Operator Approach to Linear Problems of Hydrodynamics

‘High spots’ theorems for sloshing problems

“…Let us turn to the ice-fishing problem for a single circular hole. This problem was investigated in [11] and, in particular, it was proved that two eigenfunctions of the form: which is similar to (2.9). The fundamental eigenvalue of (3.3) is also ν 1 .…”

“…where J 1 is the usual Bessel function, is shown to be positive (there is also an expression of this kernel in terms of the complete elliptic integral D), one can choose ψ 1 (r, 0) to be positive for r ∈ (0, 1); it is clear that ψ 1 (0, 0) = 0 because J 1 (0) = 0. Finally, choosing ψ 1 (1, 0) = 1 (see [11,Proposition 4]) says that the following asymptotic formula holds:…”

“…This and more general versions of the problem (it is usually referred to as the sloshing problem), have been the subject of a great number of studies over more than two centuries; see the paper [8] for a historical review and early results are described by Lamb in his Hydrodynamics [15]. Among recent works on this problem we mention the book by Kopachevsky and Krein [11], and the papers [12][13][14]. The latter two papers are, in particular, concerned with the question (related to the topic of our paper) when eigenvalues are simple.…”

“…With a time-harmonic factor removed, the velocity potential u(x, y) for the flow must satisfy the boundary value problem: holds, thus excluding the zero eigenvalue of (1.1)-(1.3), in which case the spectral parameter ν is equal to ω 2 /g, where ω is the radian frequency of the water oscillations and g is the acceleration due to gravity. It has been well known since the 1950s (these results can be found in many sources the most recent of which is the book [11]) that problem (1.1)-(1.4) has a discrete spectrum; that is, there exists a sequence of eigenvalues 0 < ν 1 ν 2 . .…”

The ice‐fishing problem: the fundamental sloshing frequency versus geometry of holes

“…The sloshing problem has been the subject of a great number of studies over more than two centuries. A historical review can be found in Fox and Kuttler's paper [3], and the monograph [4] by Kopachevsky and Krein contains an approach to problem (1)-(2) based on the theory of operators in Hilbert space. In particular, it is shown that the problem has a discrete spectrum that consists of positive eigenvalues, having ÿnite multiplicity and tending to inÿnity as the eigenvalue's number increases (see Section 3.3 in Reference [4] for details).…”