π). The Gaussian model − implies that a symmetry distribution about the still water level that corresponds to zero mean, µη = 0. 1 η2 p(η)= exp (4.29) σ √2π −2σ2 η x The probability density of water surface elevation is symmetric about the still water level where the probability density is maximum. The probability of having a particular positive elevation is the same as that for the negative elevation. The measured data on the water surface elevation have indicated that the Gaussian distribution for η is appropriate. For waves of appreciable amplitude, however, crest amplitudes are higher than the trough amplitudes and the sea-surface elevation are skewed, Chakrabarti [1]. In such a case, the expected value is diﬀerent from the still water elevation.

Probability distribution of wave height The ocean surface is considered to be composed of a large number of sinusoidal waves with different frequencies, amplitudes, and phases. When these frequencies are in a narrow frequency band about ω then it is said that it is a narrow-band ocean. The surface elevation is obtained by superposing of different waves profiles.

K H η(t)= i cos(ω t ε ) (4.30) 2 i − i i=1 X th th where Hi is the amplitude of the i harmonic wave, ωi is the frequency of the i harmonic wave, εi is the phase of the ith harmonic wave and K is the number of time harmonic waves. It can be written using the complex form as

K H η(t)= i exp i(ω t ε ) (4.31) ℜ 2 i − i (i=1 ) X h i where is stand for the real part of . . It may be rearranged as ℜ { } η(t)= c(t)exp(iωt) (4.32) ℜ{ } where K H c(t)= i exp i (ω ω)t ε (4.33) ℜ 2 i − − i (i=1 ) X n o is the slowly varying wave group for a narrow-band spectrum. If a combination of the following four sinusoidal waves is taken into account 4 H η(t)= i cos(ω t ε ) 2 i − i i=1 X Hi =2 , 2.25 , 2.75 , 3 π π π π ω = , , , i 6 5.5 5 4 π π π 2π ε = , , , 4 3 2 3 the plot of the surface elevation is as shown in Fig. 4.4. The wave proﬁle resembles a slowly varying wave group.

53 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

5 4 3 2 ] 1 m [ 0

η, −1 −2 −3 −4 −5 0 60 120 180 240 t, [s] Figure 4.4: Surface proﬁle of combination of four sinusoidal waves

It has been theoretically shown by Longuet-Higgins (1952) that the wave amplitude A has a Rayleigh distribution for a narrow-band Gaussian ocean wave whose components are in random phase. According to (4.11) and (4.12), it can be written that

π A 2 P (A)=1 exp (4.34) − − 4 µ " A # πA π A 2 p(A)= exp (4.35) 2µ2 − 4 µ A " A # 2 Since any single wave is closely sinusoidal for a narrow-band motion, the variance σA may be expressed as 2 2 the average of integrals taken over each wave. (For a single wave σA = Ai /2.) In this manner, it can be obtained that √π A = √2σ µ = σ (4.36) rms A A 2 A Therefore, we can write that:

A 2 P (A)=1 exp (4.37) − − A " rms # 4A A 2 p(A)= exp (4.38) A2 − A rms " rms # Usually, wave heights H are employed in describing ocean waves rather than the amplitude A. Assuming that H =2A, we can write that:

H 2 P (H)=1 exp (4.39) − − H " rms # 2H H 2 p(H)= exp (4.40) H2 − H rms " rms #

The probability that a wave height is greater (less) than or equal to a design wave height H0 may be found from m P (H>H )= 0 N m P (H H )=1 (4.41) ≤ 0 − N

54 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

where m is the number of waves higher than H0 and N is the total number of wave height record. We can now derive the relationships for the signiﬁcant wave height and the other wave height characteristics. According to the deﬁnition for H1/3 that the average of the highest one-third waves, we can write that: H 2 1 2 P (H H )=1 exp 0 =1 = = H =1.0481H (4.42) ≤ 0 − − H − 3 3 ⇒ 0 rms " rms #

The signiﬁcant wave height is the centeroid of the area under the probability density function for H H0. Therefore, it can be written that: ≥ ∞ Hp(H)dH H0 H¯s = H¯ = ∞ (4.43) 1/3 p(H)dH R H0 Inserting (4.40) and (4.42)R into (4.43) and doing the integration, we obtain: ¯ ¯ Hs = H1/3 =1.416Hrms (4.44) The relationships for the other wave characteristics may be obtained using the same method as describe for the signiﬁcant wave height. The following equation according, Rahman [3], can also be applied to compute the average highest pN waves, p =1/n, such as the average of the highest one-tenth waves.

H¯ 1 1/2 √π 1 1/2 1/p = ln + erfc ln (4.45) H p 2p p rms " # As an example for p =1/10, the average of the highest one-tenth waves is ¯ H1/10 1/2 10√π 1/2 = [ln (10)] + erfc [ln (10)] =1.80Hrms. Hrms 2 n o If we consider p =1, 1/3, 1/10, 1/100, the following relationships are obtained.

H¯ = H1 =0.886 Hrms (4.46) ¯ ¯ Hs = H1/3 =1.416 Hrms (4.47) ¯ H1/10 =1.80 Hrms (4.48) ¯ H1/100 =2.36 Hrms (4.49)

H¯max =2.63 Hrms (for1000wavecyclesintherecord) (4.50)

Example - 2

A wave height from an ocean wave record over a 24 hr period are as follows:

Wave height Number of waves [ft] - 0 5 5600 5 − 10 7200 10 − 15 1920 15 − 20 960 20 − 25 320 − a - Plot the wave height histogram and the Rayleigh probability density distribution. b - Compute the signiﬁcant wave height, the mean of 1/10 of highest waves, the mean of 1/100 of highest waves and the most probable maxima, Hmax.

