pRCH1E'
Proc. Camb. Phil. Soc. (1967), 63, 1167 PCPS 63-136
With 5 text-figures Printed in Great Britain
Seiches over parabolic bottomst
By E. V. LAITONE
University of California, Berkeley
(Received 12 July 1966)
Abstract. The solutions are derived for the shallow water standing waves of small
amplitude that can form in channels or lakes of varying breadth with a concave
parabolic bottom. In addition explicit solutions are given for standing waves in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls.
1. Introduction. The typical seiche in a lake or inside a harbour is a long period and
small amplitude oscillation of all of the contained water so as to form a so-called
standing wave. Since the amplitude is relatively very small when compared with the
tab.
y.
Scheepsbou'ikunde
Techrische
IiogeschGp67
Deift
Fig. i(a). Coordinate system for a rectangular cross-section channel. (b) Piane fundamental seiche with vertical dimensions stretched.
wavelength, and the motion is practically the same near the bottom as it is at the
surface for any given location, therefore the linearized shallow water wave theory is sufficient for analyzing the usual seiche motion. If, as shown in Figure 1, we let denote the ordinate of the wave profile, and let represent the horizontal displacement of the water from its equilibrium position at any given value of x, then the linearized shallow
water theory may be written as
_=....gi;
b?,=_(bhsc)
(1.1)t This work was supported by the Fluid Dynamics Branch, U.S. Off ce of Naval Research under contract N(ONR)-3656 (07).
(2.2) 1168
for a rectangular cross-section channel of varying breadth b and depth h as shown by
Lamb (6).
One of the most extensive investigations of these shallow water waves over a para-bolic bottom was carried out by Cbrystal(2) who used (1.1) in the form
S=bh
(1.2)which he then reduced to a simpler equation by introducing the new variables
U=S,
V=Jbdx
(1.3)so that (1.2) could be written as
a2
a i
U] a2uat2 = xLb1----i = gSb (1.4)
Then by using the method of Frobemus, Chrystal was able to obtain series solutions of (1.4) that would reduce to finite polynomials for a constant breadth channel with a
parabolic bottom. Haim (4) thon showed how the constant breadth channel with a
convex bottom had standing wave solutions related to hypergeometric functions, while
Chrystal's solutions for a concave parabolic bottom were given explicitly by the
Legendre polynomials. Previously Lamb (5) had already derived these Legendre
polynomial solutions for a constant breadth channel with a concave parabolic bottom. Lamb's derivation was much simpler than that of Chrystal and Haim because he found the wave proffle directly from (1.1) by writing it as
i a2
i a'(S,)
-
15- S
a-
+
,h
see also Wehausen and Laitone (7) where the limitations on these shallow water
equations are discussed. Although Chrystal's equation (1.4) may be more suitable for the numerical calculation of seiches in most actual lakes, and is still being used for that
purpose as shown by Defant (3), still (1.5) is much more suitable for investigating
shallow water seiches in bodies of water whose geometry can be expressed in terms of polynomials. As Chrystal (2) pointed out the concave parabolic bottom is probably the
best approximation for most cases so it will now be shown how the hypergeometric functions easily yield explicit polynomial solutions of (1.5) for seiches in various
varying breadth channels having a concave parabolic bottom defined by
(h/h0) = i - (x/a)2 i - (r/a)2. (1.6)
2. Seiches in rectangular cross-section channels. The simplest solutions of (1.5) can be
obtained by introducing the relations
(x,t)=iì()e,
=(x/a)
(21)into (1.5) and (16) so as to obtain
= O, À2 = a2o2/c2, c2 = gh0
where
Seiches over parabolic bottoms 1169
Then if we express the varying breadth b as a polynomial in the form
(b/b0) =
(be/b) =p(l+)-1q(l)-'
(2.4)we can reduce (2.2) to the hypergeometric equation
(1E)+[(pq)(p+q+2)fl+À2ìì = 0
(2.5)which has the following solution, see Whittaker and Watson (8), for any p > - i
(cL)(/J) /1+\
=
F(fi;
'2
) =
n=o(y)n!
2 )
(x)=x(+1)...(+nl),
(c.)0= 1,y=(p+l) >0,
(+fi) = (p+q+ 1),
OEfi = = (a2o2/c2) > 0.(2.6)
On the other hand if p is a negative integer the solution of (2.5) becomes, see Whittaker and Watson (8) ??h10 = (
l±)MF(
M,/3+M; i + M;i
(2.7) whereM = (ly) = p = 1,2,3,
...,
(+fi) =(qM+ 1),fi = À2 = (a2o2/c2) >0.
These solutions reduce to simple polynomials if either c or
fi
is a negative integer. For example, if we consider the case wherein p > - i we obtain from (2.6)z=n= 1,2,3,...,fi=(n+1+p+q), y=(p+l) >0
(28) so that the solution of (1.5) may be written for p> - i as(x,t) = ,0F(n.n+ 1+p+q; p+ 1; -(1+x/a))coso-t,l
À2= (a2a2/c2)=n(n+1+p+q).
