Wavefront Expansions Research Articles

Propagation of longitudinal waves in non-homogeneous four-parameter viscoelastic rods

AbstractThe purpose of this paper was to study the propagation of longitudinal waves in non-homogeneous four-parameter viscoelastic rods of arbitrary thickness. The rods were initially supposed to be unstressed and at rest. Apart from a sudden rising traction uniformly applied over the boundary of the opening and parallel to the faces of the plates, which is steadily maintained thereafter, the rods are otherwise free from loading. Methods for treating reflection at the free end of the finite rod and reflection and transmission at an interface between two media in the semi-infinite bi-viscoelastic rod are also presented. Asymptotic techniques are used throughout, and formal asymptotic wave-front expansions of the solution functions are obtained.

On the radiation of ultrasound into an isotropic elastic half-space via wavefront expansions of the impulse response

The problem of propagation of pulses in the radiating near zone of a large circular normal transducer directly coupled to a homogeneous and isotropic elastic half-space is re-visited. It is shown that for certain observation angles the impulse response approach is computationally inefficient. A new method based on the so-called wavefront expansions of the impulse response is developed instead. The expansions are obtained by the analytical harmonic synthesis of the high-frequency asymptotics of the transducer field. Unlike these asymptotics the wavefront expansions are expressed in terms of elementary functions only. The direct P, edge P and S waves as well as the transition regions (penumbra and axial region) are described. The uniform asymptotic expansions applicable throughout the radiating near zone are derived as well. The code based on the time convolution of the pressure input function with the wavefront expansions is compared to a direct numerical code. It is thousands of times faster but practically just as accurate except that the phenomena related to the head waves are not described. Formulas pertaining to the far field are also offered.

Wavefront expansions for non-linear axial shear wave propagation

Wavefront expansions are employed to study non-linear axial shear waves propagating from a cylindrical cavity in an unbounded medium. Depending on the nature of the boundary condition, an acceleration front or a shock front propagates from the boundary of the cavity. For an acceleration front the coefficients in the wavefront expansion satisfy a sequence of transport equations which can be solved analytically. Numerical results based on a Padé-extended version of these expansions are presented for this case. For a shock front, a wavefront analysis gives approximate formulas for the wave speed, shock front and intensity of the various field variables at the front. In order to understand the usefulness and limitations of wavefront expansions, all problems analysed using these expansions are also solved numerically by MacCormick's finite difference scheme.

Wavefront expansions for pulse scattering by a surface inhomogeneity

Much recent interest has arisen in the coupled dynamics of fluid-solid interfaces, mainly with regard to the use of the acoustic microscope as a valuable technique for material characterization and the detection of surface, and subsurface, flaws; there is also interest in structural acoustics and geophysical applications. The majority of recent work has dealt with time-harmonic waves; in this paper solutions for transient disturbances are developed and analysed in detail. The resulting fluid and solid responses are found in closed form allowing specific wavefronts to be identified and explicit results are given. Wavefront expansions for the head waves and cylindrical wave in the fluid are found explicitly together with the disturbance generated by the leaky Rayleigh wave; these are the dominant fluid responses detected experimentally and such exact solutions will be of value. The limit of light fluid loading is examined separately; it is shown that the leading-order fluid-solid results are reconstructed by considering simpler problems where the fluid loading is taken to be absent. An unambiguous interpretation of the disturbance in the fluid created by the leaky Rayleigh wave is given in this light fluid-loading limit.

Wavefront expansions and nonlinear hyperbolic waves

Asymptotic wavefront expansions are here employed in the study of nonlinear hyperbolic waves. Numerical results based upon these methods are obtained for a particular case of interest and both these results and those obtained by the method of characteristics are presented graphically. The wavefront-based method is accurate, requires an order of magnitude less computer time, and offers a clearer understanding of the underlying wave process.

Padé-extended wavefront expansions and non-linear dissipative waves

Methods of constructing asymptotic wavefront expansions for non-linear waves in continuous dissipative media are presented. We consider those cases where the dissipation mechanisms are provided through relaxation (rate dependence) and/or stratification (inhomogeneity) of the medium. Apart from the first non-constant one, the coefficients in these expansions are shown to be governed by linear transport equations. Numerical results, based on a Padé-extended version of these expansions, are presented for waves in fluid filled compliant tubes. Comparisons with results derived by the method of characteristics are made and presented graphically.

