A note on the efficiency of propulsion of bodies in waves

TECHNISCHE HOGESCHOOL DELFT LUCHTVAART. EN RUIMTEVAAPTTECHNIEK

A note on theefficiency of propulsion of bodies in waves

by R. Coene

Abs tract

Bodies which absorb, reflect or generate wave energy are submitted to mean forces. For moving bodies the mean forces in the direction of motion contribute to the drag or propulsion of the body. For slender bodies swimming in waves at a constant forward velocity LI normal to the crestsof the waves, the mean rate of working W and the mean thrust T are evaluated. It is shown that the Froude efficiency of propulsion for cases without shedding of vorticity is given by the number tJ/(U+c),

(U+c) being the phase velocity of the waves with respect to the body. The result remains valid when shedding small aiounts of vorticity. T is closely related to the radiation stress and this result for the efficiency is independent of the details of the motion.

The same efficiency can be obtained when calculating the propulsion of two-dimensional bodies oscillating in regular trains of

two-dimensional

waves. The Froude efficiency of Longuet - Higgins' wave making boat is found to be 16%. For three-dimensional cases the number U/(U+c) represents the upper limit when the outgoing waves are properly beamed.

b

Ib'I

1

1. Introduction

In [1] and [2] a slender body potential flow theory _{was worked out for the} swimming of slender bodies with tails in an oncoming _{train of regular surface} waves. Head seas as well as following seas with the crests of _{the waves}

perpendicular to the direction of motion of the body _{were included. In}

particular, the effects of the adaptation of the voluntary _{part of the swimming} motions to the waves on the efficiency of propulsion _{were discussed. As usual,} the propulsive force was assumed to balance, in the _{mean, the frictional drag} and the wave drag, although the incoming _{waves were assumed to be essentially} undisturbed by the swimming body. In [3] _{some towing tank experiments are} described which tend to confirm several results obtained _{in [1] and [2]. In} the theory developed in [1] and [2] the influence _{of the tail on the propulsion} in waves was studied in particular. In the _{present paper we shall cOncentrate} on the influence of the body-wave interactions which also exist _{when the body} has no tail, i.e. when no vorticity _{can be shed into the wake, or when the tail} is inoperative or only weakly operative. It is _{found that in contrast with the} situation in uniform oncoming flow, _{thrust can be generated on the slender} body by properly adapting its motions to the oncoming waves.

The Froude efficiency of propulsion is found _{to be independent of the details} of the voluntary motions, although the magnitude _{of the mean thrust and the} mean rate of working do depend on the details. The _{mean thrust or drag on the} body has the form of .a radiation stress and on the other hand these forces can be interpreted as Lagally forces. Moreover the efficiency _{of propulsion is} found to be strikingly similar to the Froude _{efficiency for an actuator surface.} This suggests that the results should be applicable _{to a class of swimming}

problems which is more general than the slender _{body case. And indeed, as will} be shown, the.same efficiency of propulsion _{can be realized in the 2-D case.}

In the 2-D case it remains valid even when the _{body is swimming in its own waves.} In the absence of forward speed the 2-D results _{correspond with those for wave} power machines and stationary vessels as discussed in _{e.g. [4] and [5].} The calculations are based on bilinear potential _{flow results for cases without} shedding of vorticity, i.e. constant circulation _{solutions in the 2-D case, but} with an exception for the slender body _{case. The 2-D results are complementary} to those of [6] and [7] where the shedding of _{vorticity plays an important râle} in the generation of propulsion. As in [i] _{and [2] the oncoming waves in the 2-D} problems of [7] are assumed to be unaffected _{by the presence of the body or the}

hydrofoil. In several circumstances this _{assumption is more realistic for the} slender body case than for the 2-D case when important fractions of the energy in the waves are being extracted.

slender body swimming in a regular train of waves: L (x,t) = -(. +

._) (wA) +

(. +

Here w is the resultant cross flow defined by

w*=.+U_*

wz

The * indicates a quantity at the mean swimming depth, _{being the vertical} component of the orbital velocity of the wave motion. In an oncoming wave of the form 2ir 2iTf = ac e cos -- 1x - (tJ+c)t} a w one has with = * _{cos {x - (U+c)t}} wz 2Trd 2rr

