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On modelling of long waves in the Lagrangian and Eulerian description
Coastal Engineering 53 (2006) 759 – 765 www.elsevier.com/locate/coastaleng
On modelling of long waves in the Lagrangian and Eulerian description Jaroslaw Kapinski Institute of Hydroengineering of the Polish Academy of Sciences, Gdansk, Poland Received 21 February 2005; received in revised form 25 August 2005; accepted 20 March 2006 Available online 15 May 2006
Abstract Long wave equations derived by means of the Lagrangian and Eulerian methods are discussed herein. For both approaches differences are pointed out as well as a transition from one to another description is presented. First, selected examples of the small-amplitude wave theory are given. They concern cases of a shallow-water area and a swash zone. Some analogies between the linear Lagrangian wave and the 2nd Stokes' wave are shown as well. Finally, a simple finite-amplitude model elaborated in the Lagrangian manner is presented and tentatively compared with its Eulerian counterpart. Some discrepancies in the solution for both methods are also noticed in this case. © 2006 Elsevier B.V. All rights reserved. Keywords: Lagrangian method; Eulerian method; Small-amplitude wave theory; Finite-amplitude wave theory; Long wave; Stokes' wave; Shallow-water; Swash zone; Wave run-up; Orbital motion
1. Introduction As well known, there are two alternative ways of description of the fluid motion. The Eulerian method analyses what happens at every fixed point in space whereas the Lagrangian viewpoint follows the trajectory of each individual particle. Both approaches lead to solutions, which should present the same picture of a modelled phenomenon despite different ways of derivation and usually different forms of the final equations. However, sometimes the results can diverge slightly depending on the point of view. Additionally, in some cases the solution is more preferable for one of the mentioned methods because it is more convenient or provides more comprehensive information or is easier or ever possible to obtain. Therefore, the choice of the proper method of the flow analysis should be in question first before we start the work on mathematical modelling. In this paper long wave equations derived by means of the Lagrangian and Eulerian methods are treated. A special attention is focused on the comparison of results for which in particular differences are pointed out. First, selected cases of the small-amplitude wave theory including a wave run-up E-mail address: [email protected]. 0378-3839/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2006.03.009
phenomenon are discussed. They mainly concern fundamental properties of the wave motion. The way of transition from one to another description is given as well. Moreover, some resemblance of the linear solution obtained in the Lagrangian method to the 2nd Stokes' wave has been discussed. In the second part of this paper a simple solution for a non-linear long wave is presented. Governing equations are derived here in the Lagrangian manner. Also in this case the model is compared with its Eulerian counterpart. In the paper two alternative methods of description of the fluid motion interpenetrate. However there is only a simple mathematical background used, it can make a great deal of confusion. Therefore to facilitate considerations all hydrodynamical parameters corresponding to the Lagrangian and Eulerian meaning have been distinguished by use of a different notation. 2. Examples of the small-amplitude wave modelling 2.1. Comparison of governing equations Linearized long wave equations for the ideal fluid expressed in the Eulerian co-ordinates are commonly presented in handbooks and encyclopaedias of fluid dynamics. They usually
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Nomenclature Notation c shallow-water wave celerity, g gravitational acceleration, h still water depth at fixed cross-section x, hL still water depth at moving cross-section xL, H incident wave height, k wave number, L incident wavelength, Rdown wave run-down height, Rup wave run-up height, SWL still water level, t time, T wave period, u velocity at fixed cross-section x, i.e. flow velocity, uL velocity at moving cross-section xL, i.e. water particle velocity, x initial position of particle, horizontal co-ordinate as well, xL instantaneous position of particle, z vertical co-ordinate, ζ water surface elevation at fixed cross-section x, ζL water surface elevation at moving cross-section xL, ξ displacement of particle, ξ = xL x, ω angular frequency, θ phase
take the following set of two final equations (e.g. Whitham, 1979): Bu Bf B2 f B Bf þg ¼ 0; gh ¼ 0; ð1Þ Bt Bx Bt 2 Bx Bx with unknown parameters u and ζ, where: u ζ h g t x
depth-averaged flow velocity, water surface elevation above SWL, still water depth, gravitational acceleration, time, space co-ordinate
Equations describing the wave motion in case that is equivalent to Eq. (1) but expressed in the Lagrangian coordinates are as follows (Kapinski, 2003):1 B2 n BfL B2 fL B BfL gh ¼ 0; þg ¼ 0; ð2Þ Bt 2 Bx Bx Bt 2 Bx with unknown parameters ξ and ζL which denote a depthaveraged horizontal particle displacement and a water surface elevation above SWL, respectively. The elevation is not identified here with a fixed cross-section but corresponds to the instantaneous position of a moving particle.
