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Ship waves on the surface of a floating liquid
U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain
Vo1.28,No.4,pp.102-106,1988 0
OQ41-5553/88 $lo.00+0.00 1990Pergamon Press plc
SHIP WAVES ON THE SURFACE OF A FLOATING LIQUID
P.N. ZHEVANDROV
The problem of ship waves on the surface of a liquid with particles (ice particles, for example) floating on it is studied. It is shown that the angle within which the wave field (the wedge of ship waves) is concentrated decreases as the surface density of the particles increases.
Introduction. In this paper the linear problem of waves excited by a system of pressures which moves across the surface of a liquid covered with floating particles of a certain substance is The treatment of this liquid, which is referred to in /l/ as a floating liquid, is studied. equivalent to the assumption that its free surface is ponderable and has a surface density 0 which is identical to the mass of the floating particles per unit area. The derivation of the corresponding system of equations for the velocity potential QI of the vortex-free motion is available in /l/. When the liquid fills the lower half-space (YCO, ZERZ), the problem of determining the potential of the velocities caused by the motionoftheconcentrated system of pressures, has the form 118-0,
y-=0,
a++0,
y-+-m,
~(mtsrp,)+bm.=-p-lp!(z,f). 0=%-O,
'(la) (lb)
Y=Q
(IC)
t=o.
y=O,
Here, g is the acceleration due to gravity, e=o/p, P is the density of the liquid, and PC& 0 -p-'/+ is the excess surface pressure over the equilibrium pressure. We shall assume that Sa(z,-Vt,zt)q(t),where u(s) is a smooth function with a compact carrier, S is a constant of appropriate dimensions, V is the velocity of the source and q(l) is a smooth function which is equal to zero in the neighbourhood of zero and to unity in the neighbourhood of infinity (it is introduced in order to eliminate transients associated with the establishment of the wave picture). The classical Kelvinresult Problem (1) has been studied in detail when e=O (see /2/). is the fact that, at distances L from the source such that h=gLlV2*1, the waves which are created by it are concentrated within an angle with a vertex at the point (I’t,O), the tangent The amplitude of the of half of which is equal to 6,=2-S (a wedge of ship Kelvin waves). waves at the internal points of this angle is a quantity of the order of d(h-"~). 0(X-"') on the sides and negligibly small outside it. It turns out that a similar result also holds in the case of a floating liquid. However, the angle, within which the wave field is concentrated, decreases monotonically to zero as At the same time, the tangent of half the parameter Y=sYlV' increases from zero to infinity. of this angle is defined by the formula (l-yZ)"'[(le+4~r')"'-rJl 6= Here,
I=[~(i+px)l-"* and
x=x(r)
(2)
4Y-~((~"+4Y~')"-ysl.
is the unique
root of the polynomial
6(~x3+x-1)(1+2px)[2x(1+~x)"-1]-[2~(1+~x)z-l~z[2x(~+~x)*3(~xz+x-l)(l+2~x)]-(~x*+~-~)[2x(~+~~)z-3(l+2~x)l on the beam (km), ko=[(i+4~)".-1]/(2p). Asymptotic formulae for b at small and large and, in fact, the expansions
p are readily obtained
(3)
from
(2) and
6_ (22+;;,;2~;~~;(") ;"R,",;::
.(4)
hold. The proof of these results
is presented
below.
1. Formal investigation. The solution of problem (1) is readily obtained with respect to the variable I. It has the form
*Zh.vychisl.Mat.mat.Flz.,
28,7,1110-1115,1988
102
with the help of a POurier
(3)
transform
103
where
k=(k,, kr), x=lkl,
o=[x/(~+ex)]"~, a(k)-(24t)-*
is the Fourier transform defined by the formula
of the function
jqz)e~p(-fk~)dz U(Z). The elevation
of the free surface
<(z,t)
is
1
q(r)cos o(t-5)exp(ikz-fkIV?)a(k)dr dk -
Let us consider the wave field at large distances L from the point to=(Vt,O) and at large T=LIV, by putting b=gLIV’+-, 8,6tIT
9=x/L, t’= t/T, E’(k’)=E(k’g/v2),
o'-[x'/(l+ex')l'", x’=lk’l, q’(-c’)=q(LT’lV).