-X

a*_ac_e

In the deep water case one has, nreover, in terms of the _{wave number k =}

2

kc =g.

which on substitution of (2.1) can be exressed _as

(2.6)

In (2. 1), _{A represents the virtual mass of the body section per unit length}

and S is the surface area of the section. We note that in (2.1) there _{is no} contribution from the boyancy of the body. In the absence _{of waves and body} motions there is no lateral force, the body is then in the 'stretched

straight' position. From (2.1) we obtain for the _{mean rate of working by} the body: £ =

- J

- L (x,t) dx, _(2.7) (2.3) (2.4) (2.5) (2.1) (2.2)

We have Iah +

w=pA(2.) U

1x=2.. 9. 9.

+ p

1 (A+S) - V" dx + pr.i

J -

dx at wz dx oJ ₀

The determination of the mean thrust is carried out in [2] in a coordinate system fixed with respect to the body, i.e. with

X = x, Y = y, Z = z-h (x,t) and T = t. (2.9)

Then using Bernoulli's equation and evaluating terms up to and including those of orders C2U2 and C2U2 log C one obtains for the mean thrust:

0

3

(2.8)

*

\22(\21

T = 'pA(9..) 3t - wz) vax) x=L 9'. (A+S) w*()* _{dx + PU}

I

h * dx (2.10) dx wzx w zx _oJ

A detailed study of the contribution of the tail terms, in square brackets in (2.8) and (2.10) was given in [2]. As described in [3] experimental results, with an oscillating rigid delta wing, gave a somewhat better than just

qualitative partial confirmation of the expressions (2.8) and (2.10) for a not too deeply submerged slender body. In this section we shall discuss the integrals in (2.8) and (2.10). The distinction in 'tail terms' and 'body terms'

is reminiscent of a similar distinction in vortex drag and _{wave drag in} super-sonic wing theory. It may be noticed that the integrals in (2. 10) are readily interpreted as the Lagally forces on a distribution of singularities: The vertical component of the dipole strength is proportional to (A+S)w* and the velocity gradient in the (vertical) direction of the dipole is given by

a

= _{. The second integral arises from the effectivediolestrengthwhen} wxz

sources of a .strength proportional to U - are displaced over a distance h while in that direction there is a velocity gradient (

wxz

The body integrals, yield a thrust (or a drag) and a rate of working for a

body withoutatailorwithatailwhichis

_{inoperative by virtue of the fact}

=- (U+c)

wt wx

_* =--

c*

-

* t wz 3t wz wzt

x+pU

h*

dx, (2.11) wzx dx wzx

0

{

Tb = 0" (A+S) d w

= p

1 (A+S) '' dx + p U - r'' dx. (2.12) b at wz J dx at wzx

The wave potential (2.3) is periodic in x and t and depends on the variable x - (U+c)t, so one has

For swimming motions that are not correlated with the oncoming waves the integrals vanish. For oscillatory swimming motions characterized by the frequency of encounter w = k (JU+c), and being periodic we have

= (U+c) w _(2.14)

WZX

Moreover with Wk = W

-

, one obtains wz w j* _{- = w** +} * = W.IX wZX wz

wx

= _{+ ½} WZX 3x wt = %tZX

Similarly one has

- = - (h ) - h ' wz wzt (2.16) = (U+c) h * w zx

Substituting the right hand sides of (2.14), (2.15) _{and (2.16) in (2.12) and} comparing with (2.11) then yields

Wb = (U+c)

(2.17)

which can also be expressed in terms of _{the Froude efficiency as}

(2. 13)

--TbU _U

-(2. 18)

U+c

It may be noted that (2.18) is strikincTly _{similar to the Froude efficiency} of an actuator surface

1

fl

(2.19)

where C is the axial induction _{factor. In (2.18) then, c/U plays the}

rÔle of the axial induction _{factor, while Tb in (2.17) is the radiation stress on} one may say, an actuator body.

Equation (2.18) has been derived for arbitrary _{but correlated motions. One} may have thrust or drag in the mean. For negative _{value.s of U+c thrust may be} derivedbydissipating energy. Flexible and rigid _{bodies doing no work in the} mean will experience no thrust or drag forces in the _{mean. We now consider} one example in some detail. The equations of motion _are

ah

p

J

Si dx=

1Ldx,

0

3t o 12. p j

xS dx=

1xLdx.