55 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Solution - a

1 N H = H2 rms vN i u i=1 u X t 1 = (5600 2.52 + 7200 7.52 + 1920 12.52 + 960 17.52 + 320 22.52) r 16000 × × × × × = 8.66 [ft]

H 2 P (H)=1 exp − − H " rms # 2H H 2 p(H)= exp H2 − H rms " rms #

Wave Mean Number percent of percent of Cumulative Rayleigh Rayleigh height height of waves occurrence occurrence percentage probability cumulative per foot of of occurrence density probability wave height function function

H Hm Ni - - - p(H) P (H) [ft] [ft] % % % (1) (2) (3) (4) (5) (6) (7) (8) 0 − 5 2.5 5600 35 7 35 0.0615 0.0801 5 − 10 7.5 7200 45 9 80 0.0945 0.5285 10 − 15 12.5 1920 12 2.4 92 0.0414 0.8761 15 − 20 17.5 960 6 1.2 98 0.0078 0.9833 20 − 25 22.5 320 2 0.4 100 0.0007 0.9981 16000

The distribution of waves height of the record are shown in Fig. 4.5.

Solution - b The area of the highest one-third waves of the record shown in Fig. 4.6. The signiﬁcant wave height is the centroid of the area under the probability density function for H H0 =1.0481Hrms. The signiﬁcant wave height of the record is, (4.47): ≥

H¯ = H¯ =1.416H =1.416 8.66 = 12.26 [ft] s 1/3 rms × The mean of 1/10 and 1/100 of the highest waves may be obtained by (4.48) and (4.49), respectively.

H¯ =1.80 H =1.80 8.66 = 15.59 [ft] 1/10 rms × H¯ =2.36 H =2.36 8.66 = 20.44 [ft] 1/100 rms × The total number of waves is 16000. The probability to have the maximum wave is 1 P (H H )= ≥ max 16000 1 P (H H )=1 =0.99994 ≤ max − 16000

56 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

10 (a) 5

0 0 5 10 15 20 25 30 10 (b) 5 Occ.%per Occ.%per

unit wave height unit0 wave height 0 5 10 15 20 25 30 0.1

) (c) H

( 0.05 p

0 0 5 10 15 20 25 30 H, [ft]

Figure 4.5: Distribution of waves height of the record. (a) Stem plot of the data, (b) bar plot or histogram of the data, and (c) Rayleigh probability density function of the given wave record

0.1

0.08 ) 0.06 H ( p 0.04 Area of 1/3 0.02 highest 0 0 5 10 15 20 25 H, [ft] 1

0.8

) P =2/3

H 0.6 (

P 0.4

0.2

0 0 5 10 15 20 25 H, [ft]

Figure 4.6: Rayleigh probability density function and cumulative probability function of the given wave record

The most probable maximum wave height may be obtained by ﬁnding the area under the probability density

57 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle function, p(H).

Hmax 2H H2 exp dH =0.99994 H2 −H2 Z0 rms rms Consider that: H2 2H u = 2 = du = 2 dH Hrms ⇒ Hrms therefore 2 2 Hmax 2H H2 Hmax/Hrms exp dH = exp( u)du =0.99994 H2 −H2 − Z0 rms rms Z0 H =3.118H =3.118 8.66 = 27 ft max rms × It may also be found more easily from the probability distribution P (H) as follows.

H2 1 exp max =0.99994 = H =3.118H =3.118 8.66 = 27 ft − − H2 ⇒ max rms × rms Probability distribution for the wave period The probability distribution of wave period is much more diﬃcult to deal with. On the basis of experimental data, Bretschneider (1959) has suggested that the wave length may be characterized by a Rayleigh distribution, and consequently in deep water so also the squared of wave period. The distribution of wave period in deep water is thus expressed as:

4 1 exp 0.675 T/T¯ For T 0 P (T )= − − ≥ (4.51)  h i  0 otherwise

4  2.7 T 3/T¯4 exp 0.675 T/T¯ For T 0 p(T )= − ≥ (4.52)  h i  0 otherwise  4.4 Spectral description of Ocean waves

The concept of a spectrum can be attributed to Newton, who discovered that sunlight can be decomposed into a spectrum of colors from red to violet, based on the principle that white light consists of numerous components of light of various colors (wave length or wave frequency). Energy spectrum means the energy distribution over frequency. Spectral analysis is a technique of decomposing a complex physical phenomenon into individual components with respect to frequency. Frequency analysis deals mainly with the evaluation of the distribution of wave energy among various frequencies and directions. The proﬁle of a small amplitude wave traveling at an angle θ with respect to the x axis may be presented by

η(x,y,t)= A cos k(x cos θ + y sin θ) ωt + ε (4.53) − h i where ε is a phase shift. If the observation location is selected at x&y = 0 then η(t)= A cos(ωt ε) (4.54) − The average wave energy per unit area is: 1 1 E = ρgA2 = ρgH2 (4.55) 2 8

58 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

The variance of the surface elevation of a sinusoidal wave is: 1 T A2 V ar[η(t)] = σ2 = E (η(t)¯η(t))2 = E η(t)2 = A2 cos2 (ωt ε)dt = (4.56) η T − 2 Z0 h i The simplest and the most natural representation of the confused sea surface would be the linear superposition of many time harmonic waves traveling in various directions. A simple illustration of such a representation is given in Fig. 4.7. The combination of 13 elementary components, Fig. 4.7-a, are summed a final profile, Fig 4.7-b. The surface profile can be given as:

N η(x,y,t)= A cos k (x cos θ + y sin θ ) ω t + ε (4.57) i i i i − i i i=1 X h i where the direction θi and εi are in the range of π to π. 1 2 The plot of 2 A against f, the wave frequency, is given in Fig. 4.7-c. This is called the energy spectrum or variance diagram. Since the frequency components are discrete, the plot as shown in Fig. 4.7 is line diagram. The variance diagram can be converted to variance desity spectrum by dividing to ∆f, a frequency band width. 1 A2 S(f) = 2 (4.58) η ∆f The value of ∆f depends on signal recording duration, it is considered ∆f =1 Hz in Fig. 4.8. In reality an irregular wave is composed of inﬁnite number of linear waves with diﬀerent frequency. When ∆f in (4.58) approaches zero, the variance spectrum becomes a continuous curve, Fig. 4.8.

