The cumulative interaction of finite amplitude waves in strings

A “two time scale” asymptotic expansion procedure describing the modulation of a propagating simple wave governed by a system of non-linear partial differential equations is applied to the deflection waves of non-linear elastic strings. Rapid deflection signals propagating into a general slowly varying disturbance are modulated. In addition, they themselves affect the equations for that disturbance. The two effects are separated naturally when, to prevent the cumulative growth inherent in most “high frequency” procedures, an averaging technique is introduced. The interaction of two deflection waves is given as a specific example.     

Waves on vortex cores

Here we discuss the types of waves which can be supported on compact regions of vorticity. This is a subject first studied by Lord Kelvin for waves propagating along the vortex lines of the Rankine and hollow-core vorticity distributions. Kelvin's major interest was in the stability of vortex rings and the numerous resurgencies in interest have usually been driven by practically important phenomena, e.g., the observations of vortex breakdown and waves on tornado and aircraft vortices in the early 1960's and more recently in technologically and geophysically significant flows and on the quantised vortices of super fluid HeII.

The major wave-types of interest are of varicose, helicoidal and fluted form and represent a periodic swelling and contraction, a bending and a “krinkling” of the core, respectively. The first two propagate along and the third around the vortex lines. All have been studied theoretically, experimentally and numerically in the limit of small wave amplitude and their major characteristics are now clear. Of particular interest is the extension of these results to the non-linear regime in which case the two first types are known to exhibit solitary wave or soliton characteristics in certain parameter ranges. It is these non-linear waves which often dominate observations of vortex flows both in nature and in technological applications and which have caused much controversy in the interpretation of results found under complex circumstances of flow and apparatus geometry.     

Weakly nonlocal envelope solitary waves: numerical calculations for the Klein-Gordon (φ) equation

“Weakly nonlocal” solitary waves differ from ordinary solitary waves by possessing small amplitude, oscillatory “wings” that extend indefinitely from the large amplitude “core”. Such generalized solitary waves have been discovered in capillarygravity water waves, particle physics models, and geophysical Rossby waves. In this work, we present explicit calculations of weakly nonlocal envelope solitary waves. Each is a sine wave modulated by a slowly-varying “envelope” that itself propagates at the group velocity. Our example is the cubically nonlinear Klein-Gordon equation, which is a model in particle physics (φ⁴ theory) and in electrical engineering (with a different sign). Both cases have weakly nonlocal“breather” solitons. Via the Lorentz invariance, each breather generates a one-parameter family of nonlocal envelope solitary waves. The φ⁴ breather was described and calculated in earlier work. This generates envelope solitons which have “wings” that are (mostly) proportional to the second harmonic of the sinusoidal factor. In this article, we calculate breathers and envelope solitary waves for the second, electrical engineering case. Since these, unlike the φ⁴ waves, contain only odd harmonics, the envelope solitary waves are nonlocal only via the third harmonic.     

A study of main and secondary resonances in non-linear multi-degree-of-freedom vibrating systems

A uniform study of all types of resonances that can occur in non-linear, dissipative multi-degree-of-freedom systems subject to sinusoidal excitation is presented. The theoretical investigation is based on a harmonic or multi-harmonic solution and the Ritz method. The new approach suggests that non-linear normal mode shape or the so-called “coupled” non-linear mode shapes are those which are retained in resonance conditions, no matter what type of resonancemain, or secondary, periodic or almost-periodic.

By introducing the concept of non-linear normal coordinates the response of an n-degree-of-freedom system is described, to a satisfactory degree of accuracy, by a single coordinate in the case of main or secondary-periodic resonance, or by p coordinates in the case of almost-periodic (combination) resonance with p harmonic components.

Numerical examples indicate good agreement of theoretical and analog computer results and illustrate considerable discrepancies between resonance curves calculated by the commonly used “single linear mode approach” and the suggested “single non-linear mode approach”.     

Non-linear wave modulation in a prestressed viscoelastic thin tube filled with an inviscid fluid

In the present work, the propagation of weakly non-linear waves in a prestressed thin viscoelastic tube filled with an incompressible inviscid fluid is studied. Considering that the arteries are initially subjected to a large static transmural pressure P₀ and an axial stretch λ_z and, in the course of blood flow, a finite time-dependent displacement is added to this initial field, the governing non-linear equation of motion in the radial direction is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear, dispersive and dissipative waves is examined and the evolution equations are obtained. Utilizing the same set of governing equations the amplitude modulation of weakly non-linear and dissipative but strongly dispersive waves is examined. The localized travelling wave solution to these field equations are also given.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

A probabilistic linearization method for non-linear systems subjected to additive and multiplicative excitations

A general method to obtain approximate solutions for the random response of non-linear systems subjected to both additive and multiplicative Gaussian white noises is presented. Starting from the concept of linearization, the proposed method of “Probabilistic Linearization” (PL) is based on the replacement of the Fokker–Planck equation of the original non-linear system with an equivalent one relative to a linear system subjected to additive excitation only. By means of the general scheme of the weighted residuals, the unknown coefficients of the equivalent system are determined. Assuming a Gaussian probability density function of the response process and by choosing the weighting functions in a suitable way, the equivalence of the proposed method, called “Gaussian Probabilistic Linearization” (GPL), with the “Gaussian Stochastic Linearization” (GSL) applied to the coefficients of the Itô differential rule is evidenced. In addition, the generalization of the proposed method, called “Generalized Gaussian Probabilistic Linearization” (GGPL), is presented. Numerical applications show as, varying the choice of the weighting functions, it is possible to obtain different linearizations, with a variable degree of accuracy. For the two examples considered, different suitable combinations of the weighting functions lead to different equivalent linear systems, all characterized by the exact solution in terms of variance.     

Cubically non-linear effects of plane waves in isotropic soft solid materials

In this paper we analyze mathematical properties of an isotropic soft solid model which is characterized by three elastic constants. The model was proposed to interpret measurements of weakly non-linear shear waves in gel-like and tissue-like media. In our analysis we are particularly interested in third order non-linear terms. We present for the first time the full equations of elastodynamics, as well as the equations for plane waves for this model, with cubically non-linear terms. Next, the interaction coefficients for non-linear interactions of three plane waves to produce the fourth wave are explicitly calculated. These coefficients show which of the three waves interact with each other and determine how strong the effect of interaction is on the produced fourth wave. It turns out that these coefficients are expressed in terms of some combinations of three elastic constants. The obtained results can be helpful in experimental determination of elastic constants which describe the model.     

A generalized variational principle for potentializable operators

For non-conservative mechanical systems (non-potential operators), classical energetic variational principles do not hold true. In the present paper, a generalized variational principle, valid for non-linear and non-conservative systems, is deduced by means of “potentializable” operators. A systematic inclusion or boundary conditions in problems dealing with such operators is proposed and examples from continuum mechanics are presented.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

Theory of non-linear wave propagation with application to the interaction and inverse problems

The propagation of non-linear deformation waves in a dissipativc medium is described by a unified asymptotic theory, making use of wave front kinematics and the concepts of progressive waves. The mathematical models are derived from the theories of thermoclasticity or viscoclasticity taking into account the geometric and physical non-linearities and dispersion. On the basis of eikonal equations for the associated linear problem the transport equations of the nth order are obtained. In the multidimensional case the method of matched separation of initial equations is proposed. The interaction problems which occur in head-on collisions and in reflection from boundaries or interfaces are analyzed. Conditions are also studied when the interaction of non-