Wave patterns in a stream at near-critical speed

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TECHNICAL REPORT 117-12

WAVE .PATTNS IN A STREAM AT NEAR-CRITICAL SPEED

By

G. Dagan

January

₁₉₆₈

DISTRIBUTION OF THIS DOCUNT IS UNLIMITED

Prepared Under

Office ofNaval Research Department of the Navy Contract No. Nonr-3349(OO)

TABLE OF CONTENTS

Page

ABSTRACT 1

INTRODUCTION 1

THE HOMOGENEOUS PROBLEM:. THE SOLITARY WAVE 3

FLOW PAST A SINGULARITY 12

DISCUSSION OF RESULTS AND CONCLUSIONS 21

APPENDIX - THE INTEGRATION OF EQUATION [i] 23

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LIST OF FIGURES

Figure 1 - The Solitary Wave

Figure 2 - _{A Vortex in a Uhiform Stream}

Figure 3

-

A Vortex in a Critical Speed StPeam Schematical Representation of the Free Surface Profile

Detailed Representation in the Matching Region

NOTATI ON

Amplitude of solitary wave

A,B,C,D Constants in the solitary wave equation Elevation of vortex singularity (b = b'/')

f(z) Complex potential

F' Drag force (F = F'/pg.'2 )

h' Solitary wave depth at infinity (h = h'/')

k Integration va'iable

Depth at crest(soltai'y.;.wav) and upstream (singularity).. N Outer free-surface elevation (N

u',v' Inner velocity components

I

I

(u = u'/(g,')2, v = v'/(g.t')2)

u' Velocity at solitary wave crest

A

=

7

, 2

U,V Outer velocity compOnents (dimensionless)

w Complex velocity

x',y' Horizontal and vertical inner coordinates

(x = x'/', = X,Y Outer coordinates

z Complex variable

a Dimensionless wave number

/3 Constant in the solitary wave equation 1 3.

Vortex strength (5 =

C _{Small parameter}

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-1-ABS TRA CT

The two-dimensional free-surface flow generated by a singu-larity moving with near-critical speed (i.e. with Froude number referred to the water depth. near unity)is solved by using the method of matched asymptotic expansions. In the vicinity of the

singularity the problem is solved by an infinitesimal wave ex-pansion (inner exex-pansion) while at large distances from the singu-larity shallow water theory provides the proper solution (outer

expansion). A composite expansion provides a uniformly-valid so-lution for the velocity components, the free-surface profile and

the drag force. These two basic approaches of water-wave theory are shown to be solutions of the same flow problem, but valid in different regions. Although the inner expansion satisfies linear

equations, the solution depends nonlinearly on the small parameter of the problem (the singularity strength).

INTRODUCTI ON

A singularity moving at constant speed in finite-depth water is considered herein. The flow is two-dimensional and it is

steady when referred to a moving coordinate system. In this sys-temthe standing singularity is considered to be a perturbation of a stream.of uniform. velocity.

The nonlinear free-surface problem has been solved by a first order infinitesimal wave expansion for different types of singularities (Wehausen and Laitone,

_1960,

review the solutions).

All solutions diverge, however, when the unperturbed velocity approaches its critical value, i.e. when the Froude number based on stream depth tends to unity. It is generally assumed that the linear theory fails in this case (Stoker,

_1957,

_p. 217).

The other basic approach, nonlinear shallow water theory, is able to predict the existence of waves of finite amplitude at near-critical speeds; but this theory, cannot represent flows in the vicinity of singularities, because it is based on the assump-tion that the horizontal velocity component is almost uniform and much larger than the vertical one.

The purpose of this paper is to solve the problem of criti-. cal flow by using the method of matched asymptotic expansions

(Van Dyke,

196k).

