A second approximation to the time-mean lagrangian drift beneath a series of progressive gravity waves

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Pergamon Press pic

A SECOND APPROXIMATION TO THE

TIME-MEAN

LAGRANGIAN DRIFT BENEATH A SERIES OF

PROGRESSIVE GRAVITY WAVES

C. SWAN and J. F. A. SLEATH

University of Cambridge, Department of Engineering. Trumpington Street, Cambridge, CB2 IPZ, U.K. AbstractA fourth-order solution is derived for the mean drift induced by a steady train of waves in water of constant depth. New measurements are carried Out of the drift in the body of the fluid and the drift velocity gradient at the free surface. Comparison of theory and experiment shows significantly better agreement with the present fourth-order solution than with the previous second-order solution of Longuet-Higgins, MS., 1953 [Phi!. Trans. R. Soc.

A245. 535Sl]. In particular, the present solution reproduces _{the observed tendency of the} surface drift velocity to rise in shallow water and to level-off in very deep water.

1. _INTRODUCTION

THE STEADY drift induced by water _{waves is of considerable practical interest. The}

transport of sediment, and hence the build-up or erosion of coasts, and the dispersion of oil slicks and other pollutants are all very significantly affected by this drift.

Longuet-Higgins (1953) published an analytical solution _{for the} _{mass transport"}

velocity (the time-mean Lagrangian drift), correct to a second-order of wave steepness.

This solution shows good agreement with experimental results for the mass transport

velocity in the vicinity of the bed, provided the flow is laminar and the keynolds number reasonably low. At higher Reynolds numbers the agreement is less good

because, as shown by Sleath (1972) and Isaacson (1976a,b), the higher-order terms

neglected in Longuet-Higgins' solution become progressively _{more important.}

However, there is

_{a more significant lack of agreement between theory and}

experiment away from the bed. In particular, the measurements of the mass transport

velocity at the free surface show quite different trends from those predicted by

Longuet-Higgins' solution. For large depths (relative _{to the wavelength) Longuet-Higgins'}

solution shows the non-dimensional drift velocity increasing steadily with the

depth-to-wavelength ratio whereas the measurements show a levelling out. Similarly, at small

depths Longuet-Higgins' solution predicts negative velocities whereas the measurements are strongly positive.

It is not clear whether the poor agreement between theory and experiment in the body of the fluid and the free surface is, like the discrepancies observed for the flow near the bed, just due to the neglect of the higher-order_{terms. Longuet-Higgins (1953)} suggested that there might be several quite different solutions for the mass transport velocity in the body of the fluid. If this is so, the mass transport velocity at the free

surface might also be quite different from that given _{by his so-called "conduction}

solution". It is the objective of this paper to examine this question by deriving a second approximation to Longuet-Higgins' solution valid in the _{interior of the fluid and at the}

free surface, as well as near the bed. This will _{involve the derivation of a solution}

correct to the fourth-order in wave steepness.

65 _TECHNISCHE UNIVERSIrEIT Laboratorium voor Scheepshydrome Archief Mekelweg 2, 2628 _{CD Deift}

I&.:Ol5.786pO1578l

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FIG. 1. The curvilinear coordinate _{system (rì).} 2. _{THE COORDINATE SYSTEM}

In order to obtain a solution for the flow in the boundary layer at the free surface

it is desirable to adopt curvilinear coordinates which follow the water surface.

Longuet-Higgins (1953) showed that the use of Cartesian coordinates, coupled with a Taylor series expansion about the mean water level, produces a solution which is only _valid for waves of extremely small amplitude.

The water surface profile adopted by

_{Longuet-Higgins (1953) was sinusoidal.}

Although this is adequate for the first approximation _{it is not sufficiently close} _{to the}

actual wave profile if one is interested in a higher-order solution. Consequently, in this

paper we adopt a system of coordinates (, q) in which _{the line i-=O corresponds} _to

the surface profile given by Stokes' third-order solution for irrotational _progressive waves in water of constant depth. We initially expected that _{this would represent only} the first approximation to the actual water surface profile and that, once the solution had been obtained, it would be necessary to modify the assumed surface profile _and re-calculate the solution. In fact it _{was found that no correction to the assumed surface} profile was required to the present order _{of approximation.}

The transformations necessary to create this curvilinear coordinate system are very

similar to that which was proposed by _{Benjamin (1959). The details} _{are given in}

Appendix 1. The Jacobian for the coordinate _{transformation}

a(,)

(1

3(x,z)

was obtained by successive iterations. The final _{expression obtained for J is also} _given in Appendix 1. Unlike Longuet-Higgins, who adopted curvilinear coordinates only for

the boundary layer at the free surface, _{we use the} _{system defined above for the} entire flow field (see Fig. 1).

