46

MEKHANIKA ZHIDKOSTI I GAZA

DEVELOPMENT OF SHIP WAVES IN A LIQUID OF FINITE DEPTH L. V. Cherkesov Izv. AN SSSR. M e k h a n i k a Zhidkosti i Gaza, Vol. 3, No. 4, pp. 70-76, 1968 Analogous problems for steady motion of an infinitely deep liquid were considered in [1-8], for a liquid of finite depth in [4, 5], and for unsteady motion of an infinitely deep liquid in [6].

w

We study expression (1.3) for large values of R and a function

f(x, y) which equals unity in the square [x [ - b, I y [ -< b and equals zero outside this square, assuming b to be small, p0 to be sufficiently large, and P = p0bz to b e finite. Then ~l(0) has the form

/j.~ <.* .;><'Js

I II 'I0/

q

.. "

"

02

o.#

h t

03

Assume that pressures of the form

p(x, y)

(1.1)

pol(z, g)

=

are applied beginning at t = 0 m the horizontal free surface of an undisturbed liquid stream occupying the space 0 > z > - H and flowing in the positive direction of the x - a x i s with the v e l o c i t y u. Assuming the liquid motion to be v o r t i c i t y - f r e e and the disturbances created by pressures (1.1) to be small, and considering that for t = 0 there are no disturbances, we obtain an expression for the form of the free surface, analogous to that obtained in [6]: ,0

*(m, n, t ) e x p [ i ( m x + n y ) l d m d n ,

= -~.~pg

=/(re,

n)

[

-

2azrn 2

~

-

aem2--aZ

ki.~--am=t=a,

um--a

e~,,

.At_

(I __

a-~]/grthrH, r ~ m 2 ~ n

2z~pg

cos0i ~ cos02 ~ ~ - l l / r - i t h r

g~(0) = -

% (0){~r cos 0

-

*,~(0)=

,

Ri~RH-~,

12 =

g ~ R siny,

m ~ rcosO,

f ~l (0) C ]/~t-h r exp [irR~M(O)] dO, (d) v ~ utR -i. (2.6)

M(O ) = cos (O -- y) --v (cos O - - ~-ll/r-l th r) , Table 1 ~:u/c

i.0i

81~ , 64044 " 50~ 33~ , t9~ ' tt~ ' 6oi0 ' 0o

0.1404 0.2644 0.4034 0.6046 0.8155 0.9296 .9841

t.3

1,8

0.6-357 0.7342 0.8546 0.9322 0.~848

0.8314 0.8948 0.94i2 0.9852 t

Performing the investigation of the integrals (2.5) and (2.6) by the method of stationary phase, we obtain for large R and r >- r0

c~]/gH,

(ar cos 8 -- c 'Vr th r) H 2 x ~ Rcosu

(2.5)

(d)

(1.4)

-

rl(r, O)

(2,4)

I1 = I ~l(O)urexp[irRlcos(O--y)lcosOdO, (1.2)

c)"r'th r exp [it / H (--~r cos 0 -b cyr th--~r)]},

-

(2.3)

i = l i - - 12,

k

f ~ ( 0 ) e x p [irR~ cos(0 -- y)] dO,

(2.2)

It is evident that for r > r0, cos 0i and cos 02 will be less than one, and the condition Re[-ir cos 0] -- 0 wilI be satisfied on the contour (d). Integral (1.4) now can b e written as

a/z~

I(r) =

~ ~ ~t/c.

~ro - - th ro ~ O.

] z.

,

We shall perform this deformation of the integration path for r > r0, where r0 is the single positive root of the equation

umq-a e~

Here f(m, n) is the Fourier transform of the function f(x, y) and m and n are the parameters of the Fourier transform with respect to x and y. Formula (1.2) is the exact expression for the form of the waves on the free surface for an arbitrary function f(x, y) which is representable by the Fourier integral. For a function ;~(x, y) which is s y m m e t r i c about x and y the funct i o n ] ( m , n) will be s y m m e t r i c about m and n. Consequently, fer greater convenience in further study (1.2) in this case m a y be represented in the following form:

=

(2.1)

The function gl(O) does not have singularities on the integration path - 1 / 2 1 r -< 0 -< 3/21r for any r ~ 0. Therefore we can, for convenience of the later investigation, deform the original path of integration with respect m 0 into the path (d) which bypasses the point 01 along a small s e m i c i r c l e in the lower h a l f - p l a n e and bypasses the point 0z along a small s e m i c i r c l e in the upper h a l f - p l a n e and otherwise coincides with the original integration path. Then

Fig. 1

w

.

