A SIMPLE INTRODUCTION TO WATER WAVES

1 A SIMPLE INTRODUCTION TO WATER WAVESDIMITRIOS theory of the WATER WAVES is the main subject in coastal hydro-dynamics and plays a significant role in applied mathematics and in these notes we present the basics of the WATER wave theory. Specifically,after introducing briefly the basic concepts of continuum mechanics, we derivethe physical laws describing the physics of an inviscid, incompressible fluid,namely the Euler equations. Euler equations are the governing equations ofwater WAVES , but because of the great difficulties in the theoretical and nu-merical studies of these equations, SIMPLE , approximate mathematical modelshave been derived instead.

2 The models are usually simplified so as to be validfor specific types of WAVES are usually divided into categories de-pending on their amplitude and their wavelength compared to the WATER WAVES such as the Tsunami WAVES are generated in the deep ocean aswaves of small amplitude with large wavelength, but as they approach theshoreline they grow in amplitude while their wavelength is decreased. Here,we will derive models for long WAVES with or without using the small words and phrases: WATER WAVES ; Boussinesq equations; Serre equa-tions; KdV equation; BBM equation; dispersive WAVES ; solitary to KinematicsKinematics is the branch of classical mechanics that describes the motion ofbodies and systems of bodies without consideration of the causes of the this section we present the basic tools for studying the motion of a continuousbody moving inRdwithd= 2 or 3.

3 For further reading we suggest the books[Hun06, Log06, Pat83, Whi99] on which these notes are based ~abe (the label of) a particle of a continuous body that att= 0 occupies theregionP0. By afluid motionwe mean a mapping t:P0 Pt, which maps theregionP0into the regionPt= t(P0) which is occupied by the same fluid at timet. We assume that tis represented by the formula~x=~X(~a,t)(1)withX(~a,0) =~a.~ais the Lagrangian coordinates or the particle s label att= 0while~xis the Eulerian coordinate representing the position of the same particle~aat timet.

4 Usually, the mapping~X:Rd R Rd, is also referred to as theparticlepath. We assume that~Xis a diffeomorphism ofRd, is smooth and invertiblewhere the derivativeD~a~X=( ajXi)ijis a nonsingular matrix. Roughly speaking,Date: April 16, Subject (primary), 76B07, 65M70 (secondary).Key words and element method, Solitary WAVES , Green-Naghdi system, MITSOTAKISthe last condition means that the motion does not crush a nonzero material volumeto zero regionPtoccupied by the material particles~aat timetcan be describedalso as the bounded set with smooth boundary such thatPt={~X(~a,t) :~A P0}.

5 For a given fluid motion thevelocityis defined by~U(~a,t) =~Xt(~a,t) . Then thecorresponding spatial velocity~u(~x,t) is defined by~u(~X(~a,t))=~U(~a,t).Conversely, given a smooth spatial velocity~u(~x,t), we may reconstruct the motionof~X(~a,t) by solving the system of ODEs~Xt(~a,t) =~u(~X(~a,t),t),with~X(~a,0) =~aas initial that we take measurements for the material volume at timet. Let ameasurementfbe a function of the spatial coordinates (~x,t), then the correspond-ing measurementFof the material coordinates is a function of (~a,t) and they areconnected by the formulaF(~a,t) =f(~X(~a,t),t).

6 The rate of change offat a given spatial point is given by the derivativeft(or tf), while the rate of change offfollowing a particle path is given byFt. The lastderivative is called thematerial derivativeoffand is usually denoted to the chain rule,DfDt(~X(~a,t),t) =ft(~X(~a,t),t) +~Xt(~a,t) f(~X(~a,t),t)=ft(~X(~a,t),t) +~u(~Xt(~a,t),t) f(~X(~a,t),t)The last relationship implies the compact formDDt= t+~u .We also define thevorticityas the curl of the velocity field, = ~ we present theReynolds transport theoremwhich may be thought of as ageneralization of the Leibniz rule1for differentiating one dimensional integrals withvariable endpoints:ddt Ptf d~x= Pt{ft+ (f~u)}d~x,(2)or after using the divergence theorem it can be written asddt Ptf d~x= Ptftd~x+ Ptf~u ~ndS.

7 (3)Now we have all the tools needed to derive the Euler s ( b(t)a(t)f(x, t)dx)= b(t)a(t)ft(x, t)dx+f(b(t), t) b (t) f(a(t), t) a (t).A SIMPLE INTRODUCTION TO WATER of the Euler equationsThe Euler equations consist of a set of physicalconservation lawssuch as theconservation of mass and momentum, together with the assumption that the densityof the fluid is constant, cf. [Whi99].First we derive theconservation of mass. We consider a fluid of density (~x,t).The mass contained in a material volumePtis then given by Pt d~ the mass of the material volume does not change with time, thenddt Pt d~x= , using (2), we get that Pt{ t+ ( ~u)}d~x= this equality holds for allt 0 and for an arbitrary smooth regionPt, weconclude that t+ ( ~u) = 0.

8 (4)Assuming that the medium is homogenous, that the density of the fluid isconstant, the conservation of mass reduces to the equation (~u) = 0.(5)We proceed with the derivation of the equation ofconservation of total momentum of a material volumePtis Pt ~ud~ s second law states that the rate of change of the momentum of a materialvolume is equal to the force acting on it. Taking into account that the only forcesacting on the material volume is the surface pressure forcepthat acts in the inwardnormal direction and the gravity force~F= g~k, where~kis the unit vectorperpendicular to the horizontal plane.

9 Thusddt Pt ~ud~x= Ptp~ndS+ Pt~F d~ (2), and the divergence theorem, we find that Pt{( ~u)t+ ( ~u ~u) + p ~F}d~x= 0,where the tensor product~u ~u= (uiuj)ijand therefore Pt t( ui) +d j=1 xj( uiuj) + p xi Fi d~x= this equation holds for anyt 0 and for any arbitrary smooth regionPt, weconclude that( ~u)t+ ( ~u ~u) + p=~F.(6)4D. MITSOTAKISA ssuming that the density of the fluid is constant, t=| |2= 0, and using(5), the conservation of momentum (6) reduces to~ut+ (~u )~u+1 p= g~k.(7)We close the derivation of the Euler equations by studying thevorticityof thevelocity field.

10 The vorticity is defined as~ = ~u. Taking curl on both sides ofthe conservation of momentum (7) we have~ t+~u ~ =~ ~u.(8)Ifd= 2 then = v x u y, and~ ~u= 0, so (8) reduces to the transport equation t+~u = 0.(9)From equations (8) and (9) we conclude that if the flow is irrotational initialy( ~u= 0 fort= 0) then the flow remains irrotational for all timet. Theirrotationality condition is formulated as ~u= 0(10)One main consequence of the irrotationality condition is that the momentum equa-tion (7) can be written as~ut+12 |~u|2+1 p= g~k.