3. METHOD OF SOLUTION

We are interested in the case of progressive monochromatic waves advancing in water

of constant depth. For two-dimensional _{incompressible flow the vorticity} _{equation is}

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where:

/2

32\

D=J

_,)

3-q-,

iji is the stream function, and y is the kinematic viscosity.

We adopt the usual small perturbation technique which is to expand

dependent variables as power series in sorne small parameter, e. Hence,

LJ

lJ0 + tji1 + E2J2 + i3 + J4 +

J = Jo ± eJ1 + 2J, + e3J3 + e4J4 +

u = u0 + eu1 + 2 U2 + IL3 + U4 + W = W0 + ew1 + 2 w2 + w3 + e w4 +

all of the

The power series for the relevant variables _{are then substituted into the equation of} motion and terms involving the same power of _{are equated separately. A similar} procedure is adopted for the boundary conditions.

In order to facilitate the solution, a frame of reference which moves with the wave

celerity c is adopted. Hence:

4i = O at the bed. (This implies that the vertical component of velocity is

zero at the bed.)

iji = ch at the free surface, where h is the

mean water depth.

The other boundary conditions are:

The horizontal component of velocity is zero at the bed. The normal stress at the free surface is constant.

The tangential stress at the free surface is zero.

A fluid particle at the free surface remains there. The relevant equations are given in detail in Appendix 2.

4. _{EULERIANLAGRANGIAN TRANSFORMATION}

Once the Eulerian solution has been obtained it

is in principle a reasonably

straightforward matter to calculate the velocity of the fluid particles. _{However, an}

experimental comparison requires some care. Almost all experiments have involved the

observation of dye traces or neutrally-buoyant beads. Since the dye _{trace oscillates} backwards and forwards during the course of the wave cycle it is usual to start the

measuring process when the dye trace is at one extreme of its motion and to end the measurement when the trace is again in this extreme position. The mean velocity is then obtained by dividing the observed distance moved by the time interval between the start and end of the measurement. This time interval is not quite an integer multiple

of the wave period because the drift of the fluid particle in the downstream _direction during the measurement period produces a slight shift in the phase.

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In the past, most authors have _{evaluated theoretical solutions for} _{the mass transport}

velocity by integrating _{over a wave period. While this produces} _{no significant error at}

the second-order in _{wave steepness. the difference between}

_{this method and the}

experimental method outlined above is significant at the higher-orders of approximation.

In the comparison with experiments _{in Section 6, the} _{mass transport velocity is} evaluated as: UL = 1 ft2 u(r,$) at ₍₅₎ where r = X0 + fu(r.$) at = Yo + f v(r,$)at (6)

and (x0,y0) are the initial _{coordinates of the fluid particle}

at time r1. The limits t1 and t2 are the times at which the fluid particle is in extreme forward positions in successive

wave cycles. Details of the solution are given in Appendix 2.

5. _{DESCRIPTION OF EXPERIMENTAL APPARATUS AND METHOD}

The experimental work was conducted in the Cambridge _{University Engineering} Department wave tank. This tank has an overall length of 17.5 m. a width of 0.59 rn, a maximum depth of 0.45 m. and a beach slope of 1:20. Further details are given by

Swan (1987).

Surface contamination can have a significant effect _{on the results. In order to reduce}

contamination the tank was cleaned out and re-filled with fresh water before the start of each experiment. Once _{re-filled, a sponge fitting snugly}

across the width of the tank

vas drawn along the complete length _{in an attempt to "skim off"}

any surface deposits.

