The possibility of intragenic heterogeneity between copies of the long intergenic (1 63-235 rDNA) spacer region (LISR) was investigated by specific amplification of this region from 21 Enterococcus faecalis isolates. Three copies of the LlSR (rrnA, B and C) were demonstrated by hybridization of the LlSR to genomic DNA cleaved with I-Ceul and Smal. When the LISR amplicon was digested with Tsp5091, two known nucleotide substitutions were detected, one 4 nt upstream from the 5' end of the tRNAaIa gene (allele rrnB has the TspSOSl site and rrnA and Cdo not) and the other 22 nt downstream from the 3' end of the tRNAa'* gene (rmC has the Tsp5091 site). Sequence differences at these sites were detected at the allelic level (alleles rrnA, B and C) and different combinations of these alleles were designated lsp Types. Using densitometry to analyse bands from electrophoresis gels, the intra-isolate ratios of the separate alleles (rmA:rmB:rmC) were determined in each Tsp Type: I (0:3:0), II (l:2:U), 111 (2:O:I)' IV (3:0:0)# V (2:1:0) and VI (1:l:l). Sequence variation between the three copies of the LlSR was confirmed by the detection of at least five other intra-isolate nucleotide substitutions using heteroduplex analysis by conformation-sensitive gel electrophoresis (CSGE) that were not detected by Tsp5091 cleavage. Perpendicular denaturing gradient gel electrophoresis was capable of resolving homoduplexes; six to seven out of a possible nine curves were obtained in some isolates. In the isolate where seven curves were obtained one or more further nucleotide substitutions, not detected by TspSO9l cleavage or CSGE, were detected. On the basis of LlSR sequence heterogeneity, isolates were categorized into homogeneous (only one allele sequence present) and heterogeneous (two or three allele sequences present). The transition between homogeneous and heterogeneous LlSRs may be useful in studying evolutionary mechanisms between E. faecalis isolates.
SUMMARYThe purpose of this paper is to investigate the effect of a non-uniform mesh in two dimensions (2D). A change in mesh size will, in general, result in spurious refraction (and reflection) which is entirely numerical (rather than physical) in origin. To facilitate the analysis, the mesh geometry has been highly simplified in that only a single change in mesh size is considered. The analysis is based on a finite element wave model.The domain consists of two conterminous regions discernible only by their different nodal spacings in the xdirection. The interface between the two regions is internal to the mesh and is a straight line. The model is based upon the Crank-Nicolson linear finite element scheme applied to the second order wave equation. The results of the analysis are confirmed by numerical experiments. It is shown that under particular numerical conditions total internal reflection may occur and when this is the case, the transmitted wave is evanescent. An analysis of the energy flux associated with the incident, reflected and transmitted waves shows that energy is conserved across the interface between the two regions.KEYWORDS: spurious wave refraction; total internal reflection INTRODUCTKlNAn appreciation of the effects of a non-uniform mesh is fundamental to a good understanding of the processes occumng in numerical models with varying mesh sizes. This is particularly relevant to time dependent finite element (FE) models and interactively nested finite difference (FD) models (in contradistinction to passively nested FD models). Practioners are often faced with questions such as 'how much can I change the mesh size and what are the consequences?'In 1D linear systems, these questions have been partly addressed.'-3 The analyses were completed for the simplified situation of a 1D domain consisting of two semi-infinite regions abutting at a common interfacial node. The two regions were discernible on numerical grounds (e.g. different mesh spacings or different numerical algorithms) or physical grounds (effected by an abrupt change in coefficient in the governing equation). The latter case is not considered herein and hence the inclusion of the word 'spurious' in the title of the present paper.The magnitude of the effects of a change in 1D mesh size were quantified by determining the amplitudes of the transmitted and (unwanted) reflected waves, and their associated energy fluxes due to an incident wave. In a more general sense, the incident wave may be interpreted as one of many Fourier components which can be superimposed to make up a general waveform.In ID, a change in mesh size results in both wave rejection and wave transmission. In 2D however, there is an additional process at work viz wave refraction. The interface between the two conterminous
SUMMARYThis is the first of a series of three related papers dealing with some of the consequences of non-uniform meshes in a numerical model. In this paper the accuracy of the Crank-Nicolson linear finite element scheme, which is applied to the linear shallow water equations, is examined in the context of a single abrupt change in nodal spacing. The (in)accuracy is quantified in terms of reflection and transmission coefficients. An incident wave impinging on the interface between two regions with different nodal spacings is shown to give rise to no reflected waves and two transmitted waves. The analysis is verified using three different wavelengths A AX, 4Ax, 8Ax) in three 'hot-start' numerical experiments with a mesh expansion factor of 2 and three experiments with a mesh contraction factor of 1/2. An energy flux analysis based on the concept of group velocity shows that energy is conserved across the interface.
Numerical analysis of difference schemes often reveals the presence of eigenmodes which do not feature in the continuum solution. An examination of the dispersion relation shows how the spurious and physical modes interact. The behaviour of certain wave-profiles was predicted using this analysis and the results confirmed by numerical experiment. 'New address. British Gas Corporation, Engineering Research Station, Harvey Combe, Killingworth P.O. Box 1 LH, Newcastle-upon-Tyne, NE99 ILH.
The existing theory of acoustic propagation through an oceanic internal wave field with a Garrett and Munk spectrum is modified and, by numerical computation, is shown to be consistent. The fractional sound speed computation is rederived to satisfy the Garrett and Munk spectrum and used to compute a stochastic simulation of the internal wave sound speed fluctuation field. The Garrett and Munk spectrumin (o),j) space has been normalized by 4•r, and the acoustic scattering is redefined to accommodate scattering from the internal wave phase fronts as in an acoustic phase grating. These modifications are then used to compute the coherent acoustic intensity by two methods: a first-order multiple scatter approximation and a stochastic simulation. Also, the Rytov approximation is shown to be equivalent to the firstorder multiple scatter approximation in the form of the stochastic parabolic equation method in the unsaturated region. The computational results show agreement in the weak scattering region using typical deep ocean values. The stochastic simulation method is accurate in the saturated and unsaturated regions; however, the method requires long computer execution times. Phase front fragments propagating along rays with sound speeds reduced by the stochastic internal wave field are used to discuss the computational results.
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