Near-field internal wave beams in two dimensions

Abstract

Abstract

Boundary integrals for oscillating bodies in stratified fluids

“…The forcing is thus represented by the equivalent source Such representation allows the extension of (3.6) to a viscous fluid, once the spectrum is known, namely with the wave vector and its horizontal projection. The extension follows the lines laid by Lighthill (1978, § 4.10), Hurley & Keady (1997) and Voisin (2020); it will be discussed further in § 6.2.…”

“…We proceed instead from (3.17), (4.3) and (4.4), writing Expanding these for , with , we get so that In addition to the quantities associated with the wave beams, namely their angle to the vertical, their half-width and the coordinates perpendicular to and along them, respectively, this expression involves also quantities associated with the critical segments joining, for each beam, the opposite critical points on either side of the ellipse, namely their angle to the horizontal, their half-length and the coordinates along and perpendicular to them, respectively. These quantities, introduced by Voisin (2020), are represented in figure 2.…”

“…The cylinder is thus equivalent to the source of spectrum where denotes a cylindrical Bessel function, the wave vector and the angle such that The knowledge of this spectrum provides an alternative expression of the waves, into which the effect of viscous attenuation can be added following Lighthill (1978, § 4.10) at large distance from the source, and Voisin (2020) at arbitrary distance from it.…”

“…The derivation of the waves is accompanied by that of a distribution of singularities equivalent to the body, with two consequences. First, once the spatial spectrum of the distribution is known, additional phenomena, like viscosity and unsteadiness, that affect the propagation of the waves and set their local structure, can be added into the analysis; this approach has been pioneered by Lighthill (1978, § 4.10) and Hurley & Keady (1997), and recently developed by Voisin (2020). Second, for a rigid body, the added mass of the body follows directly from the first moment of the distribution, providing immediate access to the radiated energy (called ‘conversion rate’ within the context of internal tides) and to the forces exerted on the body, without requiring the actual calculation of the waves; this aspect will be reported separately, and has been presented in summary form by Voisin (2009).…”

“…The indirect formulation of the problem as a single-layer integral is presented in § 6 and shown to lead to a simple representation of the body. The representations of a vibrating sphere in Voisin, Ermanyuk & Flór (2011) and a vibrating elliptic cylinder in Voisin (2020) are recovered as particular cases. Sections 7 and 8 discuss the behaviour of the boundary integrals at the boundary and in the far field, respectively.…”