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Double-transform methods are used to find analytic and semi-analytic solutions to time-dependent end-condition problems involving propagation and reflection of elastic waves in a semi-infinite elastic bar. A straight, circular, radially symmetric, linearly elastic bar with a stress-free lateral surface is considered. Solutions to mixed-end condition problems and pure-end -condition problems are given. (Here, a mixed-end condition is described with one component of stress and one component of displacement specified at the end of the bar. Alternatively, a pure end condition is described with two components of stress specified at the end of the bar.) The residue theory is used to invert the transforms in Z, and dispersion curves are determined.
In one application, the response of the bar to a sudden application of pressure at the end is determined. Saddle-point methods are used to determine the time-dependent response of the bar far from the end. The saddle-point calculations reveal precursor waves proceeding the arrival of the head of the pulse, the behavior of the head of the pulse (using extended third order saddle point methods in the vicinity of a pole), oscillations following the head of the pulse, the arrival of surface waves, and the arrival of higher-mode contributions.
In another application, the same double-transform method is used to calculate the reflection of a continuous train of harmonic waves off the free end of a semi-infinite bar. An end-resonance effect was found, where, within a narrow frequency range, large oscillations were shown to exist at the reflecting end. These end-resonance effects are caused by the excitation of higher-order, non-propagating modes near the end.
Agreement is shown between the calculations and experimental data. Applications of the investigation of elastic waves include non-destructive testing of materials, characterization of elastic materials, and the modeling elastic buffer rods attached to transducers that are used in harsh environments.
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