~t

"x.\

~G

4 sin (br cos 0) sin (br sin 0) . . ~ , ~r (at cos 0 -- c ]/r th r) sin 20

.

li =

at

at +

a2 -~- as,

[ -- 2n~[res~2(0)]0~

(cosy > ~ - i ? r - ~ t h r )

( 0

(cos y < ~-i ~ r - i th r)

,

n ~ r s i n 0 . (1.5) ( 2ui ires ~2 (O)]o,

Expression (1.3) yields the form of the free surface for arbitrary values of R and ), and a function f(x, y) which is s y m m e t r i c r e l a t i v e to x and y.

a2 = t 0

(cos ~ / > 0, -- cosy < ~-I ]/r-I th r) (-- cos V > ~-i 1/r-ith r)

as = R-V~di (r) exp [iBqi (r)],

' (2.7)

FLUID

47

DYNAMICS

where r is the i n t e g r a n d of (2.5); dl(r) has no singularities for r >- 0; ~ ( r ) is t e a ! for r -> 0,

lz

Now let us i n v e s t i g a t e (2.11). We find the a s y m p t o t i c e s t i m a t e for this i n t e g r a t for l a r g e values o f R by t h e s t a t i o n a r y p h a s e m e t h o d .

bi -b b z + ba,

=

bl=

(v~v~,

Table 2

0~y~n)

'

~i

i b2=

/

(COS~ > v > cos ~ - - m, ( r ) s i n y , cos ? > O) 0

(v>cosy,

v
b3 ~ R-'hdz (r) exp [iRq~ ( r ) ] ,

(2.8)

where ~03(G) is the i n t e g r a n d o f (2.6); dz and q2 a r e analogous to dl and ql

90 ~ 84 ~ 81o56 , 75 ~ 60 ~ 45 ~ 30 ~ 19028, 14028, tto32" 5~ " 2~ '

t 0.9983 0.9972 0.99 0.96 0.9378 0.92 0.8530 0.82 0.7 0.6 0.55 0.4739

mt (r) = [~Zr(th r ) - i ~ 1]-'12. (2.9)

Vi = COSy -~ mi (r) sin y, Now we find f r o m

t.005 1.01 1.039 t.t75 t.4t4 2.00 3.00 4.00 5.00 10 2O 14t

"~2

90 ~ 80~ ' (64~ ') 78 ~ (61~ " ) 68~23 ' (51~ ") 49~ ' (37~ ' ) 42~ , (33~ ') 38~ , (30~ ' ) 28~ " (2A"55") 24~ ' (23~ 20~ ' (20~ ") t9~ (19~ ') 19~ ' (19030' ) 19~ " (t9~ '} t9~ 9

T h e stationary pornts are the positive roots of t h e e q u a t i o n N l' = 0, which m a y b e written in the f o r m

(1.3), (2.4), (2.7), a n d (2.8) t h a t li

-Re

,

(2.10)

tg ? = ~ (r) ~(r)~

= ~ri f ~"~(r) exp [iRN, (r)] dr q~

1, = f f

ur,,(e)eosOexp[ir[Icos(O-- y)] dO dr,

0 -- 1 ] ~

-- I I

I~ =

c,i(O)~-r--threxp[iRrM(O)ldOdr,

I,=nifgz(r)exp[i~N~(r)]dr

(2.11)

(cosy<0,

s~Qt),

(cosy%0,

s ~ QO,

g 2 _ th r

292r - - th r - - r(ch r) -z

(v > COSy > 0),

I

.

(2,16)

S i n c e the function ~(r) behaves q u i t e d i f f e r e n t l y d e p e n d i n g on whether g -> 1 or g < 1, which has a c o n s i d e r a b l e e f f e c t on the f i n a l expression for i n t e g r a l (2.!1), we shall i n v e s t i g a t e these two cases separately. w C a s e ~ >- 1. Analysis o f the f u n c t i o n r shows t h a t for g -> 1 it decreases m o n o t o n i c a l l y with i n c r e a s e o f r (r _> 0), a p p r o a c h ing zero as r -* ~ and for r = 0 h a v i n g the largest v a l u e , e q u a l to

r

( (V - - t)-'/~ = t{+ ~

(~ > i) (~ = i)

(8.1)

q~

17 = -- ~rt f ~(r)exp[iRNt(r)] dr

Thus, (2.16) for g = 1 has a single positive root for a n y values of )'(O -< 7 "< 1/2~r), w h i l e for ~ > 1 it has a single positive root for 7 -< h and has no positive roots for 7 > )'1 < 1/27r, w h e r e

s

Is = ni i ~z (r)exp [iRNz (r)] dr

(cos y > 0, v > 0),

tg yt = (~2 _ l ) - ' h .