1 The model has been later developed to the two-dimensional, depth-averaged case. See: Kapinski (2004).
In the paper some parameters like a water depth or a water elevation are expressed both in the Lagrangian and in the Eulerian sense. Therefore, to distinguish one from another, all Lagrangian variables are signed by the superscript L. As a consequence, all Eulerian parameters should be marked by the superscript E but for simplification of the notation this mark has been dropped. Additionally, some parameters, which do not depend on the point of view like acceleration due to gravity g or time lapse t, do not have any mark attached. Now, it is important to clarify the difference between the sets of Eqs. (1) and (2). They should describe modelled phenomena likewise, however there is a slight difference in the form and in the meaning. For instance, the parameter ζ in Eq. (1) denotes a free surface elevation at a fixed location x according to the origin of the co-ordinate system, x = 0. Whereas ζL in Eq. (2) means the water level rises at the instantaneous position of a water particle xL which originally (i.e. till the initial instant, t = 0) rested in the position x. The distance between the instantaneous position xL and the original one x is the particle displacement, ξ = xL x. In reality, the magnitude of ξ in the analysed wave motion is limited considerably being even for the extreme swings several times smaller than a wavelength L. However, it will noticeably influence the properties of waves and water motion. It is worthy of mention here that the particle displacement given in Eq. (2), which is in the time domain, ξ = ξ(x, t), fully describes the horizontal kinematics of the orbital motion (trajectories, velocities, etc.). Whereas in the Eulerian method, represented by the set of Eq. (1), the information on the particle orbits is not provided directly and the integration of the flow
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velocity u can fail for more complex cases even in this linear case. Another divergence between the sets of the Eqs. (1) and (2) is the term describing the motion. In the Eulerian method it is represented by the flow velocity u, i.e. the velocity at a fixed location, which is crossed by different particles one by one with L L their own velocities: …,un1 ,unL,un+1 ,…, where n is the index of the particle. Whereas in the Lagrangian description it is the Bn velocity of a single, still the same particle, uL ¼ . Bt Now, for both methods we can formulate compatible seaward boundary conditions. In case of the small-amplitude H theory a simple harmonics cosh is normally generated, where 2 θ = kx ωt is the phase. So, the substitution of the free surface elevation ζ in the form: H f ¼ cosh; 2
ð3Þ
into (1, first part) for a constant water depth, h = const., finally gives: u¼
gHT cosh; 2L
ð4Þ
where: H, T, L, k = 2π / L and ω = 2π / T are: wave height, wave period, wavelength, wave number and angular frequency, respectively. Analogously, substitution of ζL described by: fL ¼
H cosh; 2
ð5Þ
into (2, first part) must give the particle velocity uL in the form which is identical with Eq. (4). Further integration of the equation yields: n¼
HL sinh; 4kh
ð6Þ
pffiffiffiffiffi where the relation L=T ¼ gh has been used. Eq. (6) is a factual complement of the boundary condition in the Lagrangian method. It shows that the seaward end in the pure Lagrangian model is not fixed but it is moving back and forth following the most offshore water particle. This is an analogy to the motion of a wave piston in a wave flume, which is used for generation of water surface waves. Therefore the Lagrangian description of the seaward boundary seems to be more natural approach in conducting laboratory experiments. Further in this paper, if the elevation of a free water surface is described by Eq (5), it will be called the wave in the Lagrangian sense or shorter — the Lagrangian wave. This nomenclature has been taken after Shuto (1967). Analogously, in case when the surface changes are simulated by ζ in the form of Eq. (3), it will be labelled with the Eulerian wave. It is feasible to apply the Lagrangian set of governing equations (Eq. (2)) to simulate the harmonic Eulerian wave. This way some advantages of the Lagrangian approach like the exact treatment of the moving landward boundary on a beach
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slope or easy way of watching of water particles in the whole area of motion can be utilised in the analysis of the wave properties of the Eulerian type. Water surface elevation ζL for a fixed time can be developed in the Taylor series as follows: n BfðxÞ n2 B2 fðxÞ þ 1! Bx 2! Bx2 3 3 n B fðxÞ þ þ N : ð7Þ 3! Bx3 Bf Neglecting terms smaller than n and substituting Eqs. (3) Bx and (6) into Eq. (7) we have:
fL ðxÞ ¼ fðx þ nÞ ¼ fðxÞ þ
fL ¼
H H2 2 H H2 cosh þ sin h ¼ cosh þ ð1 cos2hÞ: 2 2 4h 8h
ð8Þ
This equation given as a boundary condition at the seaward end in the Lagrangian model will generate the sinusoidal waves in the Eulerian meaning. It means that a measuring gauge mounted at any fixed location of a wave flume, which is characterised by a horizontal and smooth bottom, will record sinusoidal water surface changes while the elevations recorded by a gauge placed at the moving wave piston will be consistent with Eq. (8). It is worthy of mention here that the last term on the right-hand side of Eq. (8) is the second harmonics in the Fourier series. So in the linear Lagrangian approach, there exists explicit similarity with the Stokes' wave of the second order of non-linearity. The Taylor series can also be expanded for ζ, then: n BfL ðxÞ f ¼ fL ðxL nÞ ¼ fL ðxÞ 1! BxL 3 2 2 L n Bf ðxÞ n3 BfL ðxÞ L þ þN : Bx BxL 2! 3! ð9Þ Eq. (9) will be useful for drawing the shape as well as for more thorough analysis of the Lagrangian wave seen from the Eulerian point of view. 2.2. Analysis of wave properties at constant water depth Spreading of a harmonic wave train on a free water surface has been examined numerically. The wave 0.4 m high and 20m long was propagating over the horizontal bottom 2 m deep. Thus its period calculated from the equation: T¼
L L ¼ pffiffiffiffiffi ; c gh
ð10Þ
equals to 5.52 s. The wave parameters were intentionally exaggerated to enable the comparison with the Stokes theory and to make the visual effects of computations more clear. Lagrangian and Eulerian wave profiles obtained from numerical computations are drawn in Fig. 1. The adopted point of view is Eulerian, i.e. waves are observed from a position which is fixed in space. The example shows the free
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Fig. 1. Comparison of sinusoidal waves in the Eulerian and Lagrangian meaning.
water surfaces both as a function of the distance x and as a function of time t. So, it can be concluded that the wave shapes do not depend on the domain. The Eulerian wave (dashed line) strictly corresponds to the sinusoid, whereas the profile of the Lagrangian wave (solid line) slightly differs from this picture. It has the shorter and steeper crest and the longer and flatter trough, however the heights and periods (lengths) of both the waves are the same. So, the small asymmetry appears in the Lagrangian solution. Such shape of the linear Lagrangian wave was first observed and drawn by Shuto (1967). Certainly, from the Eulerian point of view the Lagrangian wave is not a sinusoid and vice versa: the Eulerian wave is not a sinusoidal function if the Lagrangian viewpoint is considered. The profile of the Eulerian sinusoidal wave observed from the Lagrangian viewpoint, i.e. when the observations are conducted from a moving particle, is presented in Fig. 2. Such wave has the flatter crest and steeper trough comparing to the sinusoid and, certainly, after rotation on the x-axis it is identical with the Lagrangian wave shown in Fig. 1. Magnitude of the wave height, which has been utilised in numerical computations, is a bit exaggerated as for the linear model. It has been done intentionally to allow more evident comparison of the wave profiles with the non-linear theory. Fig. 3 shows the Eulerian and the Lagrangian wave forms on which the second order Stokes' wave, described by (SPM, 1984): 2 H kH coshð2kh=LÞ f ¼ cosh þ ½2 þ coshð4kh=LÞcos2h; 2 8L sinh3 ð2kh=LÞ ð11Þ is marked. It can be easily concluded that the linear Lagrangian wave more accurately overlaps the Stokes' wave in the areas around the zero crossings. Its asymmetric profile reminds the
Fig. 2. Eulerian sinusoidal wave observed from the Lagrangian viewpoint.
Fig. 3. Eulerian and Lagrangian linear wave vs. 2nd Stokes profile.
regular wave with the non-linear characteristics, however the heights of the crest and trough are still equal to each other. In the earlier paper (Kapinski, 2003) it has been shown that the Lagrangian wave for the linear theory produces a mean current velocity equal to the 2nd order Stokes drift described by (SPM, 1984 with additional assumptions: h / L → 0 and H / h → 0): 1 H 2 c; ¯u ¼ 8 h
ð12Þ
where: ū c
averaged flow velocity at fixed cross-section, shallow-water wave celerity.