4.4.1 Spectral density function

For a continuous range of frequencies where ωi , The equation (4.57) can be put in integral form as −∞ ≤ ≤ ∞ ∞ π η(x,y,t)= A(ω,θ)cos k(x cos θ + y sin θ) ωt + ε(ω,θ) dωdθ (4.59) −∞ −π − Z Z h i where A(ω,θ) is called the directional amplitude density function. The plot of A(ω,θ) against f or ω is called the continuous directional amplitude spectrum. If the waves recorded at a certain point say at the origin, x = y = 0 and it is assumed that all the waves has a single direction, say along the x-axis, then (4.59) may be given in the form

N η(t)= A cos ω t ε . (4.60) i i − i i=1 X and for a continuous range of frequency, we can write in integral form as

∞ η(t)= A(ω)cos ωt ε(ω) dω (4.61) − Z−∞ where A(ω) is called the amplitude density function. The Fourier expansion series play a very important role in the study of spectral analysis. If η(t) is a piecewise continuous function deﬁned over d t d +2p where 2p is the period of the function η(t), then it can be written that ≤ ≤ ∞ inπt/p η(t)= cne (4.62) −∞ X

59 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

(a) 1 2 3 4 5 6 7 8 9 10 11 12 13

0 0.5 1 1.5 2 t,s

10 (b)

5 η,ft

−5 0 0.5 11.5 2 t,s

1 (c) 0.9

0.8

0.7

2 0.6 ,ft 2 0.5 A 1 2 0.4

0.3

0.2

0.1

0 1 2 3 4 5 6 7 8 9 10111213 f,Hz Figure 4.7: a.) Spectral components of the surface wave, b.) Surface elevation, c.) Energy spectrum 60 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

0.9

0.8

] 0.7 s · 2

ft 0.6 [ , 2 f

A 0.5 ∆ 1 2

0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 f, [Hz]

Figure 4.8: Energy density spectrum

where (4.62) is known as the complex Fourier series and cn are the complex Fourier coeﬃcients. The 2π π fundamental period is T = 2p and therefore the fundamental frequency is ω = T = p and fundamental ω 1 1 frequency in cycles is f = 2π = T = 2p . The complex Fourier coeﬃcients are obtained as follows:

1 d+2p c = η(t)e−inπt/pdt and n =0, 1, 2, (4.63) n 2p ± ± ··· Zd The Fourier series can be also given in the sine and cosine functions as ∞ a nπt nπt η(t)= 0 a cos + b sin (4.64) 2 n p n p 1 X where it can be shown that: 1 c = a 0 2 0 1 c = (a ib ) n 2 n − n 1 c− = (a + ib ) (4.65) n 2 n n It is of interest to know the average wave energy delivered per cycle. If E represent the average wave energy per cycle, then h i 1 E = ρg η2(t) (4.66) h i 2 h i where 1 d+2p η2(t) = η2(t)dt (4.67) h i 2p Zd

61 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle is the variance of η(t) over a wave cycle. Using Parseval’s identity, it can be written that ∞ a2 1 η2(t) = 0 + a2 + b2 h i 4 2 n n n=1 ∞X 2 = c +2 c c− 0 n × n n=1 X∞ 2 2 = c0 +2 cn . (4.68) n=1 X Substituting (4.66) in (4.68), the average wave energy per cycle can be given as

∞ 1 E = ρg c2 +2 c2 (4.69) h i 2 0 n n=1 ! X The foregoing analysis based on the Fourier expansion series should be modified for random waves due to the fact that there is no finite time period of wave. In fact the period of random waves is infinite. It is assumed that d = p and the complex Fourier series (4.62) may be given in the form − ∞ inπt/p ηp(t)= cne −∞ X 1 p 1 p c = η (t)e−inπt/pdt = η (τ)e−inπτ/pdτ n 2p p 2p p Z−p Z−p where ηp(t) is a periodic function of period 2p. Substituting the relationship for complex Fourier coefficients cn into ηp(t)