(3.2)

s

We d e s i g n a t e this root b y rl(rl = r f f y , g)) a n d we note t h a t with v a r i a t i o n of 7 / f r o m 71 to zero (for fixed g) h increases c o n t i n u o u s l y from zero to infinity. S i n c e in the steady m o t i o n (t -+ ~) the lower l i m i t of i n t e g r a l (2.11) equals zero, and t h e i n t e g r a n d for 7 -<- 71 has only the stationary point r = q, while for 7 > )'t t h e r e are no stationary points, it is obvious t h a t in t h e steady m o t i o n in t h e case g -> 1 the/basic disturbances, whose a m p l i m d e for l a r g e R has t h e order R" t ' ' , will b e c o n c e n t r a t e d within the a n g l e

I9 = - - ~g I ~.(r)exp [iRNe(r)] dr, Q~

(v>cosv,

O~y~/z~)

= R- ~/~ I d~,z(r)exp [iRq,.z(r)] dr, ro

g~ sin [~--~b ]b' gh r] sin [ ~ - ' b "~r(g~r -- th r)] ~(r) --

I~l ~< 7 ,

~r

-- th

r

N~,z:~-~(I:rthrcosy-~l/~r~--rthr.sin~).

(2.13)

Here Q I , i are positive roots of the equations v ~ cos V 4- [~r(~h r) -~ - - I]-V~ sin y .

(2.14)

Since the functions d t and dz do not h a v e singularities in the r e gion o f integration, and the integrands of the expressions Ik(k = = 6, 7, 8, 9) do not h a v e s t a t i o n a r y points on the i n t e g r a t i o n path, we find t h a t e a c h o f the integraIs Ik(k = 4 . . . . . 11) has a n order no lower than " and by virtue o f this

where 71 is defined by (3.2) We note that expression (3.2) o b t a i n e d for t h e l i m i t i n g a n g l e of the w a v e trail for g >- 1 c o i n c i d e s with t h e expression o b t a i n e d in [4], where the a n a l o g o u s p r o b l e m was considered for steady m o t i o n by the m e t h o d of s u m m a t i o n o f the w a v e effects f r o m c o n c e n t r a t e d pressure pulses applied a l o n g the course of the "ship. " Since the lower l i m i t Q1 o f i n t e g r a l (2.11) in the unsteady m o t i o n is a function of u, it is obvious t h a t this i n t e g r a l will h a v e the order R-r/~ when the stationary point r = r~ with s a t i s f a c t i o n of c o n d i t i o n (8.3) iies on t h e i n t e g r a t i o n path, i . e . , when h > Q I - In t h e r e m a i n i n g cases, i . e . , for rl < Q1 a n d 7 -< 7t or for 7 > It (2.11) wilI h a v e a n order no Iess t h a n R-t. C o n d i t i o n rt ~ Q1, as follows from (2.14), m a y b e written in the f o r m

tt

I~ :

O(R-i).

(3.3)

(2.12)

(2.15)

t:l ~ ubl(~, u ri)t,

48

MEKHANIKA sin

b~([, 7, n ) = e o s ~ +

~

V[Zrt(th rQ-~

.

Considering the foregoing, we find the asymptotic formulas for (2.11) for large values of R:

I~ =

(

zxiR-V~-A~ exp{i [RN~ (rO - a/~n]} O(R-q ~/--2g/N,'(r 0 [z(r 0 I. O(R-%)

At :

(0 < y < YO (yt < ~ ~< n ) , (3.5)

(R < ubit) (R > ubJ) .

