The finite-product method in the theory of waves and stability

Abstract

This paper presents a method of analysing the dispersion relation and field shape of any type of wave field for which the dispersion relation is transcendental. The method involves replacing each transcendental term in the dispersion relation by a finite-product polynomial. The finite products chosen must be consistent with the low-frequency, lowwavenumber limit; but the method is nevertheless accurate up to high frequencies and high wavenumbers. Full details of the method are presented for a non-trivial example, that of anti-symmetric elastic waves in a layer; the method gives a sequence of polynomial approximations to the dispersion relation of extraordinary accuracy over an enormous range of frequencies and wavenumbers. It is proved that the method is accurate because certain gamma-function expressions, which occur as ratios of transcendental terms to finite products, largely cancel out, nullifying Runge's phenomenon. The polynomial approximations, which are unrelated to Taylor series, introduce no spurious branches into the dispersion relation, and are ideal for numerical computation. The method is potentially useful for a very wide range of problems in wave theory and stability theory.

An asymptotic decoupling method for waves in layered media

“…In the notation of §2a, this is E s /E = 100, ρ s /ρ = 10 and h s /h = 1 10 ; we also take equal Poisson's ratios ν s = ν = 0.3. The real and imaginary branches of the finite-product approximations (m s , n s , m, n) = (0, 0, 0, 1), (0, 1, 1, 2), (1, 2, 3, 4) and (3,4,7,8) for real frequency are plotted as dotted lines in figure 2a-h. The exact dispersion relation is superposed as solid lines, for detailed comparison of finite-product approximations with the exact results of §3.…”

“…It follows from Chapman & Sorokin [7] that this is best achieved by taking n s = m s + 1 and n = m + 1, although the choices n s = m s and n = m are also good. This gives a two-parameter family of approximations (m s , m s + 1, m, m + 1).…”

“…Their mathematical theory, based on gamma functions and the cancelling-out of Runge's phenomenon, determines precisely their range of validity in the (frequency, wavenumber) plane [7]; moreover, finite products provide a rapidly convergent sequence of approximations in any finite region in this plane, no matter how great the size of the region, yet introduce no spurious branches into the dispersion relation in any approximation. The lowest-order members of a family of finite-product approximations recover the traditional dispersion relations referred to above [21].…”

The WKB approximation for analysis of wave propagation in curved rods of slowly varying diameter

“…A preliminary justification of whether an overlap exists between the short-wave WKB method applied to long-wave beam models is now given. In Chapman & Sorokin [14], it is demonstrated that the beam model formulation for uniform beams effectively corresponds to neglecting O(h 2 /λ 2 ) terms, where h is the local cross-wise dimension and λ is the wavelength. Denoting μ = h/λ and by introducing the relation μ = q , it can be shown that for 1/2 < q < 1 both the longwave beam assumption, μ 1, and short-wave WKB assumption, λ/l 1, are fulfilled when 1.…”

The method of finite-product extraction and an application to Wiener-Hopf theory

“…The inspiration for this work arose from reading the article (Chapman (2010)) in which the authors found that by replacing trigonometric functions by a f nite product polynomial and a Gamma function expression they were able to get very good approximations for the roots of eigenvalue equations. The replacement was exact and followed from the well known identities between the Gamma function and trigonometric functions.…”