∞ p 1 inπτ/p −inπt/p ηp(t) = ηp(τ)e dτ e −∞ 2p −p X Z ∞ p 1 inπτ/p −inπt/p π = ηp(τ)e dτ e −∞ 2π −p p X Z and deﬁning that nπ (n + 1)π π ω = , ω = , and δω = ω ω = , n p n+1 p n+1 − n p we obtain the following equation. ∞ 1 p η (t)= η (τ)e−iωnτ dτ eiωnt∆ω p 2π p −∞ −p X Z Let p goes to inﬁnity such that the non-periodic limit of ηp(t) becomes η(t). If p then ∆ω 0 and the summation over the frequency can be replaced by integral for the continuous random→ ∞ wave records→ or wave elevations. ∞ 1 ∞ η(t)= η(τ)e−iωτ dτ eiωtdω (4.70) 2π Z−∞ Z−∞ The equation (4.70) is known as the Fourier integral. We may rewrite (4.70) in the following form. 1 ∞ ∞ η(t) = η(τ)eiω(t−τ)dτdω 2π Z−∞ Z−∞ 1 ∞ ∞ = η(τ) cos ω(t τ)+ i sin ω(t τ) dτ dω 2π − − Z−∞ Z−∞ 62 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Taking into account oddness of the sine function and the evenness of the cosine function, we can write that: ∞ sin ω(t τ)dω =0 −∞ − Z ∞ ∞ cos ω(t τ)dω =2 cos ω(t τ) dω − − Z−∞ Z0 Therefore: 1 ∞ ∞ η(t)= η(τ)cos ω(t τ)dτdω. π − Z0 Z−∞ Expanding the cosine function and deﬁning: ∞ a(ω)= η(t)cos ωt dt , −∞ Z ∞ b(ω)= η(t) sin ωt dt (4.71) Z−∞ we can write 1 ∞ η(t)= a(ω)cos ωt + b(ω) sin ωt dω. (4.72) π 0 Z h i The total energy of wave per unit surface area in the wave record is 1 ∞ E = ρg η2(t)dt. (4.73) 2 Z−∞ Using (4.72), it can be written that 1 ∞ E = ρg η(t).η(t)dt 2 Z−∞ 1 ∞ 1 ∞ = ρg η(t) a(ω)cos ωt + b(ω) sin ωt dω dt 2 −∞ π 0 Z Z h i ρg ∞ ∞ ∞ = a(ω) η(t)cos ωt dt + b(ω) η(t) sin ωt dt dω 2π Z0 Z−∞ Z−∞ Using (4.71), the total wave energy per unit surface area is ρg ∞ E = a2(ω)+ b2(ω) dω (4.74) 2π Z0 which, alternately, is written as ρg ∞ E = A2(ω)dω (4.75) 2π Z0 where A2(ω)= a2(ω)+b2(ω). Comparing (4.73) with (4.75), we obtain the Parsavel’s identity for a continuous non-periodic random wave function η(t). ∞ 1 ∞ η2(t)dt = A2(ω)dω (4.76) π Z−∞ Z0 This equation really gives rise to the concept of the wave energy spectrum. The amplitude A(ω) has a unit of Length-time and is a continuous function of the frequency ω. If the wave record is obtained over the time length Ts, the mean square value (variance) of η(t) can be given as follows. 1 Ts η2(t) = η2(t)dt (4.77) h i T s Z0 63 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Therefore, the average wave energy per unit surface area can be given as

ρg 1 Ts E = η2(t)dt. (4.78) h i 2 T s Z0 Doing some manipulations, we finally obtained the average wave energy in the following form, Rahman [3]. ρg ∞ A2(ω) E = dω (4.79) h i 2π T Z0 s The spectral density function is defined as A2(ω) S(ω)= (4.80) πTs where S(ω) has a unit of Length2-time. The total energy is obtained from the area covered by the energy density curve as a function of frequency. 1 ∞ E = ρg S(ω)dω (4.81) h i 2 Z0 There are two methods to compute the spectral density function from an ocean wave records: the autocorrelation method and the fast Fourier transform (FFT) method. However, the procedure of extracting spectra from wave records is an evolving field and is beyond the scope of this lecture. Chackrabarti [1] has given a good account in extracting the wave spectral density function from an ocean wave record. There are several widely used conventions in the display of energy spectrum. We have illustrated the energy density spectrum in Fig. 4.9 that is the most popular one, S(ω). The advantage of the energy density spectrum is that the area under the curve gives the total energy of the wave system. The representation of the energy density spectrum could be in a linear scale, a log-log scale or a semilog scale. The wave spectrum may be represented as a function of cyclic frequency f or circular frequency ω. It should be considered that: S(f)=2πS(ω) (4.82) The total energy present in the ocean waves is not distributed equally throughout the range of frequencies;

) 15 ω ( S

0 0 0.3 0.6 0.9 1.2 1.5 ω Figure 4.9: A one-sided energy density spectrum instead, in every spectrum, the energy is concentrated around a particular frequency (fmax), that corresponds to a certain wind speed. The frequency range is decrease as the wind speed in increases. It is customary to represent the spectrum as a one-sided spectrum, as shown in Fig. 4.9. The two sided spectrum covered both positive and negative side of the frequency, ω. The diﬀerence between a two-sided spectrum and a one-sided spectrum as illustrated in Fig. 4.10 is quite important. Note that the two-sided

64 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle spectrum is symmetric about the origin, covering both negative and positive frequencies to account for all wave energy from 4 to +4. But, it is customary in ocean engineering to present the spectrum as a one-sided spectrum. This requires− that the spectral density ordinates of two-sided spectrum be doubled in value if only the positive frequencies are considered.

Figure 4.10: Deﬁnition of one- and two-sided wave spectrum

4.4.2 Spectral properties of ocean waves if it is assumed that the wave train is generated by constant wind blowing over a ﬁtch limited, X surface area of ocean with depth h, the frequency spectrum S(ω) of surface wave is a function of

S(ω)= f(ω,ρa,ρw,νa,νw, Υ,g,U∞,fc,t,X,h) (4.83) where:

- ρa and ρw are the density of air and water,respectively,

- νa and νw are kinematic viscosity of air and water, respectively, - Υ is the surface tension of water and air interface, - g is the gravitational acceleration,

- U∞ is the wind speed at the upper limit of the atmospheric,

- fc is the Coriolis parameter, - t is the duration of wind blowing, and - ω is the circular frequency. It is assumed that:

1. Wave motion is irrotational and the inﬂuence of molecular viscosity νw can be neglected. 2. Only the gravitational part of the spectrum S(ω) is considered and the inﬂuence of surface tension is neglected and frequency ω is restricted to the range