0 3t

A _{particular solution R (x,t), which}

was also used in [2],of (2.20) is obtained by equating the local time rate _{of change of the lateral momentum}

of the body and the local force exerted _{on the body by the time varying part of}

the

pressure of the water:

2-.

pS(x)

L (x,t).

at

The local mean rate of working is _{equal to zero:}

2

a (afi\

- - _{L (x,t) = - P S(x)}

-

-j

= 0,

and after integration with _{respect to x one has}

Wb = 0,

(2.23) (2.20)

(2.21)

so by virtue of

(2.17)

one also has, for iJ+c*0,

Tb = 0,

(2.24)

for the flexible recoil nxde. Obviously, this is also valid for bodies with

tails which, in this case then, are inoperative because at the tail S(9.)

= 0

and at the tail one has

j;

=('-+-\

_*

x2.

\3t

x)L -

wz(x=L)'

which can also be expressed as

;*

₌₀

x=9..

(2.25)

(2.26)

by virtue of (2.2).

The general case of swiiruning motions in waves can be decomposed

_{as follows:}

h(x,t) = f(x,t) + f(x,t),

(2.27)

where

_{(x,t) is the flexible recoil mode, i.e.}

_{a passive swimming mode which}

is a particular solution of the equations of rnotion.f(x,t) is

the voluntary

part which may or may not be correlated with the oncoming

waves. This swimming

problem is discussed in [1] and [2]. In (1] it

was noted that with the

additional ussurfiption

f << f

,

_(2.28)

(2.8)

and

(2.10)

yield upon retaining linear terms in f only, for the tail terms:

Wt=PA(9)

_3x)

(2.29)

I.

af\1

_(2.30)

Using

(2.13)

and

(2.25)

we have, at x=2,

U+cJ

In a coordinate system (x1, _{y1, z1, t1) with}

x1 = x-Ut, y1 = y, = z, t1 = t

which is fixed to the water at rest far from the _{surface the wavelength of the} paths of the body sections, i, is _{given by}

J U

3: _{(2. 36)}

thus relating the efficiency _{of swimming to the}

wavelength of the oncoming waves and the wavelength of the

swimming motions with respect to the water

at rest I

The previous results were obtained on the assumption _{that the oncoming waves} are not significantly affected by _{the swimming body in as far as}

and

at the mean swimming depth

are concerned. The possibility to interpret the integrals in the thrust _{T (2. 10) as Lagally forces}

suggests that similar results should be valid in _{the two-dimensional case. In the next section it will} be shown that this is indeed the case.

-7-so that (2.29) and (2.30) lead to

- (U-f-c) ₌ 0,

(2.32)

and

for f<<i

. (2.33)

Thus even with a weakly operative

tail which implies the shedding of vorticity, the number _{represents the efficiency of swimming.}

As shown in [1] the other possibilities with f _{and f>>Fi yield other efficiencies.}

It may be noted that the number

has a simple geometrical significance also. All parts of the swimming body _{in a correlated swimming}

motion oscillate at the frequency of encounter:

(2.24)

3. The two-dimensional case

We assume that a swimming body has a constant mean velocity U with respect to the waterat rest in a horizontal direction perpendicular to the crests of regular deep water waves. The body is deforinable but there is no shedding of vorticity. As indicated in the figure there is an incoming head sea of amplitude a and phase velocity c in the positive x1directiori at x1 = -At the same time at x1 = - there is an outgoing wave of amplitude a' and negative phase velocity c'. At x1 = we assume an outgoing wave of amplitude b and posItive phase velocity c. Finally, at .x1 = we have in incoming wave of amplitude b' and negative phase velocity c'.

LI

.'-4.I

)

I

- -

-'

_11.(L3

dj

-

_{- %}

,_______I%._._,

I, - - - .

-The introduction of the four waves permits a rather general discussion of the two-dimensional swimming problem in waves. The flow is periodic in a reference system moving steadily (at constant velocity U) with the swimming body. Far from the body, the incoming and outgoing waves are also periodic in the

reference system (x1, z1, t1) fixed to the water at rest far from the surface defined in (2.35).