The infinitesimal wave expansion will be shown

to provide a meaningful solution in the vicinity of the singularity and will be called, consequently the inner expansion. This expan-sion diverges at a large distance from the singularity. There, an outer expansion, derived according to matching requirements, de-scribed the flow, conditions. This outer expansion is precisely the well-known shallow water theory. A composite expansion, therefore, solves the paradoxal problem of critical flows. The results, however, go beyond this purpose; the.y show that the two basic theories of water waves are organically interrelated by the method of matched asymptotic expansions.

The present paper is an extension of previous workin which the method of matched.asymptotic expansions was applied to free-surface flows in porous media .(Dagan,

_.1967).

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-3-THE HOMOGENEOUS PROBLEN: -3-THE. SOLITARY WAVE

The derivation of the solitary wave by the method of matched asymptotic expansions will be discussed in the first stage..

With the variables of Figure 1 made dimensionless b.y re-ferring the velocity components u, v to (g2' )2 and the lengths to 2', the exac.t equations of flow are

+ v2

v - ui = 0

x

v=0

r =1

UA being the velocity at th.e crest.

The solution for the flow in the vicinity of crest A is sought by a first-order infinitesimal wave expansion whose zero order term is a uniform flow of

velocity.0

and depth TL = 1, i.e. (y =

(y=0)

[5]

(x=0)

_[6]

(lxi =oo) [r] U

+v

y

=0

[1] u y

-v

x

=0

[2]

U =

U +EUi

0

V=Vi

= 1

+ 111

U =U +U

_{A a A1}

E being the small parameter.

Bysubstituting the expansion {] into Equations [i]-[6], and discarding the condition at infinity [q], one obtains the usual linearized equations

[8]

Considering, for the sake of simplicity, only the region

x 0, the elementary solution

of

Equations [9]-[lk] tnay be written as

U1x + Vi;

=0

(0 < y < 1) [9] U

-.V

ly

i K

uu1

+

=0

=

uuA

[10]

[11]

(y 1) vi

-uTI

0 3-X 0

[12]

Vi = 0 (y

= a)

[13]

=0

(x

= a)

[1k]

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U = U +

0

-5-u1 = c (cosh.a - cos ax cash ay) ± C sin ax cosh ay + UA _[15]

v1 = -c sin ax sinh ay _- c sinh ay cos ax

= -c u cosh a (1-cos ax) - c u cosh a sin ax

so

Co

where a is. the root of the equation

tanha

₂

-u

a a

and u 1.

0

The solution is continued n the region x< 0 by replacing x by -x in Equations [l5]-[17]. The two arbitrary constants c

and C multiply, therefore, a smooth solution at the origin and

a cusped one,.respectively (equation 17).

The above well-known solution is periodic and cannot satisfy the requirements of nonperiodicity and uniformity at

lxi =

ex-pressed by Figure 1 and Equation [7]. In the vicinity of the origin,however,.and for small a (i.e. .0 close to thecritical speed u =1) the solution [15]-[l8] may be expanded in a power series as follows

a2 (x2 -y2+1)

cax+

v =

(-cay

-

ca2xy)

_[20]

22

ax

-=1+ -cuax-cu

_cc

_{so .2}

[21]

.tanha

=1 --+ ...

=2

[22] a ₃ a

with the highest order ret-ained.being

0(a2).

The solution expressed by Equations [191-1122] becomes

un-bounded a.s x , the expansion being therefore singular there.

Considering, this as an inner expansion, an outer expansion is sought according to the usual procedure (Van.Dyke,

196k)

by adopting the following outer variables

1 1

X

= EX

Y=y

U = u

v= 2V

[23]

The outer variables of [23] have been selected so that for large x the quadratic term in Equation [21] becomes of the order of magnitude 0(1) with respect to .

The inner variables may now be expanded tentatively in an E

power series

TJ=U+U1+...

.

[2k]-V =

V1

+ ...

. /

[2]

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-7-The Equations [23]-[26] represent precisely the shallow water expansion of Friedericha and Keller (Friederichs, 1948).

The outer expansion of the exact Equations [l]-[6] has been carried out systematically by Laitone (Laitone, 1960) and will not be repeated here. The first order equations are the well-known shallow water equations, the solution being (Laitone, 1960)

1

U = N 2 = const.