JAs a simple example of (2.9) let us consider the case when p = i = q so that
(b/b0) = (1x2/a2) = (h/h0),
n(xt)=ioF(_nn+3;2; l+9a)cosot,
(2.10)= (c/a) (n2 + 3n)l.
The channel and the wave proffles at t = O are shown in Figure 2 for n = 1, 2 and 3.
However, if p is a negative integer (- M), we then obtain a polynomial solution of
(1.5) by means of (27) by letting (a+M) = nso that
i+M;
1+xIa)cost
À2 = (a2rx2/c2) = (n (b/b0) = (1+x/a)_M(1_x/a),
(2.9)
h/h.
10
05
05
10
Fig. 2. Seiches in a closed channel defined by (2-lo).
As a final illustration we find that (2-11), or (2.9) after a linear transformation, gives the channel and wave profiles showii in Figure 4 for M = i = q in (2.11). For p = i = - q we find that (2.9) directly gives the mirror image of Figure 4.
The only example that has been previously given of the new family of solutions
represented by (2.4), (2.9) and (2-11) is the case of the constant breadth channel which
was first derived by Lamb (5), and later by Chrystal(2). In this case we
find that
M
= p =
O = q so that (2.9) and (2-11) both reduce to il+x/a)
= i0P(x/a)cos o-t,
o-= (c/a) (n2 + n)i
as given by Lamb (5, 6).
(2.13) where now M is a positive integer and q can be any real number. For example, if we
take M = i = q we have
(b/b0) = (1x2/a2)1 = (h0/h),
(x,t)=O(l+:/a)F(n,n+l;2; l+x!a).t
(2-12)o- = (c/a)(n2+m)l.
Seiches over parabolic bottoms
1171 3. Seiches in a circular basin. The shallow water waves in a circular basin with aparabolic bottom as defined by (1.6) have already been studied by Lamb (6). However, we shall re-derive Lamb's equations so as to present them in a simpler form. Then later in section 4 we will apply these solutions to analyze the seiches in a circular lake with a
central island; that is, a ring-type lake with a parabolic bottom having a circular
cylinder projecting vertically from its center.
where 10 05 b/b, O
05
10
15
025025
Fig. 3. Seiches in an open-ended channel defined by (2.12).
Lamb (6) page 292, shows that for a circular basin with a parabolic bottom (1 5) must be replaced by
r2( i - r2/a2) fl,.,. + (1 - 3r2/a2) r/,. + {(À + m2) (r2/a2) - m2] = O, (3.1)
Q;,
i(r, O, t) = ij(r) eimoei, À2 = a2o2/c2, c2 = gh0.
Now if we introduce the variable z r2/a2 we transform (3.1) to a recognizable form of
the hypergeometric equation that was extensively studied by Chaplygin (1) for
two-dimensional, isentropic gas flow in the hodograph plane, namely
z2(1z)+(1-2z)z+[(À2+m2)zm2]iì = 0
(3.2)with a solution that is analytic at the origin given by
=
zl''F(4,u+ m, 1 ,u+ m; i +m; z)eimoe,
as shown by Chaplygin(1), and Whittaker and Watson ((8), page 283). If we use the
positive root sign for u in (33) we obtain
= (c/a) (t2-2m2 )+
10 05 b/b0 005
10
10 0505
10
Fig. 4. Seiches in a channel closed at on.ly one end.
and a solution corresponding to that given by Lamb ((6), page 292). However, if we
use the negative root sign for 1a we can obtain the following foi m = O,
= -2n = [(1+ A)2 1] < O,
(r,t) = 't;i0F(n,n+l; 1;r2/a2)cosot
(34)= 0P(1 - 2r2/a2) cosot, o- = (c/a) (4n2 + 4n)i.
This is the solution for the symmetrical seiche, independent of O, which is indicated by the footnote in Lamb ((6), page 292). Now a more useful form of the solution for m + O may be obtained by noting that (33) reduces to a finite polynomial if we write
4a=_(2n+m)=_[(1+A2+rn2)_1]< O,
= ,/o(r/a)mF(_n,n+ i +m; i +m; r2/a2)cosmO cosot,'t (35)
Seiches over parabolic bottoms
1173 Obviously (34) is a special case of (35) when m = O, and it is found that any integervalues of m and n give the same polynomial solution as does the series expansion given
by Lamb, or the equation given in Wehausen and Laitone ((7), page 672). For any
m + O the lowest frequency is given by taking n = O in (3.5) so as to obtain
ii(r, O, t) = ìì0(r/a)m cos mO cos (2m) ct/a. (36)
The fundamental fi equency corresponds to m = i so for the lowest frequency the free surface is always plane as indicated in Figure lb.
4. Seiches in a ring-type lake. Now we can also present some explicit solutions for shallow water waves in a ring-type lake with a parabolic bottom defined by
(h/h0) = (lr2/a2) (ar?b>O),
hr(a) = - 2h0/a, hr(b) = cc.
This iepresents a circular basin with a circular cylinder placed vertically at the centre of the parabolic bottom. Consequently our additional boundary condition becomes
ia
--=--=O at rb.
