A stable and accurate numerical method to calculate the motion of an interface between two fluids is used to calculate two‐dimensional standing water waves. The general method calculates arbitrary time‐dependent motion of an interface, possibly including interfacial tension and different density ratios between the fluids. Extremely steep standing waves are determined, significantly steeper than has been previously reported. The peak crest acceleration is used as the determining parameter rather than the wave steepness as the wave steepness is found to have a maximum short of the most extreme wave. Profiles with crest accelerations up to 98% of gravity are calculated (a sequence of raster images of this profile as it evolves in time over one period may be obtained upon application to the authors: e‐mail gmercer@spam.ua.oz.au or aroberts@spam.ua.oz.au), and the shape of these extreme standing wave profiles are discussed. The stability of the standing waves is examined and growth rates of the unstable modes are calculated. It is found that all but very steep standing waves are generally stable to harmonic perturbations. However, standing waves are typically unstable to subharmonic perturbations via a sideband‐type instability.

1.
C. J.
Amick
and
J. F.
Toland
, “
The semi-analytic theory of standing waves
,”
Proc. R. Soc. London Ser. A
411
,
123
(
1987
).
2.
J. W. S.
Rayleigh
, “
Deep water waves, progressive or stationary, to the third order of approximation
,”
Proc. R. Soc. London Ser. A
91
,
345
(
1915
).
3.
W. G.
Penny
and
A. T.
Price
, “
Finite periodic stationary gravity waves in a perfect fluid, Part 2
,”
Philos. Trans. R. Soc. London Ser. A
244
,
254
(
1952
).
4.
L. W.
Schwartz
and
A. K.
Whitney
, “
A semi-analytic solution for nonlinear standing waves in deep water
,”
J. Fluid Mech.
107
,
147
(
1981
).
5.
J. W.
Rottman
, “
Steep standing waves at a fluid interface
,”
J. Fluid Mech.
124
,
283
(
1982
).
6.
W.-T.
Tsai
and
D. K. P.
Yue
, “
Numerical calculation of nonlinear axi-symmetric standing waves in a circular basin
,”
Phys. Fluids
30
,
3441
(
1987
).
7.
T. R.
Marchant
and
A. J.
Roberts
, “
Properties of short-crested waves in water of finite depth
,”
J. Aust. Math. Soc. B
29
,
103
(
1987
).
8.
M. S.
Longuet-Higgins
and
E. D.
Cokelet
, “
The deformation of steep surface waves on water. I. A numerical method of computation
,”
Proc. R. Soc. London Ser. A
350
,
1
(
1976
).
9.
T. Vinge and P. Brevig, “Breaking waves on finite water depths: A numerical study,” The Ship Research Institute of Norway, Report No. R-111.81, 1981.
10.
G. R.
Baker
,
D. I.
Merion
, and
S. A.
Orszag
, “
Generalised vortex methods for free surface flow problems
,”
J. Fluid Mech.
123
,
477
(
1982
).
11.
D. I.
Pullin
, “
Numerical studies of surface-tension effects in nonlinear Kelvin-Helmholtzand Rayleigh-Taylor instability
,”
J. Fluid Mech.
119
,
507
(
1982
).
12.
J.-M. Vanden-Broeck (private communication). Since the acceptance of this paper for publication, it has come to the attention of the authors of a similar method being applied to standing waves with similar results and with possible extensions to include capillarity.
13.
A. J.
Roberts
, “
A stable and accurate numerical method to calculate the motion of a sharp interface between fluids
,”
IMA J. Appl. Math.
31
,
13
(
1983
).
14.
M. J. D. Powell, in Numerical Methods for Non-linear Algebraic Equations, edited by P. Rabinowitz (Gordon and Breach, London, 1972).
15.
H.
Aoki
, “
Higher order calculation of finite periodic standing waves by means of a computer
,”
J. Phys. Soc. Jpn.
49
,
1598
(
1980
).
16.
G. I.
Taylor
, “
An experimental study of standing waves
,”
Proc. R. Soc. London Ser. A
218
,
44
(
1953
).
17.
P. G.
Saffman
and
H. C.
Yuen
, “
A note on numerical computations of large amplitude standing waves
,”
J. Fluid Mech.
95
,
707
(
1979
).
18.
A. J.
Roberts
, “
Highly nonlinear short-crested water waves
,”
J. Fluid Mech.
135
,
301
(
1983
).
19.
B.
Chen
and
P. G.
Saffman
, “
Steady gravity-capillary waves on deep water—I. Weakly nonlinear waves
,”
Stud. Appl. Math.
60
,
183
(
1979
).
20.
M. S.
Longuet-Higgins
, “
The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics
,”
Proc. R. Soc. London Ser. A
360
,
471
(
1978
).
21.
M. S.
Longuet-Higgins
, “
The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics
,”
Proc. R. Soc. London Ser. A
360
,
489
(
1978
).
22.
D. R.
Crawford
,
B. M.
Lake
,
P. G.
Saffman
, and
H. C.
Yuen
, “
Stability of weakly nonlinear deep-water waves in two and three dimensions
,”
J. Fluid Mech.
105
,
177
(
1981
).
23.
V. E.
Zakharov
, “
Stability of periodic waves of finite amplitude on the surface of a deep fluid
,”
Zh. Prikl. Mekh. Fiz.
2
,
86
(
1968
)
[translated in
V. E.
Zakharov
,
J. Appl. Mech. Tech. Phys.
2
,
190
(
1968
)].
24.
J. W.
McLean
, “
Instabilities of finite-amplitude water waves
,”
J. Fluid Mech.
114
,
315
(
1982
).
25.
M.
Okamura
, “
Instabilities of weakly nonlinear standing gravity waves
,”
J. Phys. Soc. Jpn.
53
,
3788
(
1984
).
26.
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
27.
R. S.
MacKay
and
P. G.
Saffman
, “
Stability of water waves
,”
Proc. R. Soc. London Ser. A
406
,
115
(
1986
).
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