The reflection coefficients of extensional guided modes from notches of different axial, circumferential and through-thickness extent in pipes of different diameters have been studied using finite element analysis. A selection of the predictions has also been validated by experiments. For part-thickness notches of a given circumferential extent and minimal axial extent, the reflection coefficient increases monotonically with depth at all frequencies, and increases with frequency at a given depth. When the wavelength is long compared to the pipe wall thickness, the reflection coefficient from part-thickness notches of a given circumferential extent is a strong function of the defect axial extent, the reflection being a maximum at an axial extent of about 25 per cent of the wavelength and a minimum at 0 and 50 per cent. The reflection coefficient is a linear function of the defect circumferential extent at higher frequencies (with frequency-diameter products greater than about 3000 kHz mm) where a ray theory analysis explains the behaviour, while at low frequencies the reflection coefficient at a given circumferential extent is reduced. In the high-frequency regime, the axial extent of a through-thickness defect has little influence on the reflection coefficient, while it is important at lower frequencies. Three-dimensional, finite element predictions in the high-frequency regime have shown that the reflection coefficient from a part-thickness, part-circumferential defect can be predicted by multiplying the reflection coefficient for an axisymmetric defect of the same depth and axial extent by that for a through-thickness defect of the same circumferential extent.
Well Posedness and Finite Element Approximability of Three-Dimensional Time-Harmonic Electromagnetic Problems Involving Rotating Axisymmetric Objects