We shall assume that the function q’(T’) is smooth, vanishes in the neighbourhood tends to unity when r'~&/2 (the eource is "slowly included"). For the function 6 in the new variables (we omit the primes), we get
of zero and
t
since the second term in (5) can be neglected at the indicated values of t. It is intuitively clear that points at which the gradients of the phase functions 'p*(z, k, t,r)=+o(t-r)+kz-k,r with respect to the variables them and equate the resulting
r and k vanish, make the main contribution expressions to zero. We then obtain aq* -&-!
ati
z,-**--((t-7)=0, L
to (6).
Let us find
(7a)
a. arpf -&- = 21 f --(t-7)=0,
(7b)
acp+ia7=ro-k,=o,
(7c)
2
*
corresponding to the N+) and N-_) indices, transform into one another when the Systems (7), We shall therefore subsequently confine ourselves to investigating sign of kl is changed. the system which corresponds to the X--D sign. System (7), in which k and T are treated as parameters, describes a family of curves in BY introducing the notation k,=xfus 6,kz=x sintl, -n/Z< BGIc/Z and expressing the (GZI) plane. x in terms of cos.0 using formula (7c), we get rr-[(z,-r)*+zz~i~~~= (t-z)[(coaxe+4p co9 ep-cos3 ei/(4p).
(8)
specified by formula (8) are homothetic with a centre of similitude The curves qp)+,=r,(e)l (t,O). We now present some simple properties of these curves: 1) each curve LI(p,) lies within the curve L&Z) when p,+2. 2) the curves Lx(p) are smooth, closed and symmetric with respect to the I, axis. are straight lines which pass through 3) the envelopes of the family of curves L,(p) the point (1,O) and have a slope to the negative direction of the 2, axis equal to p from (2). 4) just two curves of the family, corresponding to different r, pass through eachinternal point of the domain PI, bounded by the right interval of the curve Lo(p) and the envelopes while just a single curve passes through each internal point of the domain Q,, bounded by the curve L,(p). L,(p) which lie in the first quadrant are shown in Fig.1 for The parts of the curves These graphs were obtained by means of a numerical calculation. (=I,v-2.0,1.0, 0.5 and 0.1. 1) and 2) are elementary and their proof Let us prove properties l)- 4). Properties Property 4) follows from the properties of homothety and from involves direct verification. 3). Let us prove 3). The fact that the envelopes are straight lines follows from the must be satisfied on the envelope of The relationship detHessp-=O properties of homothety. function the family L,(p) (see /3/), where Hessrp- is a matrix of the second derivativesofthe
104
q-with
respect
to the variables
k and
'c:
It is readily seen that detHessrp- vanishes either as X-+” or when t=l, or when the polynomial (3) vanishes. By virtue of (7a) and (7b), the first case corresponds to the interval {O=ZZ,S:~,ZZ=~) while the second corresponds to the point (t,O). It is obvious that these sets of points are not envelopes. Formula (7~) is satisfied either when x=0 or when x>ko since lcos8/<1. The first case is excluded by virtue of (?a) and (7b) while the root of the polynomial (3) on the above-mentioned ray enables one to find the envelope. In fact, by using x found from (7c), it is possible to express k, and then to find kz and, using formulae (7a) and (7b), to find the coordinates of a point lying on the envelope. By making use of the fact that the envelope is a straight line, we obtain (2). It remains to be proved that, in the case of the polynomial (3), there is only a single root on the above-mentioned interval. We note that the equation detHesso-=O together with (7~) is, when WT, the equation for a point of inflection of the function k2=k2(kl) which is implicity specified by (7~) (this is proved by direct verification). It is readily seen that (7~) can be written as: k,‘(l-~k~Z)-2-kl’=k~Z. Since
this equation is invariant under a change in the signs of kt and k2. the behaviour of k%(k,) in the first quadrant may be considered. A graph of the function kt(k,) is shown in Fig.2. One is readily convinced by straightforward differentiation that the second derivative d’k&kP tends to -a as Irr-+k, and to +m as k,+ko8,kgo=)l+. The third derivative of the function kz(k,) has the form
(9)
Let us prove that d3kz/dki*>0 when ko=Gktckoo. &lrl Actually, when this is so zsZl and by virtue of the inequality k&f. Hence, the first and second parenthesis within the square brackets in (9) have a lower bound of zero while the third parenthesis is not less than three. We obtain the required inequality from this. Consequently, there is just a single point of inflection on the graph of &(k,) in the first quadrant while, in the case of the family L&), there are only two envelopes which are symmetrically disposed with respect to the I, axis. Property 3) is proved.