Referring to the results obtained in the appendix, we have from (A.28) for the mean excess momentum flux at x1 =

-1 (a2 + a'2),

(x =-)

e 1 At = one has (x = = p g (b2 + b'2). e 1

From (A.29) with positive c and negative c' one has for the influx of energy at x1 =

-i (x1 = - co) = p g (ca2 + c'a'2). (3.3)

Similarly, at x1 = the outflux of energy is

(3.1)

9

1

W (x1 = c)

= p g (cb2 + c'b'2). (3.4).

The mean rate at which momentum in the positive x1-direction is being increased follows from (A.26)

2 2

ui=pgu(b.;a+bI2_at2)..

C,

On the other hand, the mean rate _{at which wave energy is being added to the} flow follows from (A.27) as

= _{p g U (b2 - a2 + b'2}

- a'2). _(3.6)

We are noW in a position to evaluate

the mean thrust on the body Tb and the mean rate of working by the body in the swimming _{motions Wb. The mean thrust}

is easily obtained from (3.1), _{(3.2) and (3.5) as}

Tb = P g (b2 - a2 + b'2 - a'2) 1 (b2 2 b'2 - a'2

+pgU

-a

+

_).

The mean rate of working by the thrust _{is _TbU. so we have from (3.3).,}

(3.4)

and (3.6) for the power balance

1

Wb - TbU = P g {c(b2 - a2) + c'(b'2 - a'2)} +

+ p g U (b2 - a2 + b'2 - a'2). _(3.8) Substituting (3.7) in (3.8) gives:

r(U+C) (2U+c)

(b2 2

wb_4Pg[

C (U +

C')

(2u +

C')

(b'2 - a12)] _(3.9 (3.9) C,

It is rewarding to express the mean thrust (3.7) in a similar way:

Tb - P r(2u

_{+ C)}

2 2 _{(2U + c')} (b'2 - a'

-- g[

.

(b -a)

The expressions (3.9) and (3.10) _{permit a systematic treatment}

of the propulsion problem. We begin by considering _{some special cases.}

(3.5)

(3.7)

With undisturbed following seas and head seas, i.e. with (b'2-a'2) =0

and (b _a)=0oneobtainsib=0andb=Q.

With b'2 - a'2 = 0, (3.9) and (3.10) yield:

and

It should be noted that this remains valid without incoming sea,

a2 0, i.e. for a body swimming in its own waves With a2 = 0 these expressions also apply to Longuet-Higgins' wave making boat [4]. At a frequency of 3s' of the oscillator at the stern his boat was propelled at a speed U = 0,12 rn/s. With

2ir

w

e

gA

2ir'

we obtain c = 0,625 rn/s and A = 0,25 m, implying an efficiency

0,12

- _{- 16%.}

0,12 + 0,625

The mean thrust which balances the drag (frictional + wave resistance) follows from (3.12) as

- 1

2U+c 2

T=-pg

b.

1 (U + c) (2U + c) 2 2 (b

-a),

C 1

2U+c

2 2 (b - a ),

=-pg

U

U+c

We point out that for a given waveheight, b, the thrust is 38% larger with the estimate (3.17) than the thrust estimated upon neglecting the forward speed.

In this case Wb and Tb have the same sign and it follows that in order to generate a positive thrust at the efficiency (3.13) there can be no extraction of energy from the oncoming head sea. In (1], (2] and [31 it has been shown that this is not necessarily so for propulsion of the vorticity shedding type.

(3.14)

(3.15)

(3.16)

2 2

(iii) With (b - a ) = 0, (3.9) and (3.10) yield:

1 (U + c') (2tJ + c') (b'2 -. a'2), ' W = P g - 1 _(2tJ+

T=Pg

_ci n

-C')

(b'2 - a'2),

In (3.20), c' < 0 son> 1 with U + c' >0. For _{U + c' < 0 one} can generate thrust by extracting energy from the waves, implying a negative rate of working by the body, _{Wb < 0 at positive mean} thrust Tb > 0!

(iv) _{The relevance of the group velocity of the waves becomes evident}

by swimming at the group velocity. One then has 2U + c' = 0 because c ' = c'. The terms associated with the following sea

g 2

drop out!