(3.6)

Since the integrand in (2.11) for r = 0 (this correaponds m )' = }'i) vanishes, in spite of thefact that Ni'(O) = 0 and Ni"(O)e 0 the integral

O

g.f

O.#

g-F

0.8

/

Fig. 2 I~ along )' = ~'r has an order higher than R-~/z, From (2.10), (2.15), (3.5) we find the final expression

[=(0(/r ~t

(y=0, yt~<'~-=), 2p0

=

-- - -

A~ sin [RN~ (r d

--

tAn],

(3.7)

(8.8)

nPg where & is given by (3.6). Thus, for ~ -> 1 the basic disturbances of the free surface, which are only longitudinal waves of the form (3.8), are concentrated within the angle [Yl < )t to the left of the curve R = ublO')t, where )% and bt are given by (3.2) and (3.4), and rl is a positive root of (2.16). Since in the general case it is not possible to find the expression for tl as a function of y and ~ from (2.16), m determine the form of the wave-trail leading edge (R = ubl0", ~)t we made numerical calculations for different values of ~. The results of these calculations for g = 1.01, 1.3, 1.8 are shown in Table 1 and Fig. I. These calculations give an idea of the process of wave development on the free surface; in particular, from ahem we see that the distance of the leading edge from the coordinate origin decreases monotonically with increase of )' (for fixed ~) and that the value of bl0'l, g) will be smaller, the smaller ~, and bl(0, ~) equals unity regardless of G Table 2 shows the values of the limiting angle )'~ for various values of ~, calculated from (8.2). ~4. Let us analyze the case ~ < 1. The function r for g < 1 vanishes for r = r0 > 0 and for r = ~o has a single extremum (maximum) for r = r4(r0 < r4 < ~) which is a positive root of the equation @'(rt) = O.

(4.1)

In the region r0 -< r < r4 r increases monotonically, and for r > r4 it decreases monotonically with increase of r. From this it follows that (2.16) for y > y~, where yz = are tg ~ (r0

(4.3)

where 72 is given by (4.2) and depends on the quantity ~, Numerical calculations were performed to determine the dependence of the limiting angle 72 on the quantity ~. The results of these calculations are presented in Table 2. The corresponding values of the limiting angle y~, obtained [4] in the solution of the steady-state problem, are shown i n the same table i n parentheses. We see from a comparison of these values for 0 -< ~ -< 0.6 that the values obtained here and those obtained in the cited study for the limiting angle actually coincide (the discrepancy does not exceed one minute); for values 0.6 < ~ _< 0.82 these differences do not exceed one degree, while for 0.82 < ~ these values differ considerably, and the m a x i m u m discrepancy amounts to about seventeen degrees. This difference between the values is apparently explained by two factors: first, in [4] the solution was found by summation of the rises resulting from concentrated pressure pulses applied sequentially along a straight line on the free surface, while expression (1.3) is the exact solution in the linear formulation for the rise resulting fromthe surface pressure (1.1) which is shifted rectilinearly at constant velocity; second, the value of the limiting angle in the cited study was obtained as a result of triple sequential application of the stationary phase method to the triple integral whose integrand does not contain a large parameter, while expression (4.2) for the limiting angle was obtained as a result of a single application of the stationary phase method to (2.11), which contains a large parameter. The author believes that this yields a good basis to believe that the values of the angle Y2 presented in Table 2 are more exact than those obtained in [4]. Let us turn to a further analysis of (2.tl). in unsteady motion the lower limit Q1 of this integral is a function of u(r0 -< Ql(u) -< r therefore, the integral will contain a contribution from the stationary point rk(k = 2, 3) when rk > Q1 and will not contain this contribution when rk < Q b The condition rk ~ Qt may be written in the form

R ~ u b ~ ( [ , y,r~.)t,

(4.4)

where b k is given by (3.4) with the subscript 1 replaced by the subscript k. From the stationary phase method for large values of R we obtain

J~t~-A" 2 a]feXp{~[~gl(l'h)--l/,~J~(--l"l~]} g=2, 3

~ia'~l~A4 exp [iNN1 (r4)l

(z = 7~)

O(R -I)

(~-~-0, T p < T < n )

(4.5)

Here the coefficients A2, As, A4 are defined by the formulas

=/~2~/IN~;'(~)J ~(~) A,

( 0(R-th)

(R< ~b~t) (R > ubht) (k = 2, 3)' (4.6)

[ ~s(n)r(%)3-'/~6V.[lN/'(r,)11-'/3 A, :

(n < ~b,t)

(0(R-ph)

(R > ab~t) " (4.7)

From (2.10), (2.15), (4.5) we obtain the following expression for the surface rise g :

2=

(4.2)

has no real roots, while for ), < )'~ it has two positive roots, one of which (r = r~) decreases with decreasing y , while the second (r = r3) increases with decreasing y(rs > ra). For )' = )'~ (2.16) has the multiple root r = r4. In the steady motion the lower limit of (2.11) equals r0; therefore in the steady motion the basic disturbances of the free surface, whose amplitude is of order R-t/z and which correspond to the stationary points r~ and rs, will be concentrated within the angle