As an additional precaution _{a plastic curtain was initially}

installed near the wave

generator, and slowly dragged along the length of the wave flume until finally being

positioned at the toe of the _{beach. In this way any remnants of a surface film}

would

be trapped within the near-shore region. Finally, Teepol was added in _{approximately} 10 p.p.m. to produce a maximum reduction in the surface tension, while at the same time maintaining a minimum change in the viscosity. This successively createda situation

in which all but the finest _{deposits tended to fall through the free surface, thereby}

maintaining the required level _{of cleanliness.}

Considerable attention was also paid to the influence of the end effects on the_mass transport velocity. Longuet-Higgins (1953) suggested that the _{convection of vorticity} from the beach might _{cause a change in the mass transport velocity with time. This}

was initially found to be the case as shown in Fig. 2a. However, _{installation of the plastic} curtain (mentioned above) at the toe of the beach appeared _{to eliminate this problem,}

as shown in Fig. 2b. Figure 3 shows that with the plastic _{sheet installed the} _mass

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6O 5O Su- Drift Vetocity (mmsl) 4GO 3G0 2GO GO A 6O 500 Su-f ae Drift Ve'ocity mtnsl) 4GO 3GO 2GO 100 00 A B C O E F G H I J K measuring section. o o o A A tiOrran t2 nirt

.

t3Omir A tmir tl2Ort

FIGS 2ab. Variations in the surface drift velocity. (a) Without plastic curtain. (b) With curtain (sections AK are equally spaced along the wave flume).

this plastic curtain at the toe of the beach was originally suggested by Russell and

Osorio (1957).

Although the final experimental measurements were all taken within 50 mm of the tank centre line, the development of any three-dimensional effects across the width of the tank, or the growth of circulating cells along the length of the tank, was regularly

checked by comparing the deformation of appropriately placed dye traces. These

precautions, although tedious and resulting in a large number of aborted experiments,

ensured that a very high level of both stability and two-dimensionality was maintained. All the observations were made by observing the trace left by a crystal of dye dropped

through the water surface. For the mass transport velocity measurements in the body

B C D E F O

measuring section.

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_40030 t0.o-0-s hi-s.

4--- f-2-0-25 l-i-s.

f-40--45 l-i-s. f::&0&5 l-i-s. -20 -10 0 10 20 30 40 50 U1 (mms')

FIG. 3. The stability of the mass transport depth profile (a = 21.4 mm;/1 = 0.4 m; _{= 8.74 s' k/i = 2.94).}

of the fluid an alignment telescope was used. The telescope was mounted on a traverse

mechanism capable of both _{horizontal and vertical} _{movement. As the telescope had}

a small depth of focus (approximately 50 mm) it was possible _{to check that the sideways}

displacement of the dye streak _{was not significant. Velocity}

measurements were made at least three times at each depth. The normal procedure was for the front of the dye trace to be tracked over a distance of approximately 30 _cm.

Wave amplitude and form _{were monitored continuously throughout}

the experiment

with a resistance wave gauge. Wave period, water_{temperature, and surface temperature}

were measured at the beginning and the _{end of each test and also, on occasions, during} the course of the test.

Measurements of the gradient of the mass transport velocity at the surface were made

using the method adopted by _{Longuet-Higgins (1960). This consisted of measuring the} time taken for an initially vertical dye trace to rotate through an angle of 450 _Once again, at least three measurements were made for each test.

6. _{EXPERIMENTAL RESULTS AND COMPARISON WITH THEORY}

Figures 4ad show the results _{obtained for the gradient of the mass transport velocity}

near the free surface. Also shown _{are the theoretical solutions}

to the second- and

fourth-order in wave steepness, together with the irrotational _{solution proposed by}

Stokes (1847). Bearing in mind the experimental scatter (as indicated by the error bars

o -50 -oo Zm -150 -2aD -250 -300 -350

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-

20 dUL seC dz 1-8 1-6 14 12 1-0 08 06 04 02 2 3 4 5 2 (m2) 6 7 o 2 3 4 5 a2(m2) h 0.40 m, = 9.60 s. k1h = 3.76; h = 0.40 m, o = 8.49 s, k1h = 2.96; 6 7 dz dz 08 06 04 02 08 seC' 06 04 0-2 xiQ I I I I I I C.

----1 2 3 4 5 6 7 8 9 10 11 12 14 a2(m2)

---r

(c) h = 0.40 m, cr = 6.54 sfl', k1h = 1.84; (d) h = 0.25 m, = 5.61 s, kh = 1.04. o 1 2 5 6 7 8 9 10 11 12 13 a2(m2) x10

FIGS 4ad. The gradient of the drift velocity in the near-surface region. - - - 2nd-order solution; 4th-order solution; Stokes' irrotational solution.