P
Fig.1
Fig.2
The fact that the angle which is swept by the curves increases follows from property 1).
L,,(p) decreases
monotonically
as
2, Proof. In order to confirm the investigation of Sect.1 it is necessary to reduce integral (6) A direct investo a standard form which permits the use of the method of stationary phase. tigation of (6) with the aid of this method is impossible owing to the fact that the phase We eliminate this point from the treatment by introfunctions 'p* are not smooth when k=O. X(k) is a function ducing the factors X(k) and *-x(k) into the integrand of (6), where from C.-(R*) which is equal to unity when Ikllj;‘/r and equal to zero when /k(z’lz. Let us evaluate the integral with the factor .x(k):
q(s)ii(k)X(k)exp(ih~*)dr
Here, having
twice integrated
by parts with respect
1
dk.
to r and using the formula
105
a
(ihI&‘,*)-1 d, exp(ih~*)=exp(ihq*),
we obtain ~(k)B(k) +,-
(k,*o)-lexp(ih$)dk-
(10)
t
ah-*x
j[ +.-
j
X(k)s(k)~(k~*~)-zexp(thlg,)dr]dk,
0
$=k,(z,-t)+kzxz. (k,*o)-= It is readily seen that the function is integrable when Iklc*/~. Hence, the second term in (10) is of the order of O(l-I). In the second term, we integrate by parts with respect to k, using the formula
which is possible for r~=[(z,-t)*+zz2]‘~~6~.The in&grand in the resulting integral will be integrable with respect to k and, hence, the whole of the integral I, is of the order of 00.~I), We that is, it is small compared with the principal term (which is of the order of O(h-"I)). therefore get
q(r)[l-X(k)ls(k)exp(ta~*)dr +.-
I
0
dk+O(L-L).
We now subdivide this integral into two integrals by inroducing the factors ~,(t-r)and X,(T) is a smooth function which is equal to unity when i-x*(t-s) into the integrand, where zcs,/4 and to zero when r9&/2. Let us consider the integral with the factor ~~('-1):
I,=aa
xj[ j +.-
q(r)[l-~(k)lX,(t-r)z(k)exp(i?.q*)dz
I
(I
dk.
The phase functions of this integrand have critical points on the carrier of the integrand only when r,=&i2/2. Hence, when r1M2 in it, it may be integrated by parts as many times as may be desired by using the formula
Let us now estimate
the first "outer integral" [l--x(k)]
(k&o)ii(k)
term. Ih,p*(s,
It has the form k, t, t) I-* exp(t%)dk.
(11)
+.-
When r,~& in (ll), it can be integrated by parts with respect to k as many times as may be It follows from this that the integral in (11) is a quantity of the order of o(a--) desired. uniformly with respect to t and z when 6,
(12)
1
[i-x(k)][l-x,(t-r)la(k)q(~)exp(“b’)dr
dk+O(li-9.
The method of stationary phase can now be applied to this integral and, in fact, if z is an internal point of the domain S&then each of the systems (7) has, by virtue of property 4) from Sect.1, just two systems of roots: kl,2=I,.**(z,t), %=7*(.%t) Using well-known
formulae,
we obtain
and
=
k,,?=ki,Z*(z,
=
t),
r=z*(z,
/4/
Q(T)z(4 exp(ilp+++nHessrf*) 'p*1".
k ,ri +O(i-9,
1d&Hess
==* F=T
t).