4ρ g3 1/4 ω <<ω = w (4.84) Υ Υ

3. Nonlinear interaction between spectral components are neglected.

65 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

4. The energy transfer from the wind to the sea surface in only due to the atmospheric boundary layer adhering to the sea surface. As the characteristics height of the boundary layer is hp = U∞/fc and wave number k satisﬁes the relation khp >> 1, we obtain the following condition for the wave frequency ω:

gf ω >> c (4.85) U ∞

Assuming that (4.85) is satisfied, the Coriolis parameter fc can be omitted in (4.84). 5. In fact, the first and the second assumptions imply that the main part of the wind energy is transferred to waves by the normal stresses. Moreover, we assume that high-frequency wave components, when ω >>ωΥ, play a negligible role in distribution these stresses along the sea surface. This is probably not true for the initial stage of wave generation, when the capillary waves are generated first on the initially calm sea surface. However, by neglecting the initial stage of wave generation, i.e., t>ti(ω), the parameters νa and Υ can be neglected. 6. The influence of molecular air viscosity is important only at the initial stage of wave growth. Then the viscous sublayer may redistribute the wind stresses, which are responsible for energy transfer to surface gravity for ω <<ωΥ. However, following Phillips (1957) and Miles (1962), this mechanism (and also parameter νa) can be neglected from consideration, assuming that:

gU 1/2 ω << ∞ (4.86) ν a

If it is considered that the ratio of the air to water densities is a constant, the equation (4.83) becomes: S(ω)= f(ω,g,U,t,X,h) (4.87) The wind speed is denoted by U for brevity. It can be written in non-dimensional variables as: S(ω)g3 ωU gt gX gh = g , , , (4.88) U 5 g U U 2 U 2 Using the same arguments, the similarity law for a multi-directional sea becomes: S(ω)g3 ωU gt gX gh = g , , , , Θ (4.89) U 5 g U U 2 U 2 gt gh In the case of a fully-developed unidirectional sea in deep water, i.e. when U , U 2 , and gX → ∞ → ∞ unlimited sea surface, i.e. U 2 , the non-dimensional density spectrum depends only on the non- dimensional wind velocity. → ∞ S(ω)g3 ωU = φ (4.90) U 5 g 4.4.3 Typical wave energy density spectrum There are quite a large number of typical wave energy density spectrum. Some of them are expressed in this part.

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Pierson-Moskowitz spectrum In 1964 Pierson and Moskowitz proposed a formula for an energy spectrum distribution of wind generated sea state. The spectrum is on the basis of the similarity theory of Kitaigorodski, (4.89) and more accurate recorded data. This spectrum commonly known as P-M model. It was developed primarily for oceanographic use and in fact is a basic element in the forecasting of storm waves. It is intended to represent the point spectrum of a fully-developed sea, (4.90), that is, fetch and duration are great and no contaminating swell from other generating areas. Pierson and Moskowitz found that the function φ can be written as:

ωU ωU ωU −4 φ =0.0081 exp 0.74 (4.91) g g − g " # Substituting (4.91) in (4.90), the P-M spectrum model is obtained as:

αg2 ωU −4 S(ω)= exp 0.74 (4.92) ω5 − g " # where α =0.0081. The shape of the P-M spectrum is controlled by a single parameter wind speed U. The P-M spectrum is shown in Fig. 4.11 for ﬁve wind speed 20, 25, 30, 35, and 40 kt.

14 U = 40 kt ] s

· 12 2 m [ 10 , ) ω ( U = 35 kt S 8

4 U = 30 kt

2 U = 25 kt U = 20 kt 0 0 0.5 1 1.5 2 2.5 3 3.5 ω

Figure 4.11: Pierson-Moskowitz spectrum at diﬀerent wind speed

The experimental spectra given by P-M spectrum yield: Uω Uω Uf p =0.879, and p = p =0.13 (4.93) g 2πg g

67 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

where ωp is the peak frequency where the wave spectral energy reaches its maximum. The peak frequency dS(ω ωp is obtained by condition dω = 0. Using (4.93) and substituting in (4.92), it is obtained:

αg2 5 ω −4 S(ω)= exp (4.94) ω5 −4 ω " p #

The JONSWAP Spectrum The JONSWAP spectrum extends the P-M spectrumto include fetch-limited seas. This spectrum is based on an extensive wave measurement program (Joint North Sea Wave project) carried out in 1968 and 1969 in North Sea. The JONSWAP spectrum received almost instant recognition and become very well known in international literature. According to Hasselmann et. al 1973, the resulting spectral model takes the form:

αg2 5 ω −4 S(ω)= exp γδ (4.95) ω5 −4 ω " p # Where the term γδ is a peak enhancement factor, added to the P-M spectrum, to represent a narrow, more peaked spectrum which is typical for a growing sea. The notation δ is given in the form:

2 (ω ωp) δ = exp −2 2 (4.96) "− 2σ0ωp # Spectrum (4.95) contains ﬁve parameters, i.e. α,γ,ω , and σ = σ′ for ω ω and omega > ω , which p 0 0 ≤ p p should be know in prior. The γ parameter describes the degree of peakedness and σ0 parameter describes the width of the peak region. The mean JONSWAP spectrum yields: γ =3.3; ′ σ0 = σ0 =0.07 for ω ωp; ′′ ≤ σ = σ0 =0.09 for ω>ωp; gX −0.22 α =0.076 ; and U 2 g gX −0.33 ω =7π p U U 2 The JONSWAP spectrum are illustrated for U = 20 m/s when X = 40, 80, 120, 160 and 200 km in Fig. 4.12.

The JONSWAP spectrum and the P-M spectrum are compared for two diﬀerent regimes in Fig. 4.13-a and -b when the fetch lengths are X = 25 km and X = 200 km, respectively.

Bretschneider spectrum Bretschneider derived a spectrum model based on the assumption that the spectrum is narrow-banded and the individual wave height and wave period follow Rayleigh distribution. This spectrum is given in term of the signiﬁcant wave height and period rather than the wind speed. The Bretschneider wave density spectrum may be written in the following form.

ω4 ω −4 S(ω)=0.258 H2 s exp 1.03 (4.97) s ω5 − ω " s # where Hs is the significant wave height and ωs =2π/Ts is the significant wave frequency. The notation Ts is the significant wave period. It is defined as:

Ts =0.9528Tp (4.98)

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9 X = 20 km 8

7 s

· 160 2 6 U = 20 m/s , m

) 5 ω (

S 120 4

3 80 1

1 40

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 ω, rad/s

Figure 4.12: The JONSWAP spectrum at various limited fetch length areas

0.12 a P-M Spectrum 0.1 JONSWAP Spectrum s ·

2 0.08 , m

) 0.06 ω (

S 0.04

0.02

0 0 0.5 1 1.5 2 2.5 3 ω, rad/s

2.4 b 2 P-M Spectrum

s JONSWAP Spectrum ·

2 1.6 , m

) 1.2 ω (

S 0.8

0.4

0 0 0.5 1 1.5 2 2.5 3 ω, rad/s

Figure 4.13: Comparison of the P-M spectrum with the JONSWAP spectrum: a) X = 25 km, b) X = 200 km

where Tp is the peak period. Taking into account (4.98), the Bretschneider wave density spectrum may also be written as function of peak frequency as follows.