However, in this approach water particles draw closed orbits and thus no resultant water flow exists in the wave direction, i.e. no wave-driven current is observed. Eq. (12) was derived from the governing set of Eq. (2) for a constant water depth whereas for irregular bottoms numerical computations have to be carried out. Easy way of numerical prediction of the flow velocity, which is equivalent to the mass transport velocity, successfully allows to analyse instantaneous and mean flows for more complex conditions relating different wave phenomena like shoaling, partial reflection, etc. Both the aforementioned mathematical models have been elaborated for the shallow-water conditions, whereas the similar problems for the waves propagating at bigger depths were investigated in a wave flume by Woltering and Daemrich (1994a,b,c,d, 1995). Their experiments showed that the Lagrangian wave shapes are getting more and more asymmetrical with the depth increase reaching the shape of the 5th order Stokes' wave. They also studied wave-induced water flows and for several examples concerning an intermediate water and a deep water measured mean flow velocities equivalent to the 2nd order Stokes' wave. General conclusion from the above considerations is that the linear waves in the Lagrangian meaning give results similar to
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the Stokes' theory of higher orders. Non-linearity of their surface forms increases with the growth of the water depth. Whereas, the linear Lagrangian waves for any water depths always induce flow velocity equal to the second order Stokes drift. 2.3. Wave transformation in the swash zone One of the advantages of the Lagrangian approach is easiness of exact modelling of a wave transformation in a swash zone in fact being the solution of the moving boundary problem. Instantaneous location of a water tongue tip on a slope is represented by the most onshore water particle, for which the depth always equals to zero hL = 0 (Shuto, 1967; Goto, 1979; Kapinski, 2003). This assumption usually leads the governing equations to a simpler form making the solution feasible in the whole area. Whereas in the Eulerian method, the water surface elevation ζ will tend to infinity if the zeroth water depth is encountered, h = 0. Therefore the set of equations (Eq. (2)) has been chosen for numerical simulations of both the Lagrangian and the Eulerian wave run-up. The profile of the latter one will be obtained if Eq. (8) is used at the seaward boundary. The example presented in Fig. 4 shows two opposite stages of a standing wave in the swash zone: at maximum run-up height Rup and at maximum run-down height Rdown. Numerical simulations were carried out here for the initially progressive wave (H = 0.4m, L = 10m) approaching the shore from the area with a constant bottom depth. The slope with an inclination 1:1, assumed as smooth, impermeable and high enough to avoid overtopping, was founded at the depth of h = 2m. Solid line in Fig. 4 indicates the Lagrangian wave profile, whereas the Eulerian wave is drawn with the dashed line. Small discrepancy observed between the shapes of the water tongue depends on the method applied. However, the extreme locations of the water tip on the slope seem to be identical. Fig. 5 shows a vertical component of a displacement of a water tongue tip on a slope as a time domain. As it could be already concluded from the comparison of the wave profiles propagating at the constant water depth (cf. Fig. 1), the duration of the Lagrangian wave on the slope over SWL is a bit shorter than in the area below this level. Whereas the tongue tip of the
Fig. 4. Wave profiles in the swash zone at maximum run-up and run-down.
Fig. 5. Location of the water tongue tip on slope.
Eulerian wave spends the same period on both parts of the slope. 3. Non-linear long wave modelling 3.1. Governing equations The above-mentioned examples concern the propagation of the small-amplitude waves expressed either in the Lagrangian or in the Eulerian sense. Now, the case with large but finite amplitudes is taken into considerations. The governing equations are based on the Lagrangian momentum and mass conservation principles for the ideal fluid (cf. Fig. 6): qV L aL ¼ V L jpL ;
qV ¼ qV L :
ð13Þ
where: B2 n acceleration of water parcel, Bt 2 pL = p0 + ρg(ζL zL) vertical distribution of water pressure at xL , aL ¼
V = hdx, VL = (hL + ζL)dxL volume of parcel at initial position x and at instantaneous position xL, respectively, where: BxL B Bn dx ¼ ðx þ nÞdx ¼ 1 þ dx: ð14Þ dxL ¼ Bx Bx Bx It is worthy to note here that the Lagrangian momentum equation does not include the convective term. It arises from the fact that the material time derivative of the parcel velocityL dvL is by definition completely described by the local term Bv dt Bt
Fig. 6. Sketch of the model.
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(see e.g. Batchelor, 1970). Now, simple rearrangement of Eq. (13) yields: BfL B n ¼ g Bx ; Bn Bt 2 1þ Bx 2
fL ¼
h Bn 1þ Bx
hL ;
ð15Þ
where ξ, analogously as in Eq. (2), is a depth-averaged parameter.
Mixing of Eqs. (16) and (17) gives the single-ζL equation: L 2 L 2 ! B 2 fL Bf B2 fL Bf 3 ghb hb 2 þ hb 2 2 ¼ 0; ð21Þ Bt Bt Bx Bx fL where: b ¼ 1 þ : h The dispersion relationship for Eq. (21) is as follows: # 3 " x2 H H 3cos2 h ¼ gh 1 þ sinh 1 ; H 2h 2h sinh þ 2h k2 ð1 þ cos2 hÞ
3.2. Wave transformation at constant water depth
ð22Þ
The linearized solution (2), which is also based on the principles expressed by Eq. (13), was derived for the assumption of small gradients of the parcel displacement field B2 h j Bn Bx jb1 as well as for a gentle curvature of the bottom j Bx2 jb1 (Kapinski, 2003). Now, the only condition of the constant bottom depth, h = hL = const., is considered and not any limitations due to magnitude of wave amplitudes is introduced. However, the pressure in the water body is still determined by the hydrostatic distribution. Considering the constant water depth, the mass conservation equation (15, second part) will take the following form: Bn f ¼ Bx : Bn 1þ Bx L
h
ð16Þ
B2 n 2 B n ¼ gh Bx 3 : Bt 2 Bn 1þ Bx
ð17Þ
The single-ξ equation is of the hyperbolic type hence the finite-amplitude disturbance on the water surface propagates with the celerity: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u gh c¼u ð18Þ ; u t Bn 3 1þ Bx where the term ∂ξ / ∂x has a periodic characteristic both in space and in time. Eq. (18) needs to satisfy the condition: 1þ
Bn > 0: Bx
Eqs. (20) and (23) have been assumed as equivalent and compared with each other. It leads to the equality: 2=3 f 1=2 fL 3 1þ ¼ 1þ þ2: h h
Substitution of Eq. (16) into (15, first part) gives: 2
For small surface disturbances, H / 2h b 1, the equation is reduced to the well-known formula: ω2 / k2 = gh. By the way, it is worthy to note here that one of the parameters on the right-hand side of Eq. (22) is the phase θ which suggests existence of the amplitudinal dispersion. It is difficult to define the wave celerity from Eq. (21), however it must be identical with Eq. (20). The corresponding celerity in the Eulerian description is as follows (Whitham, 1974): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c ¼ 3 gðh þ fÞ 2 gh: ð23Þ
ð19Þ
Eq. (24) shows that the Lagrangian and Eulerian wave celerities tend to be the same only for the linear case, i.e. when the water surface elevation denoted by ζ and by ζL is much smaller than the water depth h. However, for larger wave heights H they will differ only slightly. Selected results of computations are shown in Figs. 7 and 8. Numerical simulations were performed for the finite-amplitude wave (H = 0.1m, L = 10m) in the Lagrangian meaning propagating at the constant water depth h = 0.6 m. Fig. 7 shows transformation of the wave profile for the time lapse: 4, 5, 6 and 7s since the start of the simulation. A rapid deformation is observed there because of the amplitudinal dispersion. The wave front is strongly steepening towards the destination of the wave advance but the magnitudes of the amplitudes continue to be constant. Finally, a numerical instability appears on the front of the first wave. It is caused by the forming vertical line of a
Substitution of Eq. (6) into Eq. (19) follows to the obvious conclusion that the wave amplitude H / 2 cannot be larger than the water depth h. Substitution of ∂ξ / ∂x from Eq. (16) into Eq. (18) gives: 2=3 pffiffiffiffiffi fL c¼ 1þ gh: ð20Þ h Now it is evident that for the linear case (ζL b h) the wave celerity tends to the square root of gh.