I GAZA

lvl ~< v~ ,

(3.4)

t

- -

ZHIDKOSTI

~12+ ~1~

(0 < y < Yp)

O(R -i)

(V = O, ~2 < V <<-n),

]

=

npg ~l~ = --

2p0 A~ sin [RN~ (n)1, z~gg

(k = 2, 3),

(4.8)

where Ak(k = 2, 3, 4) are given by (4.6) and (4.7). Thus the primary disturbances of the free surface (wave trail) in the case considered (~ < 1) are concentrated within the angle 171 <-

FLUID

DYNAMICS

49 Table 3 ~u/c "t

80o43' 75029 ' 59o599 47050" 40021 , 23011, 6o58' 0o

0.9983

0.9803

be.tO -*

b2.10 "4

bz.iO -1

4.t60 tt.446 27.3t2 4t.147 49.506 75.264 97.731 t00

41.60 t3.52 6.72 4.98 4.3t 3.62 3.35 3.30

1.477 3.705 4.738 7.474 9.77t tO

0.9292 bs.iO -~

~-=:~;tte[ie'(r)dr! where Is is given by (2.8) and the instant of termination of the pressure input is taken as the initial time (t = O). Considering the results of the previously performed analysis for the integral I2, we find that in the case g ~ 1 the form of the liquid free surface is given by (3.7), where A1 is defined by (3.6) with replacement of the inequality signs by their opposites (where R > ublt in (a.7), we need R < ublt, and vice

bs-lO -1

w

m

-< 7z, where 72 is given by (4.2), to the left of the curve R = ubsO')t. The free surface within this angle to the left of the curve (bz > bz) is covered by steady-state waves, which are the combination of longitudinal 77s and transverse ~2 waves. The region of the free surface which is included between the curves R = ubzt and R = ubst (b20'z) = = bsO'z)) is covered only by transverse waves, and here the transverse wave front propagates with higher velocity than the longitudinal wave front. In the general case it is not possible to find rz and rs as functions of )' and g from (2.16); therefore to determine the form of the leading front of the transverse waves (Pc = ubst) and longitudinal waves (R = = ubst) we carried out numerical calculations for several values of g. The results of these calculations are presented in Table 8 and Fig. 2. Hence we see that the distance between the leading front of the transverse waves and the coordinate origin increases monotonically with decrease of ~ from ~'~ to 0 and equals ut for ), = 0 regardless of g; the distance between the leading front of the longitudinal waves and the coordinate origin in this case decreases monotonically and for ], = 0 is the smaller, the larger the value of g. w Considering the problem of the decay of steady ship waves in a liquid of finite depth, formed by pressures of the form (1.1), after termination of the pressure input, we find the following expression for the Iiquid surface rise

b~.lO -1

14.~7 6.478 5.928 4.257 3.894 3.86t

m

3.03t 7.279 9.769 10

3"~3t t. 54t 1.329 1.324

versa). In the case ~ < 1 the free surface form is given by (4.8), and here As, As, A4 are defined by (4.6) and (4.7), again with replacement of the inequality signs by the reversed signS. From this we see that in the decaying motion in a liquid of finite depth the basic disturbances for g -> 1 are included within the angle I 7 1< 7t to the right of the curve R = ublt, while for g < 1 they lie within the angle I)' I < Yz to the right of the curve R = ubzt. With an increase in time t the free-surface rise at any fixed point approaches zero as a quantity of order t -1. I would like to thank L. N. Sretenskii for his interest in the study and V. S. Fedosenko for assistance in making the numerical calculations. P.EFERENCES 1. L. N. Stretenskii, "On the waves raised by a ship in motion along a circular path," Izv. AN SSSR, OTN, no. 1, 1946. 2. L. N. Stretenskii, "Waves on the interface of two fluids with application to the dead water phenomenon," Zh. geofiziki, vol. 4, no. 3, 1984. 3. E. Hogner, "Contribution to the theory of ship waves," Arkiv fOr matematik, astronomi och fysik, vol. 17, no. 12, 1962. 4. T. Havelock, "The propagation of group waves in dispersive media with application to waves on water produced by a travelling disturbance," Prec. Roy. Soc. (A), vol. 71, p. 398, 1908. 5. G. Crapper, "Surface waves generated by a travelling pressure point," Prec. Roy. Soc. (A), vol. 282, no. 1891, 1964. 6. L. V. Cherkisov, "Development and decay of ship waves," PMM, vol. 27, no. 4, 1968.

26 July 1967

Sevastopol