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Oblique Interactions between Internal Solitary Waves

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In this paper, we study the oblique interaction of weakly, nonlinear, long internal gravity waves in both shallow and deep fluids. The interaction is classified as weak when where Δ1=|cm/cn −cosδ|, Δ2=|cn/cm −cosδ|,cm,n, are the linear, long wave speeds for waves with mode numbers m, n, δ is the angle between the respective propagation directions, and α measures the wave amplitude. In this case, each wave is governed by its own Kortweg‐de Vries (KdV) equation for a shallow fluid, or intermediate long‐wave (ILW) equation for a deep fluid, and the main effect of the interaction is an 0(α) phase shift. A strong interaction (I) occurs when Δ1,2 are 0(α), and this case is governed by two coupled Kadomtsev‐Petviashvili (KP) equations for a shallow fluid, or two coupled two‐dimensional ILW equations for deep fluids. A strong interaction (II) occurs when Δ1 is 0(α), and (or vice versa), and in this case, each wave is governed by its own KdV equation for a shallow fluid, or ILW equation for a deep fluid. The main effect of the interaction is that the phase shift associated with Δ1 leads to a local distortion of the wave speed of the mode n. When the interacting waves belong to the same mode (i.e., m = n) the general results simplify and we show that for a weak interaction the phase shift for obliquely interacting waves is always negative (positive) for (1/2+cosδ)>0(<0), while the interaction term always has the same polarity as the interacting waves.
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Document Type: Research Article

Affiliations: Monash University Shanghai University of Technology

Publication date: July 1, 1994

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