The subject of waves is central to the fields of hydrodynamics and solid mechanics. It has a history stretching back many centuries, and attempts to observe, quantify and describe mathematically a wide range of wave motions continue to occupy researchers and engineering practitioners. The use of wave mechanics techniques to describe the behaviour of engineering systems grows as applications previously unappreciated or undiscovered are brought to our attention through engineering needs and simple curiosity. This issue of Engineering and Computational Mechanics contains two contributions that treat wave problems directly, together with another that considers nonlinear and periodic disturbance fields relating conceptually to the former group. The issue is completed by a paper from the field of materials (specifically, concrete technology) in which a constitutive model for the fracture of concrete is presented. A briefing article from the field of aerodynamics, in which the evolution of flow generated by the collision of vortex pairs with the ground is described, completes the contributions. (Readers are reminded that ‘briefing articles' provide a mechanism for rapid publication of short, topical updates that could not otherwise be published as papers, usually because they present preliminary or incomplete work that is nevertheless regarded as being useful to the readership of the journal or because the material in its present form has a limited shelf life).
The paper by Sobey1 examines mathematically the nonlinear kinematics of standing water waves. In contrast to the case of progressive waves (where the relevant theory is relatively well established), important aspects of standing waves still require resolution. The paper has as its starting point the clarification of the questionable predictive capabilities of existing analytical models. A Stokes-type analytical treatment (in which surface displacement and velocity potential are represented as perturbation expansions in a small non-dimensional parameter related to the wave height) is then developed, with solutions being shown to be correct to 5th order. Comparisons with linear solutions show that the nonlinear effects are manifested primarily in flattening of the troughs of the waves and enhanced elevations of the crests, together with periodic horizontal displacement of the nonlinear zero-crossing position. Limits on the validity of the solution are discussed in terms of the dimensionless depth and wave heights for which the expansion parameter can still be regarded as small.
Seismic excitation of structures is studied in the paper by Nielsen2, from the aspect of including Rayleigh damping (namely damping that that is proportional to a linear combination of the stiffness and mass) in the dynamic analysis of structures. The focus of the investigation is the validity of finite-element analyses of seismically excited structures when an extra term associated with Rayleigh damping is not included in the equations of motion when these are written as total displacements. Nielsen's analysis indicates that the relative error associated with the omission of this term is within the small limits required by engineering accuracy unless the system is subject to narrow-band excitation at frequencies well below its natural frequency. Such findings may have important consequences for the performance of both finite difference and finite element codes used in such structural analysis problems.
Until relatively recently, the subject of internal waves has been the preserve of atmospheric scientists and deep sea oceanographers. However, the exposure of offshore exploration and production platforms and associated infrastructure and storage deployments to large amplitude, nonlinear internal waves travelling on density interfaces has stimulated great interest in the engineering community, particularly for cases in which the amplitude of the internal wave is comparable with the water depth within which it propagates. The refraction of interfacial (internal) waves by encounters with irregular bottom boundary topography is the subject of the theoretical treatment presented in this issue by Zhu and Noor-Harun3. The authors restrict attention to stably stratified, two-layer fluids and study small amplitude, long waves propagating over a circular hump. They employ the so-called mild-slope method to investigate the refraction properties of the encounter, in terms of the relative amplitudes of the incident and disturbance field waves and consider the individual effects on the relative amplitude of varying the density difference between the layers, the ratio of the layer thicknesses and the dimensions of the bottom topography anomaly.
The paper by Grassl4 treats mathematically the problem of fracture in concrete. Two approaches have been used to obtain mesh-independent load–displacement curves applied to a uni-axial bar subjected to tension and a three-point bending test. The first relies on an adjustment of the softening modulus with respect to the finite element size, while the second approach is based on non-local averaging of damage history variables. Using this methodology, the analysis indicates that the non-local damage–plasticity model exhibits a strong boundary effect, indicating a need for more development of boundary operators that overcome this influence of boundaries on non-local averaging.
The motion of vortex pairs and the modification to their structure by collisions with solid bounding surfaces are of fundamental importance in aerodynamics, but, in view of the importance of aspects of vortex dynamics as so-called building blocks in our understanding of turbulent motion, the subject has wide relevance in fluid mechanics. The briefing article by Miller et al 5 provides laboratory evidence to show the distortion of the vortex pair by the boundary and the generation of secondary vorticity from the boundary layer activated by the approaching vortex pair. The appearance of flow features that manifest the occurrence of instability within the interaction region supports the need for further work on this problem.
