A study of the interaction of the S0 Lamb wave with a circular through-thickness hole in a plate is presented. The study is limited to the nondispersive frequency range of this wave, in which the distributions of stress and displacement are simple. This allows a Finite Element analysis to be undertaken using a two-dimensional membrane discretization. Predictions of the direct reflection of the S0 mode and the lateral scattering of the SH0 mode are made for a range of diameters of the hole. At the same time, an analytical solution based on modal superposition is developed, and this is also used to predict the reflection and scattering coefficients. Both sets of predictions are validated by experimental measurements. It is found that the trends of the reflection coefficients for different hole diameters, frequencies and distances from the hole satisfy a simple normalization. On a detailed scale, the functions exhibit undulations which are shown to result from the interference of the direct reflection with secondary reflections which arrive slightly later.

The mathematical modeling of a typical situation in ultrasonic nondestructive testing for defects is considered. The first objective is the modeling of a reasonably general type of ultrasonic probe. This is performed by prescribing the traction vector on the surface of an elastic half-space. The effective probe area may be rectangular or elliptic and the traction may or may not include the tangential part (glued or fluid-coupled probe, respectively). The probe can be of P, SV, or SH type and of any angle. The traction is either constant across the probe (piston-type source) or it may taper off toward the edges. Numerical results for some representative cases are given showing snapshots of the field beneath the probe. The second objective of the paper is to include the presented probe model into a complete model of the ultrasonic testing situation. To this end the probe field is, via a series of transformations, expressed in spherical vector waves centered at the defect. The influence of the defect is given by its transition matrix. To model the electric signal obtained from the receiving probe, a reciprocity argument is used, giving this signal essentially as a product of the spherical expansion coefficients of the transmitting probe, the transition matrix of the defect, and the spherical expansion coefficients of the receiving probe. For a defect that is a penny-shaped crack or a spherical cavity some numerical examples are given showing the received signal as a function of position on the scanning surface.

The derivation of one‐dimensional wave equations for axially symmetric waves in elastic rods is discussed. By series expansions in the radial coordinate a hierarchy of wave equations is derived. As the lowest reasonable approximation the usual simple wave equation for the rod is recovered. At the next level a fourth order wave equation is obtained. The dispersion relation and the displacements for these approximations and for Love's equation are compared with the lowest branch of the exact Pochhammer‐Chree equation. An excitation problem with a shear force is also solved and compared among the theories.

In-plane elastic wave propagation in the presence of a damaged interface is investigated. The damage is modeled as a distribution of small cracks and this is transformed into a spring boundary condition. First the scattering by a single interface crack is determined explicitly in the low frequency limit for the case of a plane wave normally incident to the interface. The transmission at an interface with a random distribution of small cracks is then determined and is compared to periodically distributed cracks. The cracked interface is then described by a distributed spring boundary condition. As an illustration the dispersion relation of the first modes in a thick plate with a damaged interface in the middle is given.

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