On the scattering of torsional elastic waves from axisymmetric defects in coated pipes

Long range ultrasonic testing is now a well established method for examining in-service degradation in pipelines. In order to protect pipelines from the surrounding environment it is common for viscoelastic coatings to be applied to the outer surface. These coatings are, however, known to impact on the ability of long range ultrasonic techniques to locate degradation, or defects, within a coated pipe. The coating dissipates sound energy travelling along the pipe, attenuating both the incident and reflected signals making responses from defects difficult to detect. This article aims to investigate the influence of a viscoelastic coating on the ability of long range ultrasonic testing to detect a defect in an axisymmetric pipe. The article focuses on understanding the behaviour of the fundamental torsional mode and quantifying the effect of bitumen coatings on reflection coefficients generated by axisymmetric defects. Reflection coefficients are measured experimentally for coated and uncoated pipes and compared to theoretical predictions generated using numerical mode matching and a hybrid finite element technique. Good agreement between prediction and measurement is observed for uncoated pipes, and it is shown that the theoretical methods presented here are fast and efficient making them suitable for studying long pipe runs. However, when studying coated pipes agreement between theory and prediction is observed to be poor for predictions based on those bulk acoustic properties currently reported in the literature for bitumen. Good agreement is observed only after conducting a parametric study to identify more appropriate values for the bulk acoustic properties. Furthermore, the reflection coefficients obtained for the fundamental torsional mode in a coated pipe show that significant sound attenuation is present over relatively short lengths of coating, thus quantifying those problems commonly encountered with the use of long range ultrasonic testing on coated pipes in the field.     

A hybrid finite element approach to modeling sound radiation from circular and rectangular ducts

A numerical model based on a hybrid finite element method is developed that seeks to join sound pressure fields in interior and exterior regions. The hybrid method is applied to the analysis of sound radiation from open pipes, or ducts, and uses mode matching to couple a finite element discretization of the region surrounding the open end of the duct to wave based modal expansions for adjoining interior and exterior regions. The hybrid method facilitates the analysis of ducts of arbitrary but uniform cross section as well the study of conical flanges and here a modal expansion based on spherical harmonics is applied. Predictions are benchmarked against analytic solutions for the limiting cases of flanged and unflanged circular ducts and excellent agreement between the two methods is observed. Predictions are also presented for flanged and unflanged rectangular ducts, and because the hybrid method retains the sparse banded and symmetric matrices of the traditional finite element method, it is shown that predictions can be obtained within an acceptable time frame even for a three dimensional problem.     

The ferromagnetic to paramagnetic phase transition of Fe studied by x-ray photoelectron spectroscopy

Valence band x-ray photoelectron spectra from Fe(100) have been measured as a function of temperature to above the Curie temperature, T_c. The room temperature data can be reconciled with the theoretical one-particle density of states (DOS). At T = 1.034T_c, the data do not resemble the paramgnetic DOS of Fe as calculated in the disordered-local-moment limit.     

Coding and enumeration of trees that can be laid upon a hexagon lattice

A tree that can be superimposed upon a hexagon lattice is called a hexagon lattice tree. A method for mechanically coding, enumerating and drawing these objects is described, and has been tested for trees with up to ten vertices. For storage and information transmission, the code uses an expanded version of theN-tuple code in which edge vector elements having one of four possible values are inserted. For establishing uniqueness, it is used in combination with a hexagon lattice reference grid whose vertices are numbered sequentially in the tightest possible outward spiral. Published rules for the derivation ofN-tuple codes by hand are commented on, and a small error pointed out.     

A note on some coefficients of the Chebyshev polynomial form of the characteristic polynomial

The characteristic polynomial of a graph, which traditionally is written down in descending powers ofX can also be expressed in the mathematically equivalent form of a linear combination of the characteristic polynomials of linear chains, and often this expression is a simpler one. Investigation of the first few coefficients reveals that in this form the even ones are of smaller magnitude because they are more closely related to the cyclomatic number of the graph. On the other hand, the early odd coefficients are the same or are more complicated in composition.