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On high-order perturbation expansion for the study of long–short wave interactions
Published online by Cambridge University Press: 11 May 2018
Abstract
In high-order analysis and simulation of long–short surface wave interaction using mode decomposition, ‘divergent’ terms of the form
$k_{S}a_{L}=O(\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D716})\gg 1$
appear in the high-order expansions, where
$k_{L,S}$
,
$a_{L,S}$
are respectively the long, short modal wavenumbers and amplitudes, with
$\unicode[STIX]{x1D6FE}\equiv k_{S}/k_{L}\gg 1$
and
$k_{L}a_{L}\sim k_{S}a_{S}=O(\unicode[STIX]{x1D716})$
finite. We address the effect of these terms on the numerical scheme, showing numerical cancellation at all orders
$m$
; but increasing ill-conditioning of the numerics with
$\unicode[STIX]{x1D6FE}$
and
$m$
, which we quantify. In the context of mode decomposition, we show theoretical exact cancellation of the divergent terms up to
$m=3$
, extending the existing result of Brueckner & West (J. Fluid Mech., vol. 196, 1988, pp. 585–592) and supporting the conjecture that this is obtained for all orders
$m$
. We show the latter by developing a theoretical proof for any
$m$
using a Dirichlet–Neumann operator and mathematical induction. The implication of the theoretical proof on the numerical simulation of long–short wave interaction is discussed.
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