Cv) _{With (b2 - a2) = 0, a'2 = 0 but a following sea, b'2 0, with a} phase speed equal to the swimming speed,

_Ic'I

_{= U, c' = -U yields} for the thrust:

Tb D = - p g _(3.21)

which reproduces the classical result for the wave drag D.

(vi) _{An interesting case arises when a body is swimming in a sea which}

has no waves in the absence of the body _{and generates a thrust} which, in the mean, precisely balances the _{'wave drag!,}

D = _{p g} b'2. _(3.22)

One has, for this case with perfect beaming, i.e. _{a' 0, or by} neglecting a'2 with respect to b'2:

Wb4

_--pg

(U+c) (2U+c)

2

b,

C 2 1 c

b --pgb'2.

Tb vanishes with (3.23)-(3.24)

(2tJ

+ C)

b2 = c b'2. (3.25)

With (3.25) the thrust generated by means of the outgoing waves precisely balances the 'steady wave drag' (3.22). We observe that, again, the net thrust is being generated at

U

In this context we note that wavemakincr propulsion is efficient when the outgoing waves are such that c << U. At low speeds only ripples yield efficient propulsion. At high speeds longer waves become attractive. It will be clear that an additional frictional drag may be balanced by generating a wave with b2 larger than follows with (3.25).

(vii) Consider a head sea which is weakly disturbed by the swimming motions of the body: Behind the body we can assume waves of the form

(atz=0):

= aCcos k Cx - Ct) + + a*ccos _{k Cx - Ct)}

_{+ P},}

(3.27) with small a*

-= 0(c).

a

Using (A.19) we have for the amplitude b of the disturbed wave

2 2 2

b =a +a*+2aa*cos=a +0(c).

(3.29)

/ ,2

At the same time one has v-) = 0(E2) for the wave which runs ahead of the body in the direction of motion of the body. In the next paragraph (viii) it will be seen that above a certain frequency one

2

has a' = 0 in this case anyway: one has (-) ₌

_{0Cc )}

at most. Upon substitution of (3.29).in the expressions for the thrust and

the rate of working we find that _{Tb = 0(c) and = 0(c), implying} than an 0(c) thrust can always be generated at an efficiency

U

=U +

c

(3.28)

13

-One may say that due to the presence of the oncoming wave, the 0(C) terms in the momentum and the energy flux are perfectly beamed opposite to the horizontal mean force on the body. With cos > 0 one has

thrust and with cos Tj < 0 one has drag. The beaming of the 0(C) terms is taken care of by the oncoming wave. To obtain thrust it is

sufficient to impress the proper phasing. Similar conclusions apply in a following sea, with, as in (iv) the efficiency Ti > 1 or even

Wb < 0 at Tb > 0!

(viii) With U = 0 and an incoming wave of amplitude a at x1 = - and no incoming wave at x1 = , b' = 0, the outgoing wave of amplitude a'

at x1 = - is the reflected wave and the outgoing wave of amplitude b at x1 = is the transmitted wave, as in [4]. In this case the flow

is periodic and one has c' = - c. The power-balance is

=pgc (b -a

Wb

2 2

and the force on the body in the direction of propagation of the incoming wave is

Tb p g (b2 - a2 - a'2). (3.31)

In the case àf power absorption, Wb < 0, one has b2 < a2 - a'2 from (3.30). Then with (3.31) one has Tb > 0. With power extraction in this case one always has a mean force on the power absorber in the direction of propagation of the incoming wave, as expected.

(ix) With U 0 the frequency of encounter of the incident waves is

w

= k(U + c)j, (3.32)

and the flow is periodic in a reference system moving steadily with the body. For the 'reflected' wave of amplitude a' the dispersion relation yields:

(3.30)

)2

(3.33)

Considering (3.33) as an equation for k' shows that the discriminant

2

cU

_e<1

g

4

For the generation of propulsion this is quite important because it follows that with

wU

e

>4'

one has no 'reflected' wave: one has a' has a following see from the start. One one always has a perfect beaming in the is being transmitted to the "wake", and at an efficiency

g

U

= U +

for, with (3.35) and (3.32):

(U + c) U >

By virtue of the dispersion relation this can also be expressed as

4(U + c) U > C2. (3.37)

Equivalently, at given speed, the frequency of oscillation w of the wave-maker should satisfy

ü) > (3.38)

for perfect beaming.