(4.2)The usual method of satisfying this additional boundary condition is to obtain a second solution of (3.1) so that its arbitrary constant will enable one to satisfy (4.2) when the
two solutions are combined. However, the solutions of (3.1) are restricted to integer
values of m because we must have
'(r,O,t)
= ì(r,2iT,t). (4.3)Consequently the second solution of (3.1) for m an integer is an infinite series which is
not convergent at r = a as shown by Whittaker and Watson (8). For the symmetrical
seiche which is independent of O so that ni = O the second solution, which is related to (3.4), is now given by the Legendre function of the second kind as
Q(1 - 2r2/a2) cos o-t (4.4)
but this solution is also infinite at r = a so it cannot be used to satisfy (4.1) and (4.2). As a matter of fact both (34) and (35) also yield an infinite series which is divergent
at r = a unless n is an integer, in which case they both reduce to a finite polynomial.
Consequently the only useful solutions provided by (3.5) are those polynomials which
are given by restricting m and n to positive integers including zero. However these
simple solutions can also be used to satisfy (4.1) and (4.2) for certain ratios of a/b by
noting that at the crest of any standing wave defined by (3.5) we always have j,. O
so that the boundary condition (4.2) is automatically satisfied at some fixed value of r = b. Therefore a solution for a seiche in a ring-type lake is obtained by setting the
derivative of (3.5), with respect to r, equal to zero so as to obtain the relation
o = (!
' {F(_nn+1+m;
1-i-m; r2/a2)(2n)(1+n+m) (r)2F(+
1,2+n+m; 2+ni;
r2/a2)]. m(1+m) a (4.1) (4.5)1174
As an example if we let n = i then a whole family of ring-type lakes is defined by
(b/a) = 0471. (46)
Other cases are easily analysed by the direct application of (45), which also may be used for symmetrical seiches (m = 0) since then the roots of the polynomial that is
given by F( - n + 1, n +2; 2; b2/a2) = 0 (4.7) i 5
05
\
r=b=05257a/
(4.9) 08 09 f/a 101 0
-Fig. 5. Comparison of the fundamental seiche in a ring-type lake (b/a = 0.5257) and either a parabolic bottom (4.8), or a flat bottom (4-9).
determine the proper ratios of b/a. For example, if n = 3 when m = 0, we obtain from (3.5) and (4.7) the seiche proffle in a ring-type lake as
(1 - 12x2 + 30x4 - 20x6) cos (ß.928ct/a), (r/a) = x 1, (b/a) = OE5257 or 08507.
This solution is compared in Figure 5, for the case where b/a = 05257 in a ring-type lake with a flat horizontal bottom and vertical walls at r = a, as well as at r = b. The
solution for such a horizontal bottom is given in Wehausen and Laitone ((7), page 672)
and may be written in terms of the Bessel functions of the first and second kind of
order zero, as
[J0(6.75x)-0.27Y(6.75x)]cos(6.75ct/a), 05257 x = (r/a) 1. (4.9)
It is seen that both the frequency and the wave profile near the island (r = b) are
nearly the same. However, the wave amplitude is much larger for the parabolic
bottom at r = a, as would be expected.
Conclusions. New solutions are derived for shallow water seiches over a parabolic
bottom. Equations (2.9) and (2.11) give explicit solutions for a rectangular cross-section channel with any varying breadth that can be represented by a polynomial. (4.8) 10 05 (48) N-. \... I I 0 1 02 03 :-04 05
Seiches over parabolic bottoms
1175 Several simple polynomial solutions are shown in Figures 2-4. The only solution thathad been previously presented is that for the constant breadth channel, (2.13).
Equation (35) gives the polynomial solutions for the seiches in a circular basin with a parabolic bottom in a more useful form than that previously given by Lamb (6). Equation (4.5) determines the numerical values of (35) that will correspond to the
seiche formed in a ring-type lake with a parabolic bottom and a central circular island that has vertical walls. These relations provide the only polynomial solutions for this
problem, and it is found that only certain ratios of island to lake radius can have
finite polynomial solutions. One example is presented in Figure 5 in comparison with
the seiche produced in a ring-type lake with a flat bottom and vertical walls.
REFERENCES
CRAPLYGIN, S. Gas jets, NACA TM 1063 (1944) (Moscow University, 1902).
CERYSTAL, G. On the hydrodynamical theory of seiches. Trans. Roy. Soc. Edinburgh 41, (1906), 599-649.
DEFANT, A. Physical oceanography, vol. u (Pergamon Press, 1961), pp. 160-173.
HALM, J. On a group of linear differential equations of the 2nd order including Professor Chrystal's seiche-equations. Trans. Roy. Soc. Edinburgh 41, (1906), 651-676.
LAMB, H. Hydrodynamics, 2nd ed. (Cambridge University Press, 1895), Section 182. LAIVn3, H. Hydrodynamics, 6th ed. (Cambridge University Press, 1932; Dover Publications,
New York, 1945), 273-277, 291-292.
WEMAUSEN, J. V. and LATTONE, E. V. Surface waves. Handbuch der Physik 9 (1960), 668-673.
WHITTAKER, E. T. and WATSON, G. N. Modern analysis, 4th ed. (Cambridge University Press, 1927), 281-286, 318.
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