106
is the difference between the number of positive eigenvalues and the number where sgnHessp* of negative eigenvalues of the matrices Hesscp'. The fact that the integrand in (12) does not have a compact carrier with respect to k is immaterial by virtue ofthe rapid decay of the This "damages" the estimation of the residue in the standard at infinity. function ii(k) formula of the method of stationary phase but has no effect on the estimation of the residue in (13). At internal points of the domain !&each of the system (7) has just a single system of roots and only a single term remains in the formula for 5 under the summation sign. Outside 5=0(X-'). of these domains the integral in (12) is a quantity of the order of O(h-") and On the side boundaries of the domain 52,; that is, on the envelopes of the family L,(P), has a rank of formula (13) is unsuitable since, there, detHessm*=O. Since the matrix Hess@ (see /4, 5/). 2 on these straight lines, t=O(h-'") at values of z lying in their neighbourhood Standard manipulations enable one to write t in the neighbourhood of these curves in terms of Airy functions, similar to the case when p=O. In concluding the author expresses his thanks to S.A. Gabov for suggesting the problem and to S.A. Yakunin for putting a package of computer graphics programs at his disposal.
REFERENCES 1. GABOV S.A., On a problem of the hydrodynamics of an ideal liquid associated with flotation, Differents. Uravneniya, 24, 1, 16-21, 1988. 2. SRBTENSKII L-N., Theory of the Wave Motions of Liquids, Nauka, Moscow, 1977. 3. PCGOF~LOV A.V., Differential Geometry, Nauka, Moscow, 1974. 4. FEDORYUK M.V., The Method of Steepest Descent, Nauka, Moscow, 1977. 5. ARNOL'D V.N., VARCHENKO A.N. and GUSEIN-ZADE S.M., Singularities of Differentiable Mappings. Monodromy and Asymptotic Forms of Integrals, Nauka, I~loscow,1984.
Translated
by E.L.S.
Vo1.28,No.4,pp.102-106,1988 0
OQ41-5553/88 $lo.00+0.00 1990Pergamon Press plc
SHIP WAVES ON THE SURFACE OF A FLOATING LIQUID
P.N. ZHEVANDROV
The problem of ship waves on the surface of a liquid with particles (ice particles, for example) floating on it is studied. It is shown that the angle within which the wave field (the wedge of ship waves) is concentrated decreases as the surface density of the particles increases.
Introduction. In this paper the linear problem of waves excited by a system of pressures which moves across the surface of a liquid covered with floating particles of a certain substance is The treatment of this liquid, which is referred to in /l/ as a floating liquid, is studied. equivalent to the assumption that its free surface is ponderable and has a surface density 0 which is identical to the mass of the floating particles per unit area. The derivation of the corresponding system of equations for the velocity potential QI of the vortex-free motion is available in /l/. When the liquid fills the lower half-space (YCO, ZERZ), the problem of determining the potential of the velocities caused by the motionoftheconcentrated system of pressures, has the form 118-0,
y-=0,
a++0,
y-+-m,
~(mtsrp,)+bm.=-p-lp!(z,f). 0=%-O,
'(la) (lb)
Y=Q
(IC)
t=o.
y=O,
Here, g is the acceleration due to gravity, e=o/p, P is the density of the liquid, and PC& 0 -p-'/+ is the excess surface pressure over the equilibrium pressure. We shall assume that Sa(z,-Vt,zt)q(t),where u(s) is a smooth function with a compact carrier, S is a constant of appropriate dimensions, V is the velocity of the source and q(l) is a smooth function which is equal to zero in the neighbourhood of zero and to unity in the neighbourhood of infinity (it is introduced in order to eliminate transients associated with the establishment of the wave picture). The classical Kelvinresult Problem (1) has been studied in detail when e=O (see /2/). is the fact that, at distances L from the source such that h=gLlV2*1, the waves which are created by it are concentrated within an angle with a vertex at the point (I’t,O), the tangent The amplitude of the of half of which is equal to 6,=2-S (a wedge of ship Kelvin waves). waves at the internal points of this angle is a quantity of the order of d(h-"~). 0(X-"') on the sides and negligibly small outside it. It turns out that a similar result also holds in the case of a floating liquid. However, the angle, within which the wave field is concentrated, decreases monotonically to zero as At the same time, the tangent of half the parameter Y=sYlV' increases from zero to infinity. of this angle is defined by the formula (l-yZ)"'[(le+4~r')"'-rJl 6= Here,
I=[~(i+px)l-"* and
x=x(r)
(2)
4Y-~((~"+4Y~')"-ysl.