= 0, unless of course one may say that with (3.35)

2D case. All the energy thrust is again generated

(3.36) (3.34)

15

-4. Discussion

Several examples of unsteady propulsion in waves have been, worked out. In the slender body case we started from expressions- for the mean rate of

working and thrust in terms of the geometry of the body and, its oscillations in relation to the oncoming waves. Then it was noted that for bodies which shed no vorticity or only relatively small amounts of it, thrust can be

generated at an efficiency Ti = U/(U + c) which is independent of the

details of' the swimming motions. On the ,other hand the magnitude of the thrust and the rate of working depend on the details of the body geometry and its oscillations. Moreover, they are of 'interaction' type and linear in the waves and the body oscillations. In view of the striking analogy between the interaction efficiency and the Froude efficiency for an

actuator surface, one might say that- the factor c/U plays the role of an axial induction factor for actuator bodies in waves. The thrust or drag

force in the direction of propagation of the wave is obtained in the form of a radiation stress. As for actuator surfaces the results for the

propulsive efficiency of actuator bodies remain valid in the two-dimensional case. This has been shown by means of the far field incoming and outgoing waves. In several cases the propulsive efficiency was found to be independent of the details of the geometry of the body and its correlated swimming

deformations.

In the slender body case the assumption was made that the oncoming wave be unaffected at the mean swimming depth. In the two-dimensional case though, the result for the propulsive efficiency remains valid in several examples where the body is swimming in its own waves! In the more general problem of a three dimensional body swimming in 'waves the solution depends on the compactness of the body and on the possibility to beam the fluxes of momentum and energy to the rear.

REFERENCES

The swimming of slender fish-like bodies. R. Coene. Swimming and Flying in Nature, Vol 2.

Plenum. Press, New York (1975).

The swimming of flexible slender bodi-es in waves, by R. Coene; 3. Fluid Mech. (1975) vol 72, part 2..

A slender delta wing oscillating in surface waves. An example in unsteady propulsion. R. Cbene;

Department of Aerospace Engineering. Report LR-257, Ship Hydrodyn. Lab. Rep. 456. Deift (1977).

The mean forces exerted by waves on fibating _{or submerged bodies with} applications to sand bars and wave power machines. Longuet - Higgins; Proc. R. Soc..London A.352 (1977).

The interaction of stationary vessels with regular waves, by J.N. Newman (1976).

Proc. 11th symposium on naval hydrodynamics. London 1976.

Some ideas about the optimization of unsteady propellers, by J.A. Sparenberg (1976).

Proc. 11th symposium on naval hydrodynamics. London 1976.

Extraction of flow energy by a wing oscillating in

waves (1972). T.Wh

2

c k = g.

a-i

Appendix

Derivation of some properties of superimposed deep water surface waves.

We begin by considering a single two-dimensional deep water surface wave of the form

kz

= ac e cos 1k(x - ct)}, w

for a wave of amplitude a, and a phase velocity c in the positive

x-direction. The wavelength A..is related to the wave number k by k = 2TrX'.

The water surface is given by

= -a sin {k(x - ct)}. _(A.2)

For deep water surface waves the dispersion relation can be expressed as

(A.1)

(A.3)

The time-averaged horizontal

component

of momentum, per unit surface, is easily calculated: 2ir

I

-

p

dz dt = -pg1

WX

wtz=O =

2rr wx o -1 2 1 2 -1

=pack=-pagc

, (A.4)

where the overbars indicate that the time average over one period has to be calculated; 1 is positive in the positive x-direction, i.e. in the direction of propagation of the wave.

The excess potential energy due to the waves per unit horizontal area is given by

- 1

21

V = p gj z dz =

p g = p g a . (A.5)

0

The kinetic energy in the waves per unit horizontal area has the same magnitude:

K =

_{w wzz=O -}

p g a2.

Thus for the total energy per unit surface one has

- - - 1 2

E=V+K=Pga

We note that

= I c,

and it will be shown that a similar relation exists for the fluxes of

momentum and energy respectively.