5 ω4 5 ω −4 S(ω)= H2 p exp (4.99) 16 s ω5 −4 ω " p # The following MATLAB m-ﬁle is used to plot the spectrum at diﬀerent sea stats. clear all;

69 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Sea state Hs,m Tp, s 2 0.3 6.3 3 0.9 7.5 4 1.9 8.8 5 3.3 9.7 6 5.0 12.4

Table 4.2: The signiﬁcan wave height and peak period at various sea state condition

Tm = [6.3 7.5 8.8 9.7 12.4]; wm = 2*pi./Tm Hs = [0.3 0.9 1.9 3.3 5.0]; w = [0:0.005:3]; for i = 1:5 B(i) = 5/16*Hs(i)^2*wm(i)^4; A(i) = -5/4*wm(i)^4; S(i, :) = B(i)./w.^5.*exp(A(i)./w.^4); plot(w, S(i, :)); hold on end A plot of the Bretschneider spectral model is given in Fig. 4.14.

4.5

4 SS 6

3.5 s

· 3 2

, m2.5 ) ω (

S 2 SS 5 1.5

0.5 SS 4

0 0 0.5 1 1.5 2 2.5 3 ω, rad/s

Figure 4.14: The Bretschneider spectrum at various sea states

ITTC spectrum The International Towing Tank Conference (1966) proposed a modiﬁcation to P-M spectrum in terms of signiﬁcant wave height and zero crossing frequency. It has been written as αg2 4αg2 S(ω)= exp (4.100) ω5 −H2ω4 s where 0.0081 g/σ α = 2 , k = (4.101) k 3p.54ωz 70 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

in which σ = √m0 = Hs/4 is the standard deviation (r.m.s. value) of the water surface elevation.

4.4.4 Directional spectral function The most widely used and accepted method of studying wind generated waves is an examination of the spectra at a single point. However, single point measurements are not sufficient to describe the full nature of wind- induced surface waves. Due to the complicated energy transfer from the atmosphere to the sea and due to wave-wave interactions the resulting surface waves are multidirectional. Only part of the wave energy aligned with the wind direction. More specifically, wave energy associated with the frequency ω = ωp is primarily propagated in the direction of the wind, whereas wave energy associated with lower and higher frequencies is distributed over a range of directions. Wave multidirectionality is also result of the superposition at a given point of various wave train, which may be generated by different remote atmospheric forcing systems, Massel [2]. Due to the limitation of the observational methods, knowledge on the directional spectrum is relatively poor compared to the frequency spectrum. The basic approach is to multiply the frequency spectrum S(ω) by an empirically determined directional spreading function D(Θ; ω,p1,p2, ) to find the directional wave spectrum. ··· S(ω, Θ) = S(ω)D(Θ; ω,p ,p , ) (4.102) 1 2 ··· Where p1,p2, represent the various parameters associated with the various directional spreading functions. There are four··· basic form of the directional spreading function: the cosine-power function, the exponential function, the exponential series and the hyperbolic function. On the basis of a field study with pitch-and-roll buoy, Louguet-Higgins et. al (1961) developed a elabo- rated for of cosine spreading function as follows. 22p−1Γ2(p + 1) 1 D(Θ,p)= cos2p ( Θ) (4.103) πΓ(2p + 1) 2 Where κ ω 4.06 for ω<ω , p = 100.99 , and κ = m (4.104) ω 2.34 for ω ωm m − ≥ A three dimensional form of the JONSWAP spectrum using the spreading function (4.103) is shown in Fig. 4.15.

4.5 Application of wave energy density spectrum

The short-term ocean wave are treated as random process. It is also assumed that the statistics of ocean waves are stationary and ergodic. It indicate that the ensemble average E(η(t1)) at same time t1 of an infinite number of finite length time series η (t ), η (t ), η (t ), , η∞(t ) is equivalent to the temporal 1 1 2 1 3 1 ··· 1 average over all times of an infinitely long single time series η (t ), η (t ), η (t ), , η (t∞). In other word, 1 1 1 2 1 3 ··· 2 a realization of η(t), say η1(t), is typical of all other possible realizations. It is also assumed that the real ocean waves composed of various sinusoidal waves of different amplitudes, frequencies, direction and phases. However, the stationary and ergodic assumptions indicate that sinusoidal components of ocean waves are the same regardless of times and places and differ from one record to another only in the phase orientation and thereby keeping the energy of the wave system constant. In other word, the statistical characteristics of the sea state, that is, the energy spectrum or wave spectrum, will remain the same. Application of wave energy density spectrum is discussed in this part.