0 0

Ui = f(X) N1 = -{u0f(x) + C] Vi = 0 [28]

The function f(X) satisfies the differential equation

fxx-

f2 ---2f=0

2U5

0 0

where C is an arbitrary constant.

Since the solution has to be used only in the outer zone, the nonlinear Equation [29] may be easily solved by iterations in the vicinity of X = (with f(co) = 0); but we may take ad-vantage of the fact that the exact solution of [29] is well known being in the nonperiodical case (Laitone, 1963)

[27] [29] f(X) = - sech2 [13 _C 0 2 (x + )] [30]

where /3 is an arbitrary constant.

The constants u , u

, , c

, c , C, U , N and /3 which

ap-o A1 s C 0 0

pear in the inner and outer solutions have to be found by matching and by using the Equations [7] and [27]. The matching is carried out by requiring that the inner limit of the outer representation

should be equal to the outer limit of the inner representation

(Van Dyke, 196'-).

The inner limit of the outer representation is obtained by replaciig the outer variables in Equations [27]-[30] by the inner variables of Equation [23] and expanding as -' 0 and X is fixed. The result for the function f is

1

3

C'2

where, for, briefness, D =

--

, A = cosh D/3 and B = tanh D/3.

U

_0I

In Equation [31] only the terms necessary for matching at the considered order have been retained.

The substitution of [31] in Equations [23]-[28], rewritten in inner variables, yields

2 N = N -

CB2

2CBD 3/2 CD2 (1-3B / [32] o - A2 _A2 1 - 2BD2x - D2(l-3B2)

x2]

+... [31] EC 2CBD 3/2 CD2(l-3B2)

U=U-

+

x+

2x2

o

UA2

UA2

UA2

0 0 0

[33]

f= -

C

U A

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-9-The comparison of Equations [32]

and

[33] with Equations [19] and [21] shows that the matching at zero order requires

N=U =u =1

₀

0 0

which means that the unperturbed flow must be near critical. More-1

over, a has to be of order

2

in order to make the matching pos-sible at higher order. For the sake of simplicity we can take a2c = and express the inner expansion [19] and [21] as

= 1

c3/2x

- x2

u = 1 + u + c

3/2

+ 1 E2X2

A1 c 2

Since terms of order

2y2

and

2

of u appear in 2U2 only and terms of order 2y of v appear in 2V2 (Laitone, 1960), they have been discarded in Equations [35] and [36].

It can be easily checked that the pair of functions

[3!]

u=cx

u=c1

c 2

v = _c v = -cxy

which appear in the inner expansion [19] and [20] are exact solu-tions of the linearized Equasolu-tions [9]-[l3] when u = 1. They are the elementary first order solutions of the homogeneous linearized

[37] [351

equations of a free-surface flow at critical speed, counterpart of the sin and cos solutions [15] and [i6] of a subcritical flow. The Equation [37] may be used directly as first order solutions of Equations [9]-[13] with u = 1. The derivation by the limiting process discussed here is useful, however, for the case of a singu-larity as well as for higher order expansions.

The matching of the outer terms [32] and [33] with the inner terms [35] and [36] is a matter of algebraic computation.

In the case of the solitary wave a solution which is smooth at the origin is sought. Hence, cc = 0 and the corresponding outer term 2CBD/UA2 = 0, i.e. B = 0, A = 1, and = 0.

