Finite‐amplitude monochromatic waves in free‐shear layers, planar jets, gravity–capillary waves, free‐convection boundary layers, and falling films are often first destabilized by disturbances twice or two‐thirds their wavelength or period. The transition to the 1/2 mode allows energy transfer to modes with lower wave numbers or lower frequencies and it precedes the onset of wide‐spectrum turbulence in the above systems. A center‐unstable manifold theory is used to derive the onset criteria for these subharmonic instabilities in terms of a critical amplitude of the fundamental or a critical value of a control parameter. It is shown that, in contrast to classical results, a finite‐amplitude wave is always unstable to disturbances with 1/2 its wave number (frequency) if the subharmonic is linearly unstable. This instability is either oscillatory or static depending on the phase difference (frequency mismatch) between the fundamental and the subharmonic and on the amplitude of the fundamental. For a given phase difference, there exists a critical amplitude beyond which the static instability triggers a most efficient energy transfer to the 1/2 mode and causes it to grow monotonically. Inefficient oscillatory interaction occurs below this critical amplitude of the fundamental. In contrast, the fundamental −3/2 interaction is always oscillatory except when there is perfect resonance (no frequency mismatch). The theory presented here contains a rigorous and explicit derivation of amplitude equations for spatially evolving open‐flow systems without recourse to Gaster transformation or multiscale expansions. This new formulation is partially demonstrated on the inviscid shear layer.

1.
W.
Eckhaus
, “
Problèmes non linéaires de stabilité dans un espace a deux dimensions. Deuxieme partie: stabilité des solutions périodiques
,”
J. Méc.
2
,
153
(
1963
).
2.
T. B.
Benjamin
and
J. E.
Feir
, “
The disintegration of wave trains on deep water
,”
J. Fluid Mech.
27
,
417
(
1967
).
3.
C. G.
Lange
and
A. C.
Newell
, “
A stability criterion for envelope equations
,”
SIAM J. Appl. Math.
27
,
441
(
1974
).
4.
S. P.
Lin
, “
Finite amplitude side-band stability of a viscous film
,”
J. Fluid Mech.
63
,
417
(
1974
).
5.
J. T.
Stuart
and
R. C.
DiPrima
, “
The Eckhaus and Benjamin-Feir resonance mechanisms
,”
Proc. R. Soc. London Ser. A
362
,
27
(
1978
).
6.
M.
Cheng
and
H.-C.
Chang
, “
A generalized sideband stability theory via center manifold projection
,”
Phys. Fluids A
2
,
1364
(
1990
).
7.
H.
Sato
, “
Further investigation on the transition of two-dimensional separated layer at subsonic speeds
,”
J. Phys. Soc. Jpn.
14
,
1797
(
1959
).
8.
R. E.
Kelly
, “
On the stability of an inviscid shear layer which is periodic in space and time
,”
J. Fluid Mech.
27
,
657
(
1967
).
9.
P. A.
Monkewitz
, “
Subharmonic resonance, pairing and shredding in the mixing layer
,”
J. Fluid Mech.
188
,
223
(
1988
).
10.
F. O.
Thomas
, “
An experimental investigation into the role of simultaneous amplitude and phase modulation in the transition of a planar jet
,”
Phys. Fluids A
2
,
553
(
1990
);
F. O.
Thomas
and
H. C.
Chu
, “
Experimental investigation of the nonlinear spectral dynamics of planar jet transition
,”
Phys. Fluids A
3
,
1544
(
1991
).
11.
R. W.
Miksad
, “
Experiments on the nonlinear stages of free-shear-layer transition
,”
J. Fluid Mech.
56
,
695
(
1972
).
12.
N.
Brauner
and
D. M.
Maron
, “
Characteristics of inclined thin films, waviness and the associated mass transfer
,”
Int. J. Heat Mass Transfer
25
,
99
(
1982
).
13.
T.
Prokopiou
,
M.
Cheng
, and
H.-C.
Chang
, “
Long waves on inclined films at high Reynolds number
,”
J. Fluid Mech.
222
,
665
(
1991
).
14.
I. Choi, “Contributions à l’étude des mechanisms physiques de la generation des ôndes de capillarite-gravité à une interface air-eau,” Ph.D. thesis, Université D’Alix Marseille, 1977.
15.
P. A. E. M.
Janssen
, “
The period-doubling of gravity-capillary waves
,”
J. Fluid Mech.
172
,
531
(
1986
).
16.
C. C.
Chen
,
A.
Lahbabi
,
H.-C.
Chang
, and
R. E.
Kelly
, “
Spanwise vortex pairing of finite-amplitude longitudinal vortices in inclined free convection boundary layers
,”
J. Fluid Mech.
231
,
73
(
1991
).
17.
E. M.
Sparrow
and
R. B.
Husar
, “
Longitudinal vortices in natural convection flow on inclined surfaces
,”
J. Fluid Mech.
37
,
251
(
1969
).
18.
A. J.
Roberts
, “
The utility of an invariant manifold description of the evolution of a dynamical system
,”
SIAM J. Math. Anal.
20
,
1447
(
1989
).
19.
H.
Chaté
and
P.
Manneville
, “
Transition to turbulence via spatiotemporal intermittency
,”
Phys. Rev. Lett.
58
,
112
(
1987
).
20.
G. I.
Sivashinsky
, “
Instabilities, pattern formation, and turbulence in flames
,”
Annu. Rev. Fluid Mech.
15
,
179
(
1983
).
21.
P.
Huerre
and
P. A.
Monkewitz
, “
Absolute and convective instabilities in free shear layers
,”
J. Fluid Mech.
159
,
151
(
1985
).
22.
A.
Michalke
, “
On spatially growing disturbances in an inviscid shear layer
,”
J. Fluid Mech.
23
,
521
(
1965
).
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