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-lo O IO ₄₀

FIG. Sa.

on each point) the agreement between the measurements and the fourth-order theory is good. In particular, at the high values of kh (where k is the wave number) there is

considerable disagreement with the second-order solution proposed by Longuet-Higgins (1953).

It was pointed out by Sleath _{(1973) that Stokes' irrotational solution appeared}

to give surprisingly good _{agreement with measurements of the surface drift velocity. The}

measurements in Fig. 4a show that the effect of the higher-order terms in the solution is indeed to move the surface velocity gradient _{away from Longuet-Higgins' solution}

and towards Stokes' irrotational solution. However, it would seem that Stokes' solution

will only be a good approximation for waves of a given steepness, and not over the whole range of wave _steepness.

Comparisons between theory and experiment of the way in which_{the mass transport} velocity varies with depth below the free surface _{are shown in Figs Sad. The} _agreement

of the experimental results with the present fourth-order _{solution is less good than that} for the velocity gradient _{at the free surface. However, the}

fourth-order solution is

clearly an improvement in every case on Longuet-Higgins' _{second-order solution. This}

suggests that if the solution was extended to still higher-orders of_{wave steepness the} discrepancy between theory and experiment might be eliminated.

Figure 6 shows _{a comparison of the theory with the measurements of Russell and}

Osorio (1957). Agreement between theory and experiment _{is far from perfect but} appears to be somewhat closer with the _{fourth-order solution.}

Finally, Fig. 7 _{compares measurements of the mass transport velocity at the surface}

with the theory. Here, k1 is _{the wave number calculated from linear wave theory. The}

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o .0 Z(mm) 80 -120 Zmm) 360 -60 o -40 -80 -120 -200 -240 280 -320 40 20 O C, -b. t J 20 60 I I I 60 80 100 120 rn/sI

FIGS Sad. The mass transport depth profile.

-Stokes' irrotationa solution. 2nd-ordersolution; 4th-order solution; (a) a = 25.0 mm, h = 0.25 m, o = 7.39 s', k1h = 1.53; (b) a = 28.0 mm, h = 0.35 m, o = 8.57 Sfl'. kh = 2.66; (c) a = 26.7 mm. h = 0.40 m, = 8.87 1, k1h _{= 3.22;} (d) a = 26.0 mm, h = 0.40 m, cr = 9.71 s, k1h = 3.85. -co 20 O 20 40 60 80 100 d. . s I s

'I

sr s si al el a 60 -40 10 o 40 Z(rnml 80 120 160 -200 2'0 280 320 -160 '200 20 2S0 320 360 I I I I I I I 20 40 60 80 100 120 11.0 ₁₆₀ ₁₈₀ ti' (mm/sl

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o -loo Z (mmi Russell 2nd Osorio (1957). -'i . 15 20 -lo o lo 20 40 rn/s

FIG. 6. The mass transport depth profile. - - - 2nd-order solution; 4th-order solution: Stokes' irrotational solution; (a = 58.4 mm, h = 0.51 rn, cr = 2.09 s. kh = 0.50).

40 (u2(ø) 30 20 rO o 10 30

FIG. 7. The surface drift velocity (large kh values). _{- - - 2nd-order solution;}

--A

o

Present me.renïfs 1O-lSah <OE25)

Ackiihonat data (Russell and Osoei 1957)

-(case A: ak1 = 0.15, case B: ak1 = 0.20, case C: ak = 0.25).

than that predicted by Longuet-Higgins' (1953) second-order solution. On the other

hand agreement with the fourth-order solution is quite good.

Figure 8 shows a more detailed comparison with the

_{measurements of other}

investigators at smaller values of k1h. The fourth-order solution clearly follows the

trend of the experimental data better than the second-order solution. To _{achieve a}

4th-order solution: s s s_. 5% s 35 30 40 (khi lo OES 15

/

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60 50 40 30 10 00 - 10 - 20 - 30 ak,O02 ak,=003 ak=004 aIç006 o + I OE25 050 0.75 Exporimentat data:

Russet and Osorio (1957)

0 _{?i et al} ₍₁₉₇₂₎

+ Dyke and Barstow (1981)

akç= 008

al010

I00 125

lçh

FIG. 8. The surface drift velocity (small kil values). - - - 2nd-order solution; 4th-order solution.

closer agreement between theory and experiment would probably require the solution

to be carried to still-higher-orders. Both Russell and Osorio (1957) and Dyke and

Barstow (1981) observed negative drift velocities. This behaviour cannot be accounted

for in terms of an irrotational solution. However, the present fourth-order solution, _as indicated on Fig. 8, does predict negative surface drifts under certain conditions.