4.5.1 Simulation of wave proﬁle The average energy per unit area is given for a wave system as 1 1 ∞ E = ρg η2(t) = ρg S(ω)dω (4.105) h i 2 h i 2 Z0 71 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

0.4

0.3 ) s 2 m ( ,

) 0.2 f,ϑ ( S

0.1

0 π

π 2 1 0.8 0 0.6 π 0.4 − 2 0.2 −π ϑ 0 f, (Hz)

Figure 4.15: The JONSWAP spectrum

where η2(t) is the wave elevation variance. The area under the wave spectrum is the variance of the ocean surfaceh elevation.i We may consider a wave profile given by a cosine function at a fixed point in space, say x = 0, as η(t)= A cos ωt (4.106) where A is the wave amplitude and the period is 2p = T =2π/ω. The variance of the cosine wave is already given in (4.56). The computation of the variance for one cycle will suffice and therefore 1 2π A2 σ2 = η(t)2 = A2 cos2 ωt d(ωt)= (4.107) η h i 2π 2 Z0 The contribution to the variance (or energy) within the frequency range ω to ω + dω is S(ω)dω. This may be written according to (4.107) as

ω +dω n 1 S(ω)dω = A 2. (4.108) 2| n| ω Xn At a frequency ω1 = 2πf1, the energy density is S(ω1) as shown in Fig. 4.17. If we consider a frequency band of width ∆ω1 as given in Fig. 4.17, the wave height at frequency ω1 is obtained as follows according to (4.105) and (4.108)

H(ω1)=2 2S(ω1)∆ω1 (4.109) and the correspondingp wave period is 2π T = (4.110) ω1

72 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

6 .s 2 5 , ft )

ω 4 ( S 3 S(ω1)

1 ∆ω

0 0 0.3 0.6ω1 0.9 1.2 1.5

rad ω, s Figure 4.16: Energy spectrum at a given wind speed and the energy density of a particular frequency

where (H,T ) is the wave height-period pair. It is assumed that H =2A where A is the wave amplitude. A phase angle associated with each pair of height and period is chosen uniformly distributed in the range of (0, 2π) by a random number generator, Rn as

ε(ω1)=2πRn. (4.111) Then, for a given horizontal coordinate, x, which is the location at which the wave proﬁle is desired, and time, t, which is incremented, the wave proﬁle is computed, (Chakrabarti [1]).

N H(i) η(x, t)= cos k x ω t + ε (4.112) 2 i − i i i=1 X th Where ki =2π/λi is the wave number correspond the the i frequency ωi. Similarly, λi is the wave length th correspond the the i frequency ωi. Since, the ocean waves are dispersive, the wave length and wave number can be obtained from the dispersion equation.

2 ωi = gki tanh kih (4.113) The quantity N is the total number of frequency bands of width ∆ω dividing the total energy density. This method in simulating the ocean surface profile requires a given spectrum model or an actual input wave energy density spectrum. The spectrum curve is divided into several equal divisions. It is not necessary to divide equally. It is done only for computational convenience. The number of divisions should be at least 50 to assure randomness. The application of 200 components duplicates the spectrum accurately. The value of wave height, of course, will differ for various value of ∆ω. However, as long as ∆ω is small, this method produce a satisfactory random wave profile.

73 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

4.5.2 Computation of the average heights characteristics The area under the wave spectrum is ∞ η2(t) = S(ω)dω = m (4.114) h i 0 Z0 and therefore, m0 is the value of the wave elevations squared or √m0 is the root mean square (rms) value of the irregular ocean surface elevations. Thus, the area under the wave spectrum yields the mean of deviation squared. The signiﬁcant wave amplitude or height or the other averages are multiples of rms, (i.e a const. √m0). Such multipliers depend on the probability distribution of ocean wave properties. For a× narrow-band Gaussian ocean wave whose components are in random phase, the wave amplitude and hence the wave height follow a Rayleigh distribution. Since any single wave is closely sinusoidal for a narrow-band motion, the variance of the surface elevation may be expressed as the average of integrals taken 2 2 over each wave. Therefore according to (4.56), it can be σi = Hi /8. In this manner we can write that ∞ η2(t) = S(ω)dω = m h i 0 0  2 R1 N 2  η (t) = σi  h i N i=1  = Hrms =2√2m0 (4.115)  2 P ⇒  2 Hi  σi = 8

 2 1 N 2  Hrms = N i=1 Hi   The probability cumulativeP distribution function and the probability density function of the wave height is given in (4.39) and (4.40) for a a narrow-band Gaussian ocean. They may be written in the following form as a function of m0. H2 P (H)=1 exp (4.116) − −8m 0 H H2 p(H)= exp (4.117) 4m −8m 0 0 For a Rayleigh distribution of wave height, the average wave height, the signiﬁcant wave height (or average of one-third highest waves), the average of one-tenth highest waves and the average of one-hundred highest waves are:

H¯ = H1 =0.886 Hrms =2.506√m0 (4.118) ¯ Hs = H1/3 =1.416 Hrms =4.005√m0 (4.119) ¯ H1/10 =1.80 Hrms =5.091√m0 (4.120) ¯ H1/100 =2.36 Hrms =6.672√m0 (4.121)

4.5.3 Arbitrary wave spectra The result for the wave distribution is for the special case of a narrow-band spectrum. The probability distribution of wave amplitude depends upon one additional parameter ǫ terms the spectral bandwidth parameter. It is defined as m2 1/2 ǫ = 1 2 (4.122) − m m 0 4 where m0 is the area under the wave spectrum and m2 and m4 are the second and fourth spectral moment, respectively. In general the n th spectral moment is defined as − ∞ n mn = ω S(ω)dω (4.123) Z0 74 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle hence ∞ ∞ 2 4 m2 = ω S(ω)dω and m4 = ω S(ω)dω. (4.124) Z0 Z0 The spectral bandwidth parameter ǫ takes values between 0 and 1 and for a narrow-band spectrum ǫ 0. Based on the definition for ǫ, the oceanographers have introduce a correction factor as follows. →

1/2 CF = 1 ǫ2 (4.125) − by which the above constant to compute the wave height characteristics or the other wave characteristics based on the narrow-band assumption should be multiplied to take into account the broadness of the spectrum.

4.5.4 Wave period The probability distribution of wave periods for an arbitrary spectrum is complex. The distribution of wave period in deep water according to Bretschneider (1969) follows a Rayleigh distribution.