ð24Þ
Fig. 7. Finite-amplitude wave profile transformation.
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The set of Eq. (26), together with the onshore boundary condition (27), can be easily solved numerically to simulate the finite-amplitude wave transformation of the Lagrangian type on an arbitrary inclined bottom. Additionally, it gives possibility to predict the tsunami run-up phenomenon in the scope of this theory. 4. Summary Fig. 8. Water particle orbit in case of finite-amplitude wave.
free water surface, which is no longer a function. The same transformation of the wave profile is also observed for the adequate theory in the Eulerian description (cf. Voltsinger et al., 1989; Whitham, 1974). Fig. 8 shows the particle trajectory for the wave presented in Fig. 7. It was computed 7 m since the origin of generation. The initially elliptical orbit is getting more and more asymmetrical together with the increase of the distance of the wave advance. It takes the egg-type profile, however the paths still remain closed. It is hard to compare here the water particle trajectories of the Lagrangian and Eulerian type because of the difficulty appearing in the derivation of the relevant equations for the latter theory. 3.3. Finite-amplitude wave run-up on slope Voltsinger et al. (1989) showed the possibility of application of the Eulerian counterpart of the finite-amplitude wave theory presented here for the prediction of tsunami advance in the shallow-water areas. In the Lagrangian approach it is also feasible to extend the theory to simulate a wave run-up phenomenon on beach slopes. At first the set of Eqs. (16) and (17) has been derived for a constant bottom depth, h = const. Now, a constant slope condition is considered: Bh BhL ¼ ¼ const: Bx BxL Thus, the set (15) will transform to the form: B2 n n Bn Bh Bn h 2 þ 1þB 2þ B2 n Bx Bx Bx Bx Bx ¼ g ; Bt 2 Bn 3 1þ Bx BðhnÞ Bh Bn þn Bx Bx : fL ¼ Bx Bf 1þ Bx
ð25Þ
ð26Þ
At the moving landward boundary, where a priori the depth always equals to zero, hL = 0, Eq. (26) simplifies to the form: Bh Bn Bn 2þ B2 n Bh Bx Bx Bx ð27Þ ¼ g fL ¼ n : 2 ; 2 Bt Bx Bn 1þ Bx
In the paper two alternative methods of description of the long wave theory have been presented and compared. Selected examples concern the small-amplitude as well as the finiteamplitude wave propagation on a constant water depth and on a constant slope inclination. It has been shown that already for very simple cases a little but noticeable discrepancies are observed depending on the approach applied. Additionally, it has been presented that the Lagrangian description allows for the simulation of the orbital motion and the moving shoreline without any special treatment. Moreover, the linear Lagrangian approximation gives the wave profile and the resultant velocity closer the higher order Stokes theories than the adequate Eulerian solution. References Batchelor, G.K., 1970. An Introduction to Fluid Dynamics. Cambridge University Press. Goto, Ch., 1979. Nonlinear equation of long waves in the Lagrangian description. Coastal Engineering in Japan 22, 1–9. Kapinski, J., 2003. Lagrangian–Eulerian approach to modelling of wave transformation and flow velocity in the swash zone and its seaward vicinity. Archives of Hydro-Engineering and Environmental Mechanics, vol. 50. IH PAS, Gdansk, pp. 165–192. Kapinski, J., 2004. Two-dimensional modelling of wave motion in shallowwater areas. Archives of Hydro-Engineering and Environmental Mechanics, vol. 51. IH PAS, Gdansk, pp. 3–24. Shore Protection Manual, 1984, U.S. Army, Coastal Engineering Research Center, Washington, D.C. Shuto, N., 1967. Run-up of long waves on a sloping beach. Coastal Engineering in Japan 10, 23–38. Voltsinger, N.E., Klevannyj, K.A., Pelinovskij, E.N., 1989. Long-wave Dynamics of the Coastal Zone (in Russian) Gidrometeoizdat, Leningrad. Whitham, G.B., 1974. Linear and Nonlinear Waves. A Wiley-Interscience Publication. Whitham, G.B., 1979. Lectures on Wave Propagation. Springer-Verlag, Heidelberg. Woltering, S., Daemrich, K.-F., 1994a. Large scale measurements of near surface orbital velocities in laboratory waves by using a surface following device. Proceedings of Coastal Dynamics, Barcelona, Spain. Woltering, S., Daemrich, K.-F., 1994b. Mass transport and orbital velocities with Lagrangian frame of reference. Proceedings of ICCE. Kobe, Jaan. Woltering, S., Daemrich, K.-F., 1994c. Regular wave investigations of wave kinematics with Lagrangeian approach. Proceedings of the International Symposium on Waves — Physical and Numerical Modelling, Vancouver, Canada. Woltering, S., Daemrich, K.-F., 1994d. Investigations on regular wave kinematics in wave channels. Schriften des Vereins der Freunde und Förderer des GKSS-Forschungszentrums Geesthacht e.V., University of Hanover. Woltering, S., Daemrich, K.-F., 1995. Wave kinematics measurements with a surface following probe. Proceedings of ISOPE-95, Hague, The Netherlands.