The inner and outer expansions, written in inner variables, become in this case

2 2 = 1 - X E ₂

u=u +u +x

0 A1 2 ;3._ .___cL_

2X2

N=N

0

k6

0 ._c_ 3

U=U

-o U 0 0 [38]

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-11-The matching at first order with the supplementary conditions 1

U = N 2 and N = H (for x co) yields

0 0 0

N=U=u=1 cC=l-H C=

_(2/3)

EUA

=1-H

[39]

The complete solution of the solitary wave should be ex-pressed by a composite expansion, but this is not necessary since the outer expansion is regular in the inner zone and coincides with the inner expansion, i.e. the overlapping zone coincides with the inner zone. Hence, the whole solution is expressed by Equa-tions [26], [28], [30] and

[39]

in inner variables as

3(1 h)

1

= h + (1-h) sech2

]

x

which is the well known solitary wave solution. The velocities at the crest and at infinite are

UA = 1 - = h

[l]

u =U =1

0O 0

or, in dimensional variables

u' = h'

u =

Vgt'

= gh?(l+at/hT)

Co

which is, again, a classical result.

_4

gh' - V l+a/ht

It seems, therefore, that the method of matched asymptotic expansions does not provide any improvement in the accuracy of the shallow water solution. This feature is, however, charac-teristic for the solitary wave which has a uniform structure in the vicinity of the crest, but will be different in the case of a moving singularity.

The derivation of the cnoidal waves is entirely similar to that of the solitary wave and will not be considered here.

The possibility of matching the outer expansion with an inner solution which has a non-zero derivative at the origin (c a)

is the key to the solution of flow past singularities.

FLOW PAST A SINGULARITY

A submerged vortex in a uniform, unperturbed stream will be considered for the sake of definiteness. Any other type of singu-larity may be treated in a similar way. (Figure

2).

It is worthwhile for the following derivations to discuss first the radiation condition. In both subcritical and super-critical flows the flow is unperturbed at x = - (Stoker,

1957).

Adopting the same condition in the case of a near-critical flow, it is convenient to take, r = 1 and u in the inner expansion as the unperturbed depth and velocity at x = - rather than at

x = 0. Since the infinitesimal wave expansion will be shown to

be regular, under the above radiation condition, in the region x < 0, the expansion may still be called the TTinner expansion. Ben.ce, the first order linearized expansion, similar to Equation

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uou1 +

1 = 0

=0

-13-u = -13-u + ()ui 0 v = [42]

= 1 + 6()i

The gauge function () represents the vortex strength

(() - 0 as

0). Its form will be found from the require-ments of matching with the already found outer solution

_[32]

and [33]. One could obviously start the inner expansion with an. power series but an unknown gauge function has to be used in this case in the quter expansion in order to rnakë the matching possible.

The variables of the inner expansion

_[42]

satisfy the Equa-tions [9], [10],

[12]

and

[13],

Equations [11] and [14 being re-placed by

As usual the velocity components are separated in regular and singular parts

r

+

1s

v1 =

vir + vj5

_[45]

The singular term is selected to represent the potential flow due to a vortex located at x = 0, y = h and confined between two solid boundaries y = 0 and y = 1 (Figure2).

The complex. potential of this flow is

sinh (z+ih)

f (z)

-sinh (z-ib)

s s s df1

where z = x + iy and w1 = u1 -

iv1

-

d

The velocity along the upper boundary is

S 1 sin 7T

u1(x,l) =

cosh irx + cos 7T

The regular parts of the velocity components have to satisfy the equation

r ₂

u u1 + vi

= - U0 U1

_{(y= 1) [48]}

which is obtained from Equations [43] and [12] by the elimination of i. The solution may be represented by integral Fourier

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r

.

r

w1

('z)

= Ui

-00

- -

1 cos k(z-s) PV / sinirb

cosh

irs +

cos irb

sinhk

1_4_+u2k2

0 _u02

O

0 kmX 00 b sin k x

k

sin k b

k

e a o a m m -k sinh 0 0

for x 0. For x has to be replaced, by -x in Equation [50],

c is an arbitrary constant. In order to ensure theradiation re-quirement, the first term is removed from Equation [50] for

x < 0 by adding to ui the proper solution of the homogeneous

equa-tion

_[48]

(Lamb, 1945), which is, in the present case

k sinh k b sink x

0 0 0

[49]

where u'2 1. Equation

[49]

will be integrated for u <1 and

u will tend to its critical value u = 1 in the final result,

0 0

after imposing the radiation condition..