PADÉ-EXTENDED RAY SERIES EXPANSIONS IN GENERALIZED THERMOELASTICITY

The Lord and Shulman theory of thermoelasticity is used to study the problem of a one-dimensional homogeneous half-space subjected to prescribed values of uniform temperature and strain at its boundary. Asymptotic wavefront expansions are generated by using the ray series method. To obtain numerical results, the region of validity of these asymptotic expansions is extended with the use of Pade approxi-mants. Numerical results obtained for some values of material parameters are displayed graphically. Our results show that the method of Pade-extended ray series expansions provides a powerful tool for obtaining necessary numerical data to test the validity of the Lord and Shulman theory.

Boundary-initiated wave phenomena in thermoelastic materials

The linear theory of Gurtin and Pipkin, and Chen and Gurtin is adopted to study one-dimensional progressive waves generated by thermal and mechanical disturbances applied at the boundary of a circular hole in an unbounded homogeneous thermoelastic medium. A ray-series approach is employed to generate asymptotic wavefront expansions for the field variables. The characteristics of the propagation process are obtained simply and directly. The solution is then specialized to the case where this theory reduces to the linearized theory of Lord and Shulman, and numerical results for various values of material parameters obtained from the ray-series solution in conjunction with the use of Padé approximants are displayed graphically.

On the propagation of transients through thermoviscoelastic media

We examine the propagation of thermal and mechanical transients through general linear thermoviscoelastic media. A linear theory of heat conduction in deformable materials with memory is employed to study the one-dimensional problem of a homogeneous thermoviscoelastic half-space subjected to thermal and mechanical disturbances at its boundary. A ray series approach is used to generate asymptotic wave front expansions for the temperature, strain, and stress response of the medium to the disturbances. General properties of the propagation process are obtained simply and directly. As an example, we specialize the solution to the case of a medium where the conduction of heat obeys Cattaneo’s law and the viscoelastic response is that of a standard linear solid. The propagation of transients through this material is depicted graphically for various values of the thermal and mechanical parameters.

On thermoelastic transients in a general theory of heat conduction with finite wave speeds

The linear Chen-Gurtin-Pipkin theory of heat conduction in a deformable material is employed to study the one dimensional problem of a homogeneous thermoelastic half space subjected to thermal and mechanical disturbances at its boundary. A ray series approach is used to generate asymptotic wavefront expansions for the temperature, strain, and stress response of the medium to the disturbances. General properties of the propagation process are obtained simply and directly. We specialize the solution to the case for which this theory reduces to that of Lord and Shulman and demonstrate that our results for this example agree with asymptotic results obtained previously by other investigators.

Point-source representation for laser-generated ultrasound

The stress-free thermal strain corresponding to the temperature rise induced in a target by an incident laser pulse constitutes a volume source of ultrasound. In the present work, the appropriate point-source representation is derived by starting from a general representation theorem for volume sources. A formal solution for the double (Hankel–Laplace) transform of the displacement potentials is obtained for axially symmetric configurations, with the point-source lying within or on the surface of a half-space or a plate. The Cagniard-de Hoop technique is used to invert these double transforms. For the source on the surface of a half-space, detailed results are presented for the wave-front expansions, the displacement along the axis of symmetry and on the surface, the directivity pattern, and the partition of energy between longitudinal, transverse, and surface waves. For the source within or on the surface of a plate, only the epicentral displacement is considered in detail.

Padé extensions to ray series methods and analysis of transients in inhomogeneous viscoelastic media of hereditary type

In this paper we demonstrate that wavefront expansions for the analysis of transient phenomena are far from adequate when numerical information back of the wavefront is required. However, by employing Padé approximants together with ray series methods, we can obtain directly and greatly extend the range of validity of these expansions. The procedure, which is straightforward and requires very little computing time, is here applied to a non-trivial problem involving impact-generated shear transients in inhomogenoues viscoelastic media whose stress-strain laws are given in integral form. For a special combination of the material parameters an exact solution is recovered and used to check the validity of our approximate Padé-extended wavefront solution. We also compare our results from the extended wavefront solution with numerical solutions obtained using Bellman's approximate inversion scheme for Laplace transforms. A further advantage of our approach is that, unlike transform techniques, it does not depend upon being able to find tabulated special function equations for the transformed dependent variables. All numerical results are presented graphically for ease of comparison.