4 1 exp 0.675 T for T 0 P (T )= − − T¯ ≥ (4.126) ( 0h i otherwise 3 4 2.7T exp 0.675 T for T 0 p(T )= T¯4 − T¯ ≥ (4.127) ( 0h i otherwise Where T¯ is a mean period deﬁned as m T¯ = 0 . (4.128) m1 The more appropriate distribution of the wave period is derived by Longuet-Higgins. It can be found in Chakrabarti [1]. The expression for the average zero upcrossing period T¯z is given as 1 m T¯ = =2π 0 . (4.129) z N m z r 2

The crest-to-crest period T¯c is defined as m T¯ =2π 2 . (4.130) c m r 4 There are also two other characteristic period often used in statistics. One is the peak period, Tp, defined as the period at which the energy density spectrum peaks. The other one is the significant wave period, Ts, defined as the average of the highest one-third waves in the record. m T =2π 0 . (4.131) s m r 1 Taking into account (4.129) and (4.130), the spectral width parameter ǫ may be given as

1/2 T¯ 2 ǫ = 1 c (4.132) − T¯ " z #

75 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Example - 3

The International Towing Tank Conference (1966) proposed a theoretical spectrum in terms of significant wave height and zero crossing frequency. αg2 4αg2 S(ω)= exp ω5 −H2ω4 s where 0.0081 g/σ α = 2 , k = k 3p.54ωz in which σ = √m0 = Hs/4 is the standard deviation (r.m.s. value) of the water surface elevation. The relationship between wind speed and significant wave height is defined by a curve having the following ordinates: wind speed Significant wave height kt ft 20 10 30 17.2 40 26.5 50 36.6 60 48 The significant wave height versus the wind speed is shown in Fig. 4.17.

35 , ft s 30 H

10 10 20 30 40 50 60 Wind speed kt

Figure 4.17: Signiﬁcant wave height versus the wind speed

1. Plot the ITTC spectral model for the wave density energy for a wind speed of 31 kt. 2. Write a computer program to compute the following wave characteristics:

76 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

the one-tenth of highest H¯ ; • 1/10 the one-hundred of highest H¯ ; • 1/100 zero upcrossing average period T¯ ; and • z the average crest to crest period T¯ . • c Assumed that the wave height probability distribution is Rayleigh. 3. If the wave height histogram does not follow Rayleigh distribution, ﬁnd the the signiﬁcant wave height Hs. 4. Find the most probable largest wave amplitude.

Solution-1

The ITTC spectrum is αg2 2αg2 S(ω)= exp ω5 −H2ω4 s where 0.0081 g/σ α = 2 , k = k 3p.54ωz in which σ = √m0 = Hs/4 is the standard deviation (r.m.s. value) of the water surface elevation. The signiﬁcant wave height for a wind speed of 31 kt is:

Hs = 18.5 ft It is taken into account that k = 1. The ITTC spectrum for a wind speed of 31 kt is given in Fig. 4.18.

.s 40 2 , ft ) 30 ω ( S

0 0 0.5 1 1.5 2 2.5 3 ω, s−1

Figure 4.18: ITTC wave Energy density spectrum at the wind speed of 31 kt

77 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Solution-2

The following MATLAB m.ﬁle is written to obtain the solution for the wave height averages and wave periods. a = 0.0081; %a = alpha g = 32.2; Hs = 18.5; A = a*g^2; B = 4*a*g^2/Hs^2; w = [0.1:0.001:10]; Sw = A./w.^5.*exp(- B./w.^4); plot(w, Sw) clear w Sw Sw = @(w) A./w.^5.*exp(- B./w.^4); m0 = quad(Sw, 0, 10) Sw2 = @(w) A./w.^3.*exp(- B./w.^4); m2 = quad(Sw2, 0, 10) Sw4 = @(w) A./w.^1.*exp(- B./w.^4); m4 = quad(Sw4, 0, 10) H_ave = 2.506*sqrt(m0) H_s = 4.005*sqrt(m0) H_110 = 5.091*sqrt(m0) H_1100 = 6.672*sqrt(m0) Tz = 2*pi*sqrt(m0/m2) Tc = 2*pi*sqrt(m2/m4) e = sqrt(1 - m2^2/m0/m4) CF = sqrt(1 - e^2) H_s1 = H_s*CF

The solution are:

2 m0 = 21.3904 ft 2 −2 m2 = 11.8363 ft .s 2 −4 m4 = 22.9997 ft .s H = 11.5902 ft

Hs = 18.523 ft

H1/10 = 23.5458 ft

H1/100 = 30.8578 ft

Tz =8.4466 s

Tc =4.5074 s

Solution-3

In the case that the wave height probability distribution is not a Rayleigh distribution, the wave signiﬁcant height is

m2 1/2 ǫ = 1 2 =0.8457 − m m 0 4 1/2 CF = 1 ǫ2 =0.5336 − H = H .CF = 18.523 0.5336 = 9.8846 ft s1 s ×

78 Chapter 4 Real ocean Waves Dynamics of Marine Vehicle

Solution-4

It is assumed that the number of recorded waves are N = 1000. Therefore,

H =2.63H =7.44√m =7.44 √21.3904 = 34.4 ft max rms 0 × It is when the wave height probability distribution is Rayleigh. Since it is not follow the Rayleigh distribution exactly, the probable maxima is H∗ = H .CF = 34.4 0.5336 = 18.36 ft max max ×

79 Bibliography

[1] Chakrabarti, S. K., Hydrodynamics of offshre structures, Springer-Verlag Berlin, Heidelberg, 1987 [2] Massel, Stanislaw R., Ocean surface waves: Their physics and prediction, World Scientific publishing Co. Pte. Ltd., 1996 [3] Rahman, M., Water waves, relating modern theory to advanced engineering applications, Oxford Uni- verisit press, 1994 [4] Sarpkaya, T. and Isaacson, M., Mechanics of wave forces on offshore structures, Van Nostrand Reinhold Company, 1981 [5] Wikipedia, The free encyclopedia,, http:en.wikipedia.org/wiki/Beaufort-scale