The details of the integration for 1r (x,1) are given in the

Appendix.

The final result for Ui(x,l)

is

r

S

u(x,l)

= Ui

(x,i) + Ui (x,i)

+ c [50]

u

u

1 2 2 1 _{2 2}

sinhk 1---+u k

_{0 ' 2 0 0}

_'m=l

₂

+k u -lsink

_{m 0 'rn}

00

k

The 1ithitirg case of u0 1 thay be computed bytakinginto accOunt that k i a root of the following equation

tanhk

k-1+

u 2

Ic Ic o

Hence u2 = 1

_k2/3

+ and the final expressions of u1 (x,1) when u ' 1 becOme 0 0O _kmX

sinkbe

(x,i) 3bx Ic (x _[52] m=l kthX sin k.b e

k

( a) m=i

the higher terms (in k2, Ic4 ) being neglected

The complete solution of r ., according to Equations [11-2],

[k31, tk], [],

arid

[3]

iEi Co m=l 1

+)

00

m1

k sin Ic; m - m

kx

kb e

m Ic sin Ic. m -k x -. , -m sin Ic b e m (, [5'] (x . [5,5]

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where tan k = k and k > 0.

m m m

The inner solution is singular as x co. Using the previous

outer expansion (Equation 23), the inner solution may be matched with the outer solution [32] and [33] in the region x > 0.

The outer limit of the inner representation is found from Equation [55], through the usual procedure, to be

= 1 -

3bô()x

[56]

1

-ic 2 X

m

where terms containing e have been discarded since they

are exponentially small.

The outer limit of the inner representation of u(x,y) should be determined by integrating Equation

{k6]

for arbitrary y. It

is known, however, that at x 1 the u distribution is almost uniform and u(x,l) may be substituted for u(x,y). Hence,

ac-cording to Equations [k2] and [5k]

sinkb

-kx

m m

u=u

+3b5x-)

k

sink

e 0 -17-m=l

and the outer limit is

u = u + 3bx.

0

m m

[57]

The matching of the inner solution [56] and [58] with the outer solution [32] and

_[33]

at zero order implies N = U = 1. The matching at order requires that U0 and N should be

de-pendent (a similar situation occurs in the problem of inviscid flow past an airfoil (Van Dyke, l964)). Hence,

N = 1 + CB2 0 [58a]

0

U = u + U A2 0

These two relationships and Equation [27] show that, at the order considered here,

0 = . +

(3B

1) [59]

where the relationship 1/A2 = 1-lB2 has been used.

The matching of the terms dependent on x provides the ad-ditional relationship

0 = 3

From Equations [59] and [60] we finally find

(1

- )

(i-u)

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Equations [61] and [62] permit the determination of B (also A and 3) and C as functions of 5 and 1-u.

Two extreme cases are of particular interest: In a first case assume that 5 0, but 1-u 0. Three solutions are then

possible: (i) B = 1, i.e. /3 = , which means a uniform flow;

(ii) B = 0 and 5/B = 0 which Is again a uniform flow since

EC = 0; (iii) B = 0 and 5/B 0, which represents a solitary

wave. The last case is nontrivial but has to be excluded from

physical considerations. It has been considered as a mathemati-cal possibility by Filippov (1960) using other methods.

In the second case U = 1 and S 0, which means that the unperturbed upstream flow is exactly critical. In this case

1/3 2/3

B2 = A2 = 3 EC = 3(--) (bs) [63]

The outer solution for the free-surface profile in the general case according to Equations [26], [28] and [30], is given by 1 = N - EC tanh2

ri

3C

Luo6

)(x

+ [62]

[6]

2

=

1+2 () (b5)3

+

All the parameters appearing in Equation [6k] may be determine as functions of (1-u) and S from Equations [58], [59], [61] and

[62]. A composite expansion with a uniform representation of r

may be written by combining the inner and outer solutions (Van Dyke, 196k). In the simple case u0 = l.the composite expansion

of

i,

forx

0, is 00

sinkb

-kx

m m e. -k sin k m m m=l -3(1)

(bS)

tanh2

[(,)*

x + cosh

32]

In Figure

3

the shape of the free-surface is represented for the

particular case b = 0.6 and 5 = 0.01. The series in Equation [65] has been summed by using an electronic computer with the values of k from Carslaw and. aaeger (1959). The matching is possible, at the order considered here, only for 5 << 1.