Wavefront Analysis in the Nonseparable Elastodynamic Quarter-Plane Problems, I—Part 1: The General Method

This work concerns the transient response in elastodynamic quarter-plane problems. In particular, the work focuses on the nonseparable problems, developing and using a new method to solve basic quarter-plane cases involving nonmixed edge conditions. Of initial interest is the classical case in which a uniform step pressure is applied to one edge, along with zero shear stress, while the other edge is traction free. The new method of solution is related to earlier work on nonseparable wavequide (semi-infinite layer) problems in which long time, low-frequency response was the objective. The basic idea in the earlier work was the exploitation of a solution boundedness condition in the form of an infinite set of integral equations. These equations were generated by eliminating the residues at an infinite set of poles stemming from the zeros of the Rayleigh-Lamb frequency equation. These poles were associated with exponentially unbounded contributions. The solution of the integral equations determined the unknown edge conditions in the problem, hence giving the long time solution. In the new method for the quarter-plane problems the analog of the Rayleigh-Lamb frequency equations is the Rayleigh function that yields just three zeros, i.e., say the squares of cR1, cR2, and cR3 speeds for the Rayleigh surface waves. Of these six roots (and corresponding poles) only one is physical, i.e., cR1. The other five are eliminated since they correspond to: (1) exponentially unbounded waves, or (2) waves with speeds cR2 > cd, cR3 > cd, all inadmissible. They analogously (to the layer case) give four integral equations for the edge unknowns, hence the formal solution for the problems. To date the method has been successful in obtaining all the wavefront expansions for the uniform step pressure case mentioned earlier. The dilatational and equivoluminal wavefronts were obtained for the interior and both edges and the Rayleigh wavefronts for both edges. This asymptotic solution has been verified and hence establishes the credibility of the new method.

Ray analysis for shear transients in inhomogeneous viscoelastic media

Formal ray methods are developed for generating asymptotic wavefront expansions for rotary shear stress transients in inhomogeneous viscoelastic media whose behaviour is governed by integral stress-strain laws expressed in terms of either creep or relaxation functions. General results are obtained for arbitrary creep functions which depend on the radial distance from the axis of a cylindrical hole. Specific solutions are then presented for a creep functionJ(r, t) which is an analogue, for inhomogeneous media, of that first introduced by Jeffreys [1]. As a check on our results we show, how, in various limiting cases, they reduce to known exact and asymptotic results obtained previously by other investigators.

A note on the propagation of precursors in poroelastic media

Summary. In this note we discuss the behaviour of Green’s functions for Biot’s equation in the neighbourhood of the wavefronts which provide precursors. Uniform wavefront expansions are obtained which are valid for all possible values of the physical parameters. The results of this note complete the work of Burridge & Vargas.

A mathematical theory of shock-wave formation in arterial blood flow

Theoretical and experimental evidence is available for the occurrence of shock waves in arterial blood flow. A sudden increase of pressure at the entrance of a fluid-filled semi-infinite elastic tube is considered as a model to investigate mathematically the possibility of shock-wave formation. By use of the method of wavefront expansions explicit results are obtained about the circumstances under which shocks may form and about the time and distance at which this may occur.

The hypergeometric ansatz in viscoelasticity

Eason's technique [1] is applied and solutions, in terms of confluent hypergeometric functions, are obtained for transient cylindrical shear wave propagation in inhomogeneous viscoelastic solids. The results contain sufficient arbitrary constants for them to have a wide application to particular impact loading problems. The efficacy of the method is illustrated through the construction of some particularly simple closed-form exact solutions. Asymptotic wavefront expansions for a number of problems are also presented and it is pointed out that the solutions in terms of confluent hypergeometric functions for viscoelastic problems reduce in the appropriate way to solutions in terms Bessel functions of the second kind for the corresponding purely elastic problems.

Wellenausbreitung ausgehend von einem zylindrischen Hohlraum

Here we study impact-initiated disturbances propagating from a cylindrical cavity in orthotropic cylindrically anisotropic inhomogeneous elastic materials. Employing a transform technique we ascertain conditions on the various parameters of the medium which, when satisfied, enable us to express the solutions for impact problems in terms of modified Bessel functions. From these we generate asymptotic wavefront expansions and exact closed-form solutions. A formal technique for directly generating such expansions is also employed to solve a variety of problems. Comparisons of results obtained by the two methods are included and an appendix contains a rigorous verification of the leading term in the formal expansion.

Transient solutions for longitudinally impacted inhomogeneous conical shells

Here we study transient elastic wave propagation in inhomogeneous conical shells. The uniaxial theory is employed and two separate techniques are used for extracting information about the stress field for impact problems. Firstly the formal Karal-Keller method is used enabling us to directly determine asymptotic wavefront expansions for the stress field. A transform technique, based on Eason's [1], is then used to obtain conditions on the physical parameters of the medium which give solutions to the governing equation in terms of Bessel functions and some particular problems are discussed. Previously unknown simple closed-form solutions are obtained. Our results have application to the propagation of waves in filamentary cones of constant wall thickness or in homogeneous cones having an axial temperature gradient.

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.