Residue-series computation of lit-region fields via a novel function P*(ξ,q,u)

2002 ◽  
Author(s):  
P.E. Hussar
Keyword(s):  
Residue Series

An interpretative model of carbon and nitrogentransformations applied to a residue incubation experiment

1998 ◽  
Vol 49 (3) ◽  
pp. 537 ◽  
Author(s):  
B. R. Trenbath ◽  
A. J. Diggle
Keyword(s):  
Organic Matter ◽  
Plant Residues ◽  
Published Data ◽  
Incubation Experiment ◽  
Nitrogen Immobilisation ◽  
Carbon Nitrogen Ratio ◽  
Nitrogen Ratio ◽  
Parameter Values ◽  
Interpretative Model ◽  
Residue Series

A simple model of 3 equations was devised to simulate the rates through time of gross mineralisation of nitrogen, nitrogen immobilisation, and microbial respiration relating to individual inputs into soil of plant residues of any age or type. Using published data from an incubation experiment carried out in Iowa, we applied the model to a residue newly added to soil, to the original soil organic matter (SOM), and to a mixture of these. Manipulation of the model allowed the derivation from the Iowa data of a net mineralisation index which seemed to summarise the nitrogen release characteristics of the residue in all treatments of the experiment. The equations and parameter values developed for the added residue were applied to SOM using results from unamended soil. The balance between respiration and mineralisation was found not to correspond to that expected for old organic matter near an equilibrium carbon/nitrogen ratio. Rate constants of mineralisation and respiration for SOM were adapted to overcome this apparent anomaly. To model the dynamics SOM and added residue simultaneously, the 2 sets of 3 equations were applied in parallel to 4 extreme treatments in the with-residue series (lowest and highest nitrate levels with low and high residue additions). To achieve the fits presented, only 2 of the 12 parameters required in each set of equations needed to differ between the set for SOM and that for added residue. The model reproduces well most of the primary Iowa data and also some derived results. Use of the model helped to interpret divergences between simulations and data.

Creeping modes in a shadow

1968 ◽  
Vol 64 (1) ◽  
pp. 171-192 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Asymptotic Series ◽  
Shadow Region ◽  
Two Dimensional ◽  
Dimensional Theory ◽  
Smooth Curves ◽  
Original Series ◽  
Mode Expansion ◽  
In Series ◽  
Watson Transformation ◽  
Residue Series

AbstractIn the two-dimensional theory of diffraction by smooth curves there are certain canonical problems that can be solved explicitly in series form. The series converge slowly at short wavelengths but they can be transformed by Watson's transformation into another form (the residue series or creeping-mode expansion) which has been much used in shadow regions. It is found that throughout the shadow region the first few terms of the residue series are exponentially small and decrease rapidly, and these have often been used as an estimate of the wave potential without further justification. Leppington's recent work on the shadow of an ellipse has shown, however, that in part of the shadow some of the later terms of the residue series are exponentially large. In other words, the complete residue series in part of the shadow is even more slowly convergent than the original series.In the present paper the Watson transformation is re-examined in the light of this result. The original series is expressed as the sum of a finite number of terms of the residue series and of a remainder. It is shown that throughout the shadow the remainder is at short wavelengths asymptotically of smaller order than the last term retained in the finite residue series. It follows that the residue series (which is a convergent infinite series) is also an asymptotic series, and this fact is sufficient to justify most of the usual applications. The proof is given in detail for the circle and in outline for the ellipse; it makes use of the theory of conformal mapping.

7.—High-frequency Scattering in a Certain Stratified Medium

1974 ◽  
Vol 72 (2) ◽  
pp. 93-107
Author(s):  
W. G. C. Boyd
Keyword(s):  
Real Axis ◽  
High Frequency ◽  
Fourier Integral ◽  
Stratified Medium ◽  
Plane Boundary ◽  
Airy Functions ◽  
Asymptotic Field ◽  
The Real ◽  
Asymptotic Nature ◽  
Residue Series

SynopsisThe propagation of scalar waves in a certain stratified medium is studied; the field is due to a line source situated on an opaque plane boundary. The exact field can be expressed in terms of a Fourier integral involving Airy functions. By deforming the real axis, which is the contour of integration of the Fourier integral, only in the neighbourhood of the real axis, it is possible to give a simple but rigorous derivation of the asymptotic nature of the field. Two separate cases are considered: the point of observation lying in the illuminated region, when a steepest descents analysis is appropriate, or lying in the shadow region, when the asymptotic field is given by a residue series.