6. CONCLUSIONS

A fourth-order solution has been derived for the mass transport velocity induced by progressive waves in water of constant depth. The conclusions to be drawn from a comparison of this solution with the existing second-order solution of Longuet-Higgins

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io2 Fi F3 loi 10° lo3 F2,06 io2 loi loo F2 (kh) lOi -i Öl i3 I J I I 05 10 10 15 20 35 30 35 ¿cO (kh)

FIG. 9a-c. The fourth-order coefficients.

The fourth-order solution shows a much closer agreement with the measurements

of the surface drift velocity than the second-order theory. In particular, the tendency

of the drift velocity to level off at large values of k1h and the increase at the small

k1h are correctly predicted by the _{new solution. The fourth-order solution also}

provides a better description of the _{measured near-surface velocity gradients.}

Although Stokes' irrotational solution gives _{surprisingly close agreement with}

existing measurements of surface drift velocity, it would seem that exact agreement

would only occur for a particular wave steepness at any given value of k1h. Agreement of the fourth-order theory with measurements of the mass transport velocity in the body of the fluid is less close than for the surface drift. However, since the agreement with the fourth-order theory is generally better than with the

second-order theory it seems likely that the discrepancy might be eliminated if the solution were continued to still higher-orders of _{approximation.}

REFERENCES

BENJAMIN. T.B. 1959. Shearing flow over _{a wavy boundary. J. Fluid Mec/i. 6. 161-205.}

DYKE, P.P. and BARSTOW, S.F. 1981. Wave induced_{mass transport: theory and experiments. J. Hydraul.}

Res. 19, 89-106.

02 03

-10 20 30 40 05 10 15 20 25 30 35 40

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IsAACS0N, M.D. Sto. 1976a. Mass transport in the bottom boundary layer of cnoidal waves. J. Fluid Mech.

74, 401-413.

ISAACSON, M.D. StO. 1976b. The second approximation to mass transport in cnoidal waves. J. Fluid Mech.

78, 445-457.

LONGUET-HIGGINS, MS. 1953. Mass transport in water waves. Phil. Trans. R. Soc. A245, 535-81.

LONGUET-HIGGINS, MS. 1960. Mass transport in the boundary layer at a free oscillating surface. J. Fluid

Mccli. 8, 293-305.

MEl, C.C., Lw, P.L-F. and CARTER, T.G. 1972. Mass transport in water waves. MIT, Department of Civil Engineering. Ralph M. Parsons Laboratory Report No. 146.

RUSSELL, R.C.H. and OsoRio, J.D.C. 1957. An experimental investigation of drift profiles in a closed channel. Proceedings of the 6th Conference on Coastal Engineering 1, 293-305. ASCE. New York. SLEATH, J.F.A. 1972. A second approximation to mass transport by water waves. J. Mar. Res. 30, 295-304. SLEATH, J.F.A. 1973. Mass transport in waves of very small amplitude. J. Hvdraul. Res. 11. 369-83. STOKES, G.G. 1847. On the theory of oscillatory waves. Trans. Ca,nb. Phil. Soc. 8, 441-55.

SWAN, C. 1987. The hi2her order dynamics of progressive waves. Ph.D. dissertation. Cambridge University. Cambridge, U.K.

APPENDIX 1THE CURVILINEAR COORDINATE SYSTEM

The coordinate transformations

The (, 'q) coordinates shown in Fig. i were established by the repeated application of the

following transformations:

A coshnk(h+ì1,,)cos(nk,,)

=

-sinh (nkh) B,,

A,, sinh2k(h+1,,) sin(2k,,) B

fl+1 -

'

-sinh(2kh) n.

At each order of the perturbation the coefficients A,, and B,, where chosen such that they

eliminate the largest terms within the description of the surface profile. As a result 113 =0 contains

only fourth-order departures from the third-order irrotational surface profile, and thus (,'q) represents the required coordinate system.