is the unique
root of the polynomial
6(~x3+x-1)(1+2px)[2x(1+~x)"-1]-[2~(1+~x)z-l~z[2x(~+~x)*3(~xz+x-l)(l+2~x)]-(~x*+~-~)[2x(~+~~)z-3(l+2~x)l on the beam (km), ko=[(i+4~)".-1]/(2p). Asymptotic formulae for b at small and large and, in fact, the expansions
p are readily obtained
(3)
from
(2) and
6_ (22+;;,;2~;~~;(") ;"R,",;::
.(4)
hold. The proof of these results
is presented
below.
1. Formal investigation. The solution of problem (1) is readily obtained with respect to the variable I. It has the form
*Zh.vychisl.Mat.mat.Flz.,
28,7,1110-1115,1988
102
with the help of a POurier
(3)
transform
103
where
k=(k,, kr), x=lkl,
o=[x/(~+ex)]"~, a(k)-(24t)-*
is the Fourier transform defined by the formula
of the function
jqz)e~p(-fk~)dz U(Z). The elevation
of the free surface
<(z,t)
is
1
q(r)cos o(t-5)exp(ikz-fkIV?)a(k)dr dk -
Let us consider the wave field at large distances L from the point to=(Vt,O) and at large T=LIV, by putting b=gLIV’+-, 8,6tIT
9=x/L, t’= t/T, E’(k’)=E(k’g/v2),
o'-[x'/(l+ex')l'", x’=lk’l, q’(-c’)=q(LT’lV).
We shall assume that the function q’(T’) is smooth, vanishes in the neighbourhood tends to unity when r'~&/2 (the eource is "slowly included"). For the function 6 in the new variables (we omit the primes), we get
of zero and
t
since the second term in (5) can be neglected at the indicated values of t. It is intuitively clear that points at which the gradients of the phase functions 'p*(z, k, t,r)=+o(t-r)+kz-k,r with respect to the variables them and equate the resulting
r and k vanish, make the main contribution expressions to zero. We then obtain aq* -&-!
ati
z,-**--((t-7)=0, L
to (6).
Let us find
(7a)
a. arpf -&- = 21 f --(t-7)=0,
(7b)
acp+ia7=ro-k,=o,
(7c)
2
*
corresponding to the N+) and N-_) indices, transform into one another when the Systems (7), We shall therefore subsequently confine ourselves to investigating sign of kl is changed. the system which corresponds to the X--D sign. System (7), in which k and T are treated as parameters, describes a family of curves in BY introducing the notation k,=xfus 6,kz=x sintl, -n/Z< BGIc/Z and expressing the (GZI) plane. x in terms of cos.0 using formula (7c), we get rr-[(z,-r)*+zz~i~~~= (t-z)[(coaxe+4p co9 ep-cos3 ei/(4p).