On modelling of long waves in the Lagrangian and Eulerian description Jaroslaw Kapinski Institute of Hydroengineering of the Polish Academy of Sciences, Gdansk, Poland Received 21 February 2005; received in revised form 25 August 2005; accepted 20 March 2006 Available online 15 May 2006
Abstract Long wave equations derived by means of the Lagrangian and Eulerian methods are discussed herein. For both approaches differences are pointed out as well as a transition from one to another description is presented. First, selected examples of the small-amplitude wave theory are given. They concern cases of a shallow-water area and a swash zone. Some analogies between the linear Lagrangian wave and the 2nd Stokes' wave are shown as well. Finally, a simple finite-amplitude model elaborated in the Lagrangian manner is presented and tentatively compared with its Eulerian counterpart. Some discrepancies in the solution for both methods are also noticed in this case. © 2006 Elsevier B.V. All rights reserved. Keywords: Lagrangian method; Eulerian method; Small-amplitude wave theory; Finite-amplitude wave theory; Long wave; Stokes' wave; Shallow-water; Swash zone; Wave run-up; Orbital motion
1. Introduction As well known, there are two alternative ways of description of the fluid motion. The Eulerian method analyses what happens at every fixed point in space whereas the Lagrangian viewpoint follows the trajectory of each individual particle. Both approaches lead to solutions, which should present the same picture of a modelled phenomenon despite different ways of derivation and usually different forms of the final equations. However, sometimes the results can diverge slightly depending on the point of view. Additionally, in some cases the solution is more preferable for one of the mentioned methods because it is more convenient or provides more comprehensive information or is easier or ever possible to obtain. Therefore, the choice of the proper method of the flow analysis should be in question first before we start the work on mathematical modelling. In this paper long wave equations derived by means of the Lagrangian and Eulerian methods are treated. A special attention is focused on the comparison of results for which in particular differences are pointed out. First, selected cases of the small-amplitude wave theory including a wave run-up E-mail address: [email protected]. 0378-3839/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.coastaleng.2006.03.009
phenomenon are discussed. They mainly concern fundamental properties of the wave motion. The way of transition from one to another description is given as well. Moreover, some resemblance of the linear solution obtained in the Lagrangian method to the 2nd Stokes' wave has been discussed. In the second part of this paper a simple solution for a non-linear long wave is presented. Governing equations are derived here in the Lagrangian manner. Also in this case the model is compared with its Eulerian counterpart. In the paper two alternative methods of description of the fluid motion interpenetrate. However there is only a simple mathematical background used, it can make a great deal of confusion. Therefore to facilitate considerations all hydrodynamical parameters corresponding to the Lagrangian and Eulerian meaning have been distinguished by use of a different notation. 2. Examples of the small-amplitude wave modelling 2.1. Comparison of governing equations Linearized long wave equations for the ideal fluid expressed in the Eulerian co-ordinates are commonly presented in handbooks and encyclopaedias of fluid dynamics. They usually
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J. Kapinski / Coastal Engineering 53 (2006) 759–765
Nomenclature Notation c shallow-water wave celerity, g gravitational acceleration, h still water depth at fixed cross-section x, hL still water depth at moving cross-section xL, H incident wave height, k wave number, L incident wavelength, Rdown wave run-down height, Rup wave run-up height, SWL still water level, t time, T wave period, u velocity at fixed cross-section x, i.e. flow velocity, uL velocity at moving cross-section xL, i.e. water particle velocity, x initial position of particle, horizontal co-ordinate as well, xL instantaneous position of particle, z vertical co-ordinate, ζ water surface elevation at fixed cross-section x, ζL water surface elevation at moving cross-section xL, ξ displacement of particle, ξ = xL x, ω angular frequency, θ phase
take the following set of two final equations (e.g. Whitham, 1979): Bu Bf B2 f B Bf þg ¼ 0; gh ¼ 0; ð1Þ Bt Bx Bt 2 Bx Bx with unknown parameters u and ζ, where: u ζ h g t x
depth-averaged flow velocity, water surface elevation above SWL, still water depth, gravitational acceleration, time, space co-ordinate
Equations describing the wave motion in case that is equivalent to Eq. (1) but expressed in the Lagrangian coordinates are as follows (Kapinski, 2003):1 B2 n BfL B2 fL B BfL gh ¼ 0; þg ¼ 0; ð2Þ Bt 2 Bx Bx Bt 2 Bx with unknown parameters ξ and ζL which denote a depthaveraged horizontal particle displacement and a water surface elevation above SWL, respectively. The elevation is not identified here with a fixed cross-section but corresponds to the instantaneous position of a moving particle.