The drag fo'ce on the singularity may be found bya mo-mentum balance between the sections x

= -

oo and x = co The flow

depth and velocity at x = + (downstream) are

[65] = N

- 0

=

1-0 cC

u=U

=1+-0 2 2 [66]

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-21-The dimensionless drag forces is, therefore, given at order, by

2EC

A2 A2 A A

At tha critical speed u 1, F has the simple expression

2/3 F = 2(i) (b5) or in a dimensiOnal forth -I'--, 2/3 -I-/i 3 b - 2

DISCUSSION OF RESULTS AND CONCLUSIONS

Three basic lengths are present in the problems discussed in the foregoing sectiOns: the flow depth 2', the amplitude (associated with e) and the ratio

u'2/g.

In the infinitesimal wave expansion the last two lengths are referred to 2' and the periodic solution of the homogeneous equations in the

subcriti-cal range provides the well-known dispersion relationship be-tween Fr = ut/(g2r) and the wave length L' (Equation

18).

For

small values of a, where a = 2T2'/L', Equation

_[18]

yields

H2 3(1-Fr2).

1167]

[68]

A.s Frl, L' -

and the homogeneous problem has a non

periodical solution divergent at inffnity (Equations

37).

The

shallow water solution is round by relating x' to L' rather

than to £

and by imposing a nonlinear relationship. between the

amp1itude andthe wavelength (in the case of the solitary.wave.

a

Tn, this. sense the results obtãined by using the method

of matched asythptotic expansiois aDe related. .to other derivations

of the first orer equations of shallow flow (see, for instance,

Benjamin

(1967), p 561-562).

Although the flow has been found in the vicinityof the

singularity by linear equations, the dependence of the solution

on the problemT a small parameter (1-h .in the case of the solitary

wave and

is.that of the singuiariti) is nonlinear,

. even

at first

order (Equations [kO],

[65]

and [68]).

In this sense the

solu-tions may beregarded.as nonlinear.

The soThtions for the :higher ordei' ternis of the expansions

and their matching should provide verr valuable information and

inight into the problem, but the computations become tedious as

the order is increased in the inner expansions.

The method of matched asymptotic expansions proves itself

to be a powerful tool in solving notilinear

roblems of fluid

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cosk (z-s) cosh rs + cos 7rb -00

Hence, Equation [49] becomes

00

r

2 . r 1 I k sinh kb cos kz

w1 (z)=u1- lvi =--pV

I

sinh k(k cosh k - sinh k)

[71]

On the upper boundary.z = x + i the"real part of

[72] is

00

r

. ikx

Ui"(X,l)

= -

_/ k sinh kb dk

_[73]

71 j sinh k(k - 2 tanh k) -00 _U 0

The integration of Equation

[73]

follows Lamb's procedure. For x > 0 the integration is carried out in the upper k complex

plane. The poles of the integrand have the following locations:

-23-APPENDIX

THE INTEGRATION OF EQUATION [49]

One of the integrals of [49] may be evaluated in the fol-lowing closed form (Gradshteyn and .Ryzhik, pp. 505)

00

- sinh kb cos kz

-sin irb sinh k

[70]

cos k n n=1

The sarne.sumis obtained from the int?gral

- CO

two poles k = ± k on the real axes, roots of the equation tanh k= u2k; a row of poles on the imaginary axes k = 1km (= 1,2,...), roots of the equation tan k = u2k and a row of Poles:k = in (n = 1,2,...).