(8)
specified by formula (8) are homothetic with a centre of similitude The curves qp)+,=r,(e)l (t,O). We now present some simple properties of these curves: 1) each curve LI(p,) lies within the curve L&Z) when p,+2. 2) the curves Lx(p) are smooth, closed and symmetric with respect to the I, axis. are straight lines which pass through 3) the envelopes of the family of curves L,(p) the point (1,O) and have a slope to the negative direction of the 2, axis equal to p from (2). 4) just two curves of the family, corresponding to different r, pass through eachinternal point of the domain PI, bounded by the right interval of the curve Lo(p) and the envelopes while just a single curve passes through each internal point of the domain Q,, bounded by the curve L,(p). L,(p) which lie in the first quadrant are shown in Fig.1 for The parts of the curves These graphs were obtained by means of a numerical calculation. (=I,v-2.0,1.0, 0.5 and 0.1. 1) and 2) are elementary and their proof Let us prove properties l)- 4). Properties Property 4) follows from the properties of homothety and from involves direct verification. 3). Let us prove 3). The fact that the envelopes are straight lines follows from the must be satisfied on the envelope of The relationship detHessp-=O properties of homothety. function the family L,(p) (see /3/), where Hessrp- is a matrix of the second derivativesofthe
104
q-with
respect
to the variables
k and
'c:
It is readily seen that detHessrp- vanishes either as X-+” or when t=l, or when the polynomial (3) vanishes. By virtue of (7a) and (7b), the first case corresponds to the interval {O=ZZ,S:~,ZZ=~) while the second corresponds to the point (t,O). It is obvious that these sets of points are not envelopes. Formula (7~) is satisfied either when x=0 or when x>ko since lcos8/<1. The first case is excluded by virtue of (?a) and (7b) while the root of the polynomial (3) on the above-mentioned ray enables one to find the envelope. In fact, by using x found from (7c), it is possible to express k, and then to find kz and, using formulae (7a) and (7b), to find the coordinates of a point lying on the envelope. By making use of the fact that the envelope is a straight line, we obtain (2). It remains to be proved that, in the case of the polynomial (3), there is only a single root on the above-mentioned interval. We note that the equation detHesso-=O together with (7~) is, when WT, the equation for a point of inflection of the function k2=k2(kl) which is implicity specified by (7~) (this is proved by direct verification). It is readily seen that (7~) can be written as: k,‘(l-~k~Z)-2-kl’=k~Z. Since
this equation is invariant under a change in the signs of kt and k2. the behaviour of k%(k,) in the first quadrant may be considered. A graph of the function kt(k,) is shown in Fig.2. One is readily convinced by straightforward differentiation that the second derivative d’k&kP tends to -a as Irr-+k, and to +m as k,+ko8,kgo=)l+. The third derivative of the function kz(k,) has the form
(9)
Let us prove that d3kz/dki*>0 when ko=Gktckoo. &lrl Actually, when this is so zsZl and by virtue of the inequality k&f. Hence, the first and second parenthesis within the square brackets in (9) have a lower bound of zero while the third parenthesis is not less than three. We obtain the required inequality from this. Consequently, there is just a single point of inflection on the graph of &(k,) in the first quadrant while, in the case of the family L&), there are only two envelopes which are symmetrically disposed with respect to the I, axis. Property 3) is proved.
P
Fig.1
Fig.2
The fact that the angle which is swept by the curves increases follows from property 1).
L,,(p) decreases
monotonically
as
2, Proof. In order to confirm the investigation of Sect.1 it is necessary to reduce integral (6) A direct investo a standard form which permits the use of the method of stationary phase. tigation of (6) with the aid of this method is impossible owing to the fact that the phase We eliminate this point from the treatment by introfunctions 'p* are not smooth when k=O. X(k) is a function ducing the factors X(k) and *-x(k) into the integrand of (6), where from C.-(R*) which is equal to unity when Ikllj;‘/r and equal to zero when /k(z’lz. Let us evaluate the integral with the factor .x(k):
q(s)ii(k)X(k)exp(ih~*)dr
Here, having
twice integrated
by parts with respect
1
dk.
to r and using the formula
105
a
(ihI&‘,*)-1 d, exp(ih~*)=exp(ihq*),
we obtain ~(k)B(k) +,-
(k,*o)-lexp(ih$)dk-
(10)
t
ah-*x
j[ +.-
j
X(k)s(k)~(k~*~)-zexp(thlg,)dr]dk,
0
$=k,(z,-t)+kzxz. (k,*o)-= It is readily seen that the function is integrable when Iklc*/~. Hence, the second term in (10) is of the order of O(l-I). In the second term, we integrate by parts with respect to k, using the formula
which is possible for r~=[(z,-t)*+zz2]‘~~6~.The in&grand in the resulting integral will be integrable with respect to k and, hence, the whole of the integral I, is of the order of 00.~I), We that is, it is small compared with the principal term (which is of the order of O(h-"I)). therefore get
q(r)[l-X(k)ls(k)exp(ta~*)dr +.-
I
0
dk+O(L-L).