1 The model has been later developed to the two-dimensional, depth-averaged case. See: Kapinski (2004).
In the paper some parameters like a water depth or a water elevation are expressed both in the Lagrangian and in the Eulerian sense. Therefore, to distinguish one from another, all Lagrangian variables are signed by the superscript L. As a consequence, all Eulerian parameters should be marked by the superscript E but for simplification of the notation this mark has been dropped. Additionally, some parameters, which do not depend on the point of view like acceleration due to gravity g or time lapse t, do not have any mark attached. Now, it is important to clarify the difference between the sets of Eqs. (1) and (2). They should describe modelled phenomena likewise, however there is a slight difference in the form and in the meaning. For instance, the parameter ζ in Eq. (1) denotes a free surface elevation at a fixed location x according to the origin of the co-ordinate system, x = 0. Whereas ζL in Eq. (2) means the water level rises at the instantaneous position of a water particle xL which originally (i.e. till the initial instant, t = 0) rested in the position x. The distance between the instantaneous position xL and the original one x is the particle displacement, ξ = xL x. In reality, the magnitude of ξ in the analysed wave motion is limited considerably being even for the extreme swings several times smaller than a wavelength L. However, it will noticeably influence the properties of waves and water motion. It is worthy of mention here that the particle displacement given in Eq. (2), which is in the time domain, ξ = ξ(x, t), fully describes the horizontal kinematics of the orbital motion (trajectories, velocities, etc.). Whereas in the Eulerian method, represented by the set of Eq. (1), the information on the particle orbits is not provided directly and the integration of the flow
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velocity u can fail for more complex cases even in this linear case. Another divergence between the sets of the Eqs. (1) and (2) is the term describing the motion. In the Eulerian method it is represented by the flow velocity u, i.e. the velocity at a fixed location, which is crossed by different particles one by one with L L their own velocities: …,un1 ,unL,un+1 ,…, where n is the index of the particle. Whereas in the Lagrangian description it is the Bn velocity of a single, still the same particle, uL ¼ . Bt Now, for both methods we can formulate compatible seaward boundary conditions. In case of the small-amplitude H theory a simple harmonics cosh is normally generated, where 2 θ = kx ωt is the phase. So, the substitution of the free surface elevation ζ in the form: H f ¼ cosh; 2
ð3Þ
into (1, first part) for a constant water depth, h = const., finally gives: u¼
gHT cosh; 2L
ð4Þ
where: H, T, L, k = 2π / L and ω = 2π / T are: wave height, wave period, wavelength, wave number and angular frequency, respectively. Analogously, substitution of ζL described by: fL ¼
H cosh; 2
ð5Þ
into (2, first part) must give the particle velocity uL in the form which is identical with Eq. (4). Further integration of the equation yields: n¼
HL sinh; 4kh
ð6Þ
pffiffiffiffiffi where the relation L=T ¼ gh has been used. Eq. (6) is a factual complement of the boundary condition in the Lagrangian method. It shows that the seaward end in the pure Lagrangian model is not fixed but it is moving back and forth following the most offshore water particle. This is an analogy to the motion of a wave piston in a wave flume, which is used for generation of water surface waves. Therefore the Lagrangian description of the seaward boundary seems to be more natural approach in conducting laboratory experiments. Further in this paper, if the elevation of a free water surface is described by Eq (5), it will be called the wave in the Lagrangian sense or shorter — the Lagrangian wave. This nomenclature has been taken after Shuto (1967). Analogously, in case when the surface changes are simulated by ζ in the form of Eq. (3), it will be labelled with the Eulerian wave. It is feasible to apply the Lagrangian set of governing equations (Eq. (2)) to simulate the harmonic Eulerian wave. This way some advantages of the Lagrangian approach like the exact treatment of the moving landward boundary on a beach
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slope or easy way of watching of water particles in the whole area of motion can be utilised in the analysis of the wave properties of the Eulerian type. Water surface elevation ζL for a fixed time can be developed in the Taylor series as follows: n BfðxÞ n2 B2 fðxÞ þ 1! Bx 2! Bx2 3 3 n B fðxÞ þ þ N : ð7Þ 3! Bx3 Bf Neglecting terms smaller than n and substituting Eqs. (3) Bx and (6) into Eq. (7) we have:
fL ðxÞ ¼ fðx þ nÞ ¼ fðxÞ þ
fL ¼
H H2 2 H H2 cosh þ sin h ¼ cosh þ ð1 cos2hÞ: 2 2 4h 8h
ð8Þ
This equation given as a boundary condition at the seaward end in the Lagrangian model will generate the sinusoidal waves in the Eulerian meaning. It means that a measuring gauge mounted at any fixed location of a wave flume, which is characterised by a horizontal and smooth bottom, will record sinusoidal water surface changes while the elevations recorded by a gauge placed at the moving wave piston will be consistent with Eq. (8). It is worthy of mention here that the last term on the right-hand side of Eq. (8) is the second harmonics in the Fourier series. So in the linear Lagrangian approach, there exists explicit similarity with the Stokes' wave of the second order of non-linearity. The Taylor series can also be expanded for ζ, then: n BfL ðxÞ f ¼ fL ðxL nÞ ¼ fL ðxÞ 1! BxL 3 2 2 L n Bf ðxÞ n3 BfL ðxÞ L þ þN : Bx BxL 2! 3! ð9Þ Eq. (9) will be useful for drawing the shape as well as for more thorough analysis of the Lagrangian wave seen from the Eulerian point of view. 2.2. Analysis of wave properties at constant water depth Spreading of a harmonic wave train on a free water surface has been examined numerically. The wave 0.4 m high and 20m long was propagating over the horizontal bottom 2 m deep. Thus its period calculated from the equation: T¼
L L ¼ pffiffiffiffiffi ; c gh
ð10Þ
equals to 5.52 s. The wave parameters were intentionally exaggerated to enable the comparison with the Stokes theory and to make the visual effects of computations more clear. Lagrangian and Eulerian wave profiles obtained from numerical computations are drawn in Fig. 1. The adopted point of view is Eulerian, i.e. waves are observed from a position which is fixed in space. The example shows the free
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Fig. 1. Comparison of sinusoidal waves in the Eulerian and Lagrangian meaning.
water surfaces both as a function of the distance x and as a function of time t. So, it can be concluded that the wave shapes do not depend on the domain. The Eulerian wave (dashed line) strictly corresponds to the sinusoid, whereas the profile of the Lagrangian wave (solid line) slightly differs from this picture. It has the shorter and steeper crest and the longer and flatter trough, however the heights and periods (lengths) of both the waves are the same. So, the small asymmetry appears in the Lagrangian solution. Such shape of the linear Lagrangian wave was first observed and drawn by Shuto (1967). Certainly, from the Eulerian point of view the Lagrangian wave is not a sinusoid and vice versa: the Eulerian wave is not a sinusoidal function if the Lagrangian viewpoint is considered. The profile of the Eulerian sinusoidal wave observed from the Lagrangian viewpoint, i.e. when the observations are conducted from a moving particle, is presented in Fig. 2. Such wave has the flatter crest and steeper trough comparing to the sinusoid and, certainly, after rotation on the x-axis it is identical with the Lagrangian wave shown in Fig. 1. Magnitude of the wave height, which has been utilised in numerical computations, is a bit exaggerated as for the linear model. It has been done intentionally to allow more evident comparison of the wave profiles with the non-linear theory. Fig. 3 shows the Eulerian and the Lagrangian wave forms on which the second order Stokes' wave, described by (SPM, 1984): 2 H kH coshð2kh=LÞ f ¼ cosh þ ½2 þ coshð4kh=LÞcos2h; 2 8L sinh3 ð2kh=LÞ ð11Þ is marked. It can be easily concluded that the linear Lagrangian wave more accurately overlaps the Stokes' wave in the areas around the zero crossings. Its asymmetric profile reminds the
Fig. 2. Eulerian sinusoidal wave observed from the Lagrangian viewpoint.
Fig. 3. Eulerian and Lagrangian linear wave vs. 2nd Stokes profile.
regular wave with the non-linear characteristics, however the heights of the crest and trough are still equal to each other. In the earlier paper (Kapinski, 2003) it has been shown that the Lagrangian wave for the linear theory produces a mean current velocity equal to the 2nd order Stokes drift described by (SPM, 1984 with additional assumptions: h / L → 0 and H / h → 0): 1 H 2 c; ¯u ¼ 8 h
ð12Þ
where: ū c
averaged flow velocity at fixed cross-section, shallow-water wave celerity.