The contribution of the last series of poles is'

00

-kx

sJn k b e fl

-

n sinh kb 2ir

J

sinhk

-00 i kx dk

which may. be integrated in a closed form (Gradshteyn and -Ryzhik, p. 507), the result being

-k x

sinkbe.n

n

sinb

cos k - 2 cosh x + cos b n

n= 1

The contribution of the poles k is glven by the sum of residues, i.e.

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-kx

k sinkbe m

m 1 - 1 ± k rn 2 0 2

_sink

rn

The contribution of the poles ± k. is evaluated by in-tegrating

[73]

along two semi-circles, in order to obtain the principal va1ue. The result is

k .si.nh k b sin k x 0 sinh k 1 - --- +. u

2k 2

0 2 0 0 U 0

These partial results are summed in Equation [50]

When x < 0 the resu1ts ai-e similar, and are obtained from those above byrepiacing xby -x.

R.JHENCES

Benjamin, T. B., "Internal Waves of Permanent Form in Fluids of Great Depths, Journ. of Fluid Mech., Vol.

29,

_pp.

559-592,

1967.

Carsiaw, C. S., and Jaeger, T. C., "Conduction of Heat in Solids," Oxford at the Clarendon Press,

510

_p.,

1959.

Dagan, G., "Second Order Theory of Shallow Free-Surface Flow in Porous Media,".Quart. Journ. of Mech. and Appl. Math., Vol. XIX, pp.

517-526, 1967.

Fillippov, I. G., "Solution of the Problem of the Motion of a Vortex Under the Surface. of a Fluid for Froude Number Near Unity,"PMM, Vol. 2, pp.

k78-1-90, 1960.

Friederichs., K. 0., "On the Derivation of the Shallow Water Theory,"Comm. on Pure and Appl. Math., Vol. 1, _pp.

_81-85,.

l9k8.

Friederichs, K. 0. and Hi-erS, D. N., "The Existence of Solitary Waves," Comm. on Pure and Appl. Math., Vol.

_7,

_pp.

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1952+.

Gradshteyn, I. S., and Ryzhik, I. M., "Table of Integrals, Series and Products," Academic Press, New York and London, p.

1086,

1965.

Keller, T. B., "The Solitary Wave and Periodic Waves in Shallow Water," Comm. on Pure and Appl. Math., Vol. 1, _pp.

_323-339,

192+8.

Keulegan, G. H.,"Wave Motion"(in Engineering Hydraulics,edited by B. Rouse), T. Wiley and Sons, Chap. IX, _pp.

_711-756,

1950.

Laitone, V. E., "The Second Approximation to Choidal and Soli-tary Waves," Journ. of Fluid Mech., Vol.

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Van Dyke., M., "Perturbations Method in Fluid Mechanics, Academic

Press, 19'6.1.

Wehausen, V. T., and Laitone, E. V., "Surface Waves" (in Encyclo-pedia of Physics, Vol. 9), 1960.

iJ

(xi) XI I

-

V $

-- 11

= o

-

0 A

FIGURE 1 - THE SOLITARY WAVE

4

b y 77

:1

X

0.. 96 U' = 0 DRAG FORCE F' =0.0756pg '7' =V = 0.6

V

(a) 'i'0.962 ' u' = 1.038

\/I

-

-'ly' VORTEX STRENGTH

6' =0,.01.t' COMPOSITE EXPANSION (eq. 65

I I I I I I I I I -1.00 0 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 (b)

FIGURE 3 - A VORTEX IN A CRITICAL SPEED STREAM

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WAVE PATTERNS IN A STREAM.AT NEAR-CRITICAL SPEED

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The two-dimensional free-surface flow generated by a singularity oving with near-critical speed (i.e. with Froude number referred to the water depth near unity) is solved by using the method of matched

asymptotic expansiOns. In the vicinity of the singularity the

prob-:lem is solved by an infinitesimal wave expansion (inner expansion) while at large distances from the singularity shallow water theory

ri.rovides the proper solution (outer expansion). A

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