The mean flux of horizontal momentum across a vertical plane x = const per unit length in the direction parallel to the crests of the waves is given by

In the deep-water case one has

- = 0, so, to second order, with =

(p + p dz. (A.9)

Subtracting the flux in the absence of waves we obtain the mean excess

flux due to the waves: (With

p = p0 +

_{e =}

- pg z +

FJ

_JPgzdz.

(A.10)

- 0

From Bernoulli's integral we have

Pe+

1

(2

_{wx wz}) + p = c(t), (A.11)

and for periodic flows

= -

i(

42 ).

wz

(A.12)

We note that p does nOt dpend on x but 2 , vhioh also ppaar--4ft

e. wx

To second order (A.1O) can be evaluated as

0 C

F'eJ

4p

(22

)

dz_Jpcdz_PgJZdZ.

wz

(A.13)

- g

=

wtz=O'

- 1 -, 2 1 2

F=-pg

The mean value of the excess momentum flux does not depend on the direction of propagation of the wave.

The mean value of the energy flux across a vertical plane x = const per unit length in the direction of the crests of the waves is given by

o 0

W=JPe=PJtwx)C1Z

(A.15)

-

-correct to second order. Substitution of (A.1) then gives:

= - p (2k) 1:W

wzz=O =

p g a2

C.

(A.16)

Comparison of (A.14) and (A.16) shows:

W=F

C.

e

We now calculate these quantities for a superposition of two waves, with the same phase speeds. We first consider two waves propagating in the same direction: = w1

w2'

(A.17) with, at z 0,

Jwi

= a1cos k (x

-(A.18) w2 = a2Ccos {k(x - Ct) +

Upon substitution of (A.17) and (A.18) in the expressions derived above 2 it is found that in the time-averaged values one can simply replace a

by b2 with

a-3

.2 2 2

b = a1 + a2 + 2a1 a2 cos

b being the amplitude of the resulting wave.

So we have in this case a time averaged interaction and we can write

(A. 14)

simply

{

(

= _{p g}

c'

1

E12-2pgb

1 = _{p g} b2 1

cb2.

\

_{1+2 =} p g

Next we consider two waves propagating in opposite directions:

= w1 +

w2'

Upon substitution of (A.25) in the time averaged expressions involving terms proportional to there arise interference terms proportional to

a1a2 sin k Cx - Ot) cos k (x₊

Ct)

cos P which depend on x but vanish upon

taking the average over one wave length. At the same time, then, the phase angle vanishes from the expressions.The result is,for these double

averages:

11+2

=pgc

-1

(a12

-a2)

2

E12

T p g

(a12

₊

a22)

Fe12

= p g +

a22)

= P g C

(a12

-

a22)

For the time and space averaged energy of the waves and the flux of

momentum we have addition without interference in this case. On the other hand, for the momentum and the energy flux we have subtraction.

It will be clear that in cases of superposition of waves with different wave speeds the long-time averages of the interaction terms will vanish.

(A.26) (A. 27) with, at z = 0, w1 = a1ccos k (x -w2 =

a2cos

{k(x

+

Ct),

Ct) +

(A.25)

a-i

For waves propagating in the same direction (A.20) to (A.23) then become:

_1

2-1

2-1

1+2 - p g (a1 c1 + a2 c2

1 2 2

= _{p g (a1} + a2

_1

2 2

1+2

-i-pg (a1

+a2

W12

= p g (a12c1 + a22c2) (A. 33)

Equations (A.30) to (A.33) now remain valid, for the case of waves with different wavelengths and phasevelocities. in opposite directions, say with c1 > 0 and c2 <0. This is evident when (A.26) to (A.29) are compared with (A.30) to (A.33).

It is interesting to note that (A.19) implies that a small perturbation to an oncoming wave is more significant in the direction of the phase velocity of the oncoming wave than in the opposite direction. With a <<a

a2 2

say with - = 0(C) where C is a small number, the interaction of waves

a1

propagating in the same direction at the same phase velocity is 0(E)

whenever cos j 0. In the case of waves propagating in opposite directions

there is no 0(C) interaction. For weak perturbations with respect to an oncoming wave one should expect a strong directional bias which depends on phasing. We observe that in view of the symmetry of the interaction term in a1a, it follows that the same conclusion is valid for the 0(C)

ai