However, in this approach water particles draw closed orbits and thus no resultant water flow exists in the wave direction, i.e. no wave-driven current is observed. Eq. (12) was derived from the governing set of Eq. (2) for a constant water depth whereas for irregular bottoms numerical computations have to be carried out. Easy way of numerical prediction of the flow velocity, which is equivalent to the mass transport velocity, successfully allows to analyse instantaneous and mean flows for more complex conditions relating different wave phenomena like shoaling, partial reflection, etc. Both the aforementioned mathematical models have been elaborated for the shallow-water conditions, whereas the similar problems for the waves propagating at bigger depths were investigated in a wave flume by Woltering and Daemrich (1994a,b,c,d, 1995). Their experiments showed that the Lagrangian wave shapes are getting more and more asymmetrical with the depth increase reaching the shape of the 5th order Stokes' wave. They also studied wave-induced water flows and for several examples concerning an intermediate water and a deep water measured mean flow velocities equivalent to the 2nd order Stokes' wave. General conclusion from the above considerations is that the linear waves in the Lagrangian meaning give results similar to
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the Stokes' theory of higher orders. Non-linearity of their surface forms increases with the growth of the water depth. Whereas, the linear Lagrangian waves for any water depths always induce flow velocity equal to the second order Stokes drift. 2.3. Wave transformation in the swash zone One of the advantages of the Lagrangian approach is easiness of exact modelling of a wave transformation in a swash zone in fact being the solution of the moving boundary problem. Instantaneous location of a water tongue tip on a slope is represented by the most onshore water particle, for which the depth always equals to zero hL = 0 (Shuto, 1967; Goto, 1979; Kapinski, 2003). This assumption usually leads the governing equations to a simpler form making the solution feasible in the whole area. Whereas in the Eulerian method, the water surface elevation ζ will tend to infinity if the zeroth water depth is encountered, h = 0. Therefore the set of equations (Eq. (2)) has been chosen for numerical simulations of both the Lagrangian and the Eulerian wave run-up. The profile of the latter one will be obtained if Eq. (8) is used at the seaward boundary. The example presented in Fig. 4 shows two opposite stages of a standing wave in the swash zone: at maximum run-up height Rup and at maximum run-down height Rdown. Numerical simulations were carried out here for the initially progressive wave (H = 0.4m, L = 10m) approaching the shore from the area with a constant bottom depth. The slope with an inclination 1:1, assumed as smooth, impermeable and high enough to avoid overtopping, was founded at the depth of h = 2m. Solid line in Fig. 4 indicates the Lagrangian wave profile, whereas the Eulerian wave is drawn with the dashed line. Small discrepancy observed between the shapes of the water tongue depends on the method applied. However, the extreme locations of the water tip on the slope seem to be identical. Fig. 5 shows a vertical component of a displacement of a water tongue tip on a slope as a time domain. As it could be already concluded from the comparison of the wave profiles propagating at the constant water depth (cf. Fig. 1), the duration of the Lagrangian wave on the slope over SWL is a bit shorter than in the area below this level. Whereas the tongue tip of the
Fig. 4. Wave profiles in the swash zone at maximum run-up and run-down.
Fig. 5. Location of the water tongue tip on slope.
Eulerian wave spends the same period on both parts of the slope. 3. Non-linear long wave modelling 3.1. Governing equations The above-mentioned examples concern the propagation of the small-amplitude waves expressed either in the Lagrangian or in the Eulerian sense. Now, the case with large but finite amplitudes is taken into considerations. The governing equations are based on the Lagrangian momentum and mass conservation principles for the ideal fluid (cf. Fig. 6): qV L aL ¼ V L jpL ;
qV ¼ qV L :
ð13Þ
where: B2 n acceleration of water parcel, Bt 2 pL = p0 + ρg(ζL zL) vertical distribution of water pressure at xL , aL ¼
V = hdx, VL = (hL + ζL)dxL volume of parcel at initial position x and at instantaneous position xL, respectively, where: BxL B Bn dx ¼ ðx þ nÞdx ¼ 1 þ dx: ð14Þ dxL ¼ Bx Bx Bx It is worthy to note here that the Lagrangian momentum equation does not include the convective term. It arises from the fact that the material time derivative of the parcel velocityL dvL is by definition completely described by the local term Bv dt Bt
Fig. 6. Sketch of the model.
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(see e.g. Batchelor, 1970). Now, simple rearrangement of Eq. (13) yields: BfL B n ¼ g Bx ; Bn Bt 2 1þ Bx 2
fL ¼
h Bn 1þ Bx
hL ;
ð15Þ
where ξ, analogously as in Eq. (2), is a depth-averaged parameter.
Mixing of Eqs. (16) and (17) gives the single-ζL equation: L 2 L 2 ! B 2 fL Bf B2 fL Bf 3 ghb hb 2 þ hb 2 2 ¼ 0; ð21Þ Bt Bt Bx Bx fL where: b ¼ 1 þ : h The dispersion relationship for Eq. (21) is as follows: # 3 " x2 H H 3cos2 h ¼ gh 1 þ sinh 1 ; H 2h 2h sinh þ 2h k2 ð1 þ cos2 hÞ
3.2. Wave transformation at constant water depth
ð22Þ
The linearized solution (2), which is also based on the principles expressed by Eq. (13), was derived for the assumption of small gradients of the parcel displacement field B2 h j Bn Bx jb1 as well as for a gentle curvature of the bottom j Bx2 jb1 (Kapinski, 2003). Now, the only condition of the constant bottom depth, h = hL = const., is considered and not any limitations due to magnitude of wave amplitudes is introduced. However, the pressure in the water body is still determined by the hydrostatic distribution. Considering the constant water depth, the mass conservation equation (15, second part) will take the following form: Bn f ¼ Bx : Bn 1þ Bx L
h
ð16Þ
B2 n 2 B n ¼ gh Bx 3 : Bt 2 Bn 1þ Bx
ð17Þ
The single-ξ equation is of the hyperbolic type hence the finite-amplitude disturbance on the water surface propagates with the celerity: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u gh c¼u ð18Þ ; u t Bn 3 1þ Bx where the term ∂ξ / ∂x has a periodic characteristic both in space and in time. Eq. (18) needs to satisfy the condition: 1þ
Bn > 0: Bx
Eqs. (20) and (23) have been assumed as equivalent and compared with each other. It leads to the equality: 2=3 f 1=2 fL 3 1þ ¼ 1þ þ2: h h
Substitution of Eq. (16) into (15, first part) gives: 2
For small surface disturbances, H / 2h b 1, the equation is reduced to the well-known formula: ω2 / k2 = gh. By the way, it is worthy to note here that one of the parameters on the right-hand side of Eq. (22) is the phase θ which suggests existence of the amplitudinal dispersion. It is difficult to define the wave celerity from Eq. (21), however it must be identical with Eq. (20). The corresponding celerity in the Eulerian description is as follows (Whitham, 1974): pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi c ¼ 3 gðh þ fÞ 2 gh: ð23Þ
ð19Þ
Eq. (24) shows that the Lagrangian and Eulerian wave celerities tend to be the same only for the linear case, i.e. when the water surface elevation denoted by ζ and by ζL is much smaller than the water depth h. However, for larger wave heights H they will differ only slightly. Selected results of computations are shown in Figs. 7 and 8. Numerical simulations were performed for the finite-amplitude wave (H = 0.1m, L = 10m) in the Lagrangian meaning propagating at the constant water depth h = 0.6 m. Fig. 7 shows transformation of the wave profile for the time lapse: 4, 5, 6 and 7s since the start of the simulation. A rapid deformation is observed there because of the amplitudinal dispersion. The wave front is strongly steepening towards the destination of the wave advance but the magnitudes of the amplitudes continue to be constant. Finally, a numerical instability appears on the front of the first wave. It is caused by the forming vertical line of a
Substitution of Eq. (6) into Eq. (19) follows to the obvious conclusion that the wave amplitude H / 2 cannot be larger than the water depth h. Substitution of ∂ξ / ∂x from Eq. (16) into Eq. (18) gives: 2=3 pffiffiffiffiffi fL c¼ 1þ gh: ð20Þ h Now it is evident that for the linear case (ζL b h) the wave celerity tends to the square root of gh.