On the self-induced motion of a helical vortex

1999 ◽  
Vol 384 ◽  
pp. 263-279 ◽  
Author(s):  
J. BOERSMA ◽  
D. H. WOOD
Keyword(s):  
Closed Form ◽  
Asymptotic Expression ◽  
Helical Line ◽  
Complete Representation ◽  
Small Pitch ◽  
Helical Vortex ◽  
Induced Motion ◽  
A Minor ◽  
Minor Correction ◽  
Residue Series

The velocity field in the immediate vicinity of a curved vortex comprises a circulation around the vortex, a component due to the vortex curvature, and a ‘remainder’ due to the more distant parts of the vortex. The first two components are relatively well understood but the remainder is known only for a few specific vortex geometries, most notably, the vortex ring. In this paper we derive a closed form for the remainder that is valid for all values of the pitch of an infinite helical vortex. The remainder is obtained firstly from Hardin's (1982) solution for the flow induced by a helical line vortex (of zero thickness). We then use Ricca's (1994) implementation of the Moore & Saffman (1972) formulation to obtain the remainder for a helical vortex with a finite circular core over which the circulation is distributed uniformly. It is shown analytically that the two remainders differ by 1/4 for all values of the pitch. This generalizes the results of Kuibin & Okulov (1998) who obtained the remainders and their difference asymptotically for small and large pitch. An asymptotic analysis of the new closed-form remainders using Mellin transforms provides a complete representation by a residue series and reveals a minor correction to the asymptotic expression of Kuibin & Okulov (1998) for the remainder at small pitch.

XVI.—Creeping Waves in an Inhomogeneous Medium

1971 ◽  
Vol 69 (3) ◽  
pp. 227-245
Author(s):  
W. G. C. Boyd
Keyword(s):  
Refractive Index ◽  
Asymptotic Expansion ◽  
Stationary Phase ◽  
High Frequency ◽  
Asymptotic Methods ◽  
Plane Boundary ◽  
High Frequency Field ◽  
Airy Functions ◽  
Residue Series ◽  
Asymptotic Developments

SynopsisThis paper is concerned with high-frequency scattering in a medium, the square of whose refractive index varies linearly with height from a plane boundary. Two asymptotic methods are examined, namely the method of stationary phase and evaluation by residue series. The first of these corresponds to geometric optics and gives the high-frequency field in the illuminated region, while the second complements the first in the sense that if thsre is no point of stationary phase, the residue series is an asymptotic expansion of the field. The Airy functions in the residue series can be replaced by their asymptotic developments in terms of exponentials, and when this is done only the first term or first creeping wave is of genuine significance.

Supersonic wing-body interference

1970 ◽  
Vol 68 (3) ◽  
pp. 719-729 ◽  
Author(s):  
R. W. Clark
Keyword(s):  
Cross Section ◽  
Velocity Potential ◽  
Formal Solution ◽  
Boundary Surface ◽  
Summation Formula ◽  
Inviscid Flow ◽  
Poisson Summation Formula ◽  
Infinite Plane ◽  
The Laplace Transform ◽  
Residue Series

We consider here the supersonic inviscid flow past a wing-body combination consisting of a semi-infinite plane wing symmetrically placed about an infinite convex cylinder. When the cylinder is circular in cross-section a formal solution for the Laplace transform of the velocity potential exists (Neilson(l), and Stewartson, 1951, unpublished). This solution is given in the form of a series which is only slowly convergent near the boundary surface separating the disturbed and the undisturbed regions. However, this slow convergence has been overcome by Waechter(4) using the Poisson Summation formula on the series solution. This enables the solution to be expressed in terms of integrals which can be evaluated as a convergent residue series, taking on different forms in the different regions of the interaction.

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