The expanded Jacobian

After successive iterations Equation (1) yields:

= 1.0

J =

_2bk0sh[k(h+11)] _cos(k)

sinh(kh)

= h2k2cosh[2k(h+11)] [2sinh (k/i) + 3] cos (2k) h2k2 cos (2k)

2sinh4 (kh) + 2sinh2 (kh)

cosh[2k(h +'rì)]2}

+ b2 k2

2sinh2 (kh)

= b3k3 {cosh(2kh)+2} sinh[k(h+-q)I cos(k)

sinh (kh) sinh (2kh) cosh3 fk(h+-q)]

+ b3k3 {2sinh (kh) +3}cos (k)

sinh5 (kh)

+ b3k3cosh{k(h+11)] {l0sinh4 (kh) + 4sinh2 (kh)+9} cos (kg) 8sinh5 (kh)

J4 = b4k tAcosh4 [k(h+-q)] + Bcosh2 [k(h+-q)]+C} {cosh(2kh) + 2) sinh[2k(h+'q)]

- b4k4

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where 1 ₃ ₉ A = sinh4(kh) + sinh6(kh) + 4i (kh)' B= 51 ₉

4sinh2(kh) _sinh4(kh) _8sinh6(kh) _{4sinh8 (kh)'}

5

3 9 ₂₇

C

- 8sinh2(kh) _8sinh4(kh) + _{16sinh (k/i) + 32sinh} _(kh)

APPENDIX 2THE FOURTH-ORDER _SOLUTION

The governing equations

We substitute Equations (4) and (8) into Equation (2) and _{collect powers of}_{e. Integrating the} resulting equations and making the usual boundary layer approximation that k(oív)°5, _where

k is the wave number, o- is the wave frequency, and y is the kinematic _{viscosity, we obtain:}

ej1 = {A1e _{+ B1 cosh (k-q) + C1 sinh (ki)}e}

= {A2E11 + B2cosh(2ki-1) + C2sinh(2k)}E2' _{+ D,i-13} + E, i2+ F-, + G, + PI-,

E3LJ = {A31 e' + B11 cosh(ki-) + C31 _sinh(kq)}e' _{+ PJ1} e4k154 = D43+ E4q2 + _{F4 + G4 + PI4}

(9) where c=(o-/2v)°5, cxi=(cr/v)°5, _{and PI is the particular integral}

corresponding to the specific characteristics of the field equation.

The boundary conditions

The boundary conditions are listed in Section 3. The zero velocity_{conditions at the bed} _are

straightforward:

(a

(aL

0 -

-

-At the free surface (-q=O), the_{conditions are:} Zero tangential stress:

(Jv2p

{aJ)

{aJPF

ai1 aq a a ₎ 0

-Constant normal stress:

FJ05

-

a 1 (21

_i ₀

a

L \th1/

jJ0

where F is the resolved component of the gravity vector. Kinematic free surface condition:

(13) Applying these boundary conditions to evaluate the _{constants and making use of Equations}

(5)

and (6) [further details _{are given by Swan (1987)]}

we obtain the solution for the mass transport velocity:

UL = UL2 + UL4

(14) (10)

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where LTL2 is identical to the "conduction solution"_{proposed by Longuet-Higgins (1953). In the}

body of the fluid UL4 is given by:

= a4k3 [F1_{cosh4k(h+z) + F,cosh2k(h+z) + F3}

+ G1{3 + + _{± G2sinh2k(h+z) + G3}

-+ G4 sinh2k(h-+z) ()]. ₍₁₅₎

Here a is the wave amplitude, and the coefficients F1, F2, F, and G1, G2, G3, G4 are functions of the kh value as shown in Figs 9ac.

The solution for the boundary layer at the bed is identical with that obtained by Sleath (1972) except that the amplitudes of the velocity components just outside the boundary layer are reappraised at each order of the perturbation.

There is also a boundary layer at the free surface. To the second-order in wave steepness the mass transport velocity within this layer is given by the sum of the conduction solution assessed on the line _{=O, and an additional boundary layer term. This additional term is of the form:}

b2kosinhk(h+z)

_e'

16

2LJBL - _sinh(kh)

It is interesting to note that the mass transport velocity is more positive just outside the boundary layer than it _{is at the surface itself. Existing measurements of surface drift velocity} relate to the value just outside the surface boundary layer. Further details of the solution are