ð24Þ
Fig. 7. Finite-amplitude wave profile transformation.
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The set of Eq. (26), together with the onshore boundary condition (27), can be easily solved numerically to simulate the finite-amplitude wave transformation of the Lagrangian type on an arbitrary inclined bottom. Additionally, it gives possibility to predict the tsunami run-up phenomenon in the scope of this theory. 4. Summary Fig. 8. Water particle orbit in case of finite-amplitude wave.
free water surface, which is no longer a function. The same transformation of the wave profile is also observed for the adequate theory in the Eulerian description (cf. Voltsinger et al., 1989; Whitham, 1974). Fig. 8 shows the particle trajectory for the wave presented in Fig. 7. It was computed 7 m since the origin of generation. The initially elliptical orbit is getting more and more asymmetrical together with the increase of the distance of the wave advance. It takes the egg-type profile, however the paths still remain closed. It is hard to compare here the water particle trajectories of the Lagrangian and Eulerian type because of the difficulty appearing in the derivation of the relevant equations for the latter theory. 3.3. Finite-amplitude wave run-up on slope Voltsinger et al. (1989) showed the possibility of application of the Eulerian counterpart of the finite-amplitude wave theory presented here for the prediction of tsunami advance in the shallow-water areas. In the Lagrangian approach it is also feasible to extend the theory to simulate a wave run-up phenomenon on beach slopes. At first the set of Eqs. (16) and (17) has been derived for a constant bottom depth, h = const. Now, a constant slope condition is considered: Bh BhL ¼ ¼ const: Bx BxL Thus, the set (15) will transform to the form: B2 n n Bn Bh Bn h 2 þ 1þB 2þ B2 n Bx Bx Bx Bx Bx ¼ g ; Bt 2 Bn 3 1þ Bx BðhnÞ Bh Bn þn Bx Bx : fL ¼ Bx Bf 1þ Bx
ð25Þ
ð26Þ
At the moving landward boundary, where a priori the depth always equals to zero, hL = 0, Eq. (26) simplifies to the form: Bh Bn Bn 2þ B2 n Bh Bx Bx Bx ð27Þ ¼ g fL ¼ n : 2 ; 2 Bt Bx Bn 1þ Bx
In the paper two alternative methods of description of the long wave theory have been presented and compared. Selected examples concern the small-amplitude as well as the finiteamplitude wave propagation on a constant water depth and on a constant slope inclination. It has been shown that already for very simple cases a little but noticeable discrepancies are observed depending on the approach applied. Additionally, it has been presented that the Lagrangian description allows for the simulation of the orbital motion and the moving shoreline without any special treatment. Moreover, the linear Lagrangian approximation gives the wave profile and the resultant velocity closer the higher order Stokes theories than the adequate Eulerian solution. References Batchelor, G.K., 1970. An Introduction to Fluid Dynamics. Cambridge University Press. Goto, Ch., 1979. Nonlinear equation of long waves in the Lagrangian description. Coastal Engineering in Japan 22, 1–9. Kapinski, J., 2003. Lagrangian–Eulerian approach to modelling of wave transformation and flow velocity in the swash zone and its seaward vicinity. Archives of Hydro-Engineering and Environmental Mechanics, vol. 50. IH PAS, Gdansk, pp. 165–192. Kapinski, J., 2004. Two-dimensional modelling of wave motion in shallowwater areas. Archives of Hydro-Engineering and Environmental Mechanics, vol. 51. IH PAS, Gdansk, pp. 3–24. Shore Protection Manual, 1984, U.S. Army, Coastal Engineering Research Center, Washington, D.C. Shuto, N., 1967. Run-up of long waves on a sloping beach. Coastal Engineering in Japan 10, 23–38. Voltsinger, N.E., Klevannyj, K.A., Pelinovskij, E.N., 1989. Long-wave Dynamics of the Coastal Zone (in Russian) Gidrometeoizdat, Leningrad. Whitham, G.B., 1974. Linear and Nonlinear Waves. A Wiley-Interscience Publication. Whitham, G.B., 1979. Lectures on Wave Propagation. Springer-Verlag, Heidelberg. Woltering, S., Daemrich, K.-F., 1994a. Large scale measurements of near surface orbital velocities in laboratory waves by using a surface following device. Proceedings of Coastal Dynamics, Barcelona, Spain. Woltering, S., Daemrich, K.-F., 1994b. Mass transport and orbital velocities with Lagrangian frame of reference. Proceedings of ICCE. Kobe, Jaan. Woltering, S., Daemrich, K.-F., 1994c. Regular wave investigations of wave kinematics with Lagrangeian approach. Proceedings of the International Symposium on Waves — Physical and Numerical Modelling, Vancouver, Canada. Woltering, S., Daemrich, K.-F., 1994d. Investigations on regular wave kinematics in wave channels. Schriften des Vereins der Freunde und Förderer des GKSS-Forschungszentrums Geesthacht e.V., University of Hanover. Woltering, S., Daemrich, K.-F., 1995. Wave kinematics measurements with a surface following probe. Proceedings of ISOPE-95, Hague, The Netherlands.