We now subdivide this integral into two integrals by inroducing the factors ~,(t-r)and X,(T) is a smooth function which is equal to unity when i-x*(t-s) into the integrand, where zcs,/4 and to zero when r9&/2. Let us consider the integral with the factor ~~('-1):
I,=aa
xj[ j +.-
q(r)[l-~(k)lX,(t-r)z(k)exp(i?.q*)dz
I
(I
dk.
The phase functions of this integrand have critical points on the carrier of the integrand only when r,=&i2/2. Hence, when r1M2 in it, it may be integrated by parts as many times as may be desired by using the formula
Let us now estimate
the first "outer integral" [l--x(k)]
(k&o)ii(k)
term. Ih,p*(s,
It has the form k, t, t) I-* exp(t%)dk.
(11)
+.-
When r,~& in (ll), it can be integrated by parts with respect to k as many times as may be It follows from this that the integral in (11) is a quantity of the order of o(a--) desired. uniformly with respect to t and z when 6,
(12)
1
[i-x(k)][l-x,(t-r)la(k)q(~)exp(“b’)dr
dk+O(li-9.
The method of stationary phase can now be applied to this integral and, in fact, if z is an internal point of the domain S&then each of the systems (7) has, by virtue of property 4) from Sect.1, just two systems of roots: kl,2=I,.**(z,t), %=7*(.%t) Using well-known
formulae,
we obtain
and
=
k,,?=ki,Z*(z,
=
t),
r=z*(z,
/4/
Q(T)z(4 exp(ilp+++nHessrf*) 'p*1".
k ,ri +O(i-9,
1d&Hess
==* F=T
t).
106
is the difference between the number of positive eigenvalues and the number where sgnHessp* of negative eigenvalues of the matrices Hesscp'. The fact that the integrand in (12) does not have a compact carrier with respect to k is immaterial by virtue ofthe rapid decay of the This "damages" the estimation of the residue in the standard at infinity. function ii(k) formula of the method of stationary phase but has no effect on the estimation of the residue in (13). At internal points of the domain !&each of the system (7) has just a single system of roots and only a single term remains in the formula for 5 under the summation sign. Outside 5=0(X-'). of these domains the integral in (12) is a quantity of the order of O(h-") and On the side boundaries of the domain 52,; that is, on the envelopes of the family L,(P), has a rank of formula (13) is unsuitable since, there, detHessm*=O. Since the matrix Hess@ (see /4, 5/). 2 on these straight lines, t=O(h-'") at values of z lying in their neighbourhood Standard manipulations enable one to write t in the neighbourhood of these curves in terms of Airy functions, similar to the case when p=O. In concluding the author expresses his thanks to S.A. Gabov for suggesting the problem and to S.A. Yakunin for putting a package of computer graphics programs at his disposal.
REFERENCES 1. GABOV S.A., On a problem of the hydrodynamics of an ideal liquid associated with flotation, Differents. Uravneniya, 24, 1, 16-21, 1988. 2. SRBTENSKII L-N., Theory of the Wave Motions of Liquids, Nauka, Moscow, 1977. 3. PCGOF~LOV A.V., Differential Geometry, Nauka, Moscow, 1974. 4. FEDORYUK M.V., The Method of Steepest Descent, Nauka, Moscow, 1977. 5. ARNOL'D V.N., VARCHENKO A.N. and GUSEIN-ZADE S.M., Singularities of Differentiable Mappings. Monodromy and Asymptotic Forms of Integrals, Nauka, I~loscow,1984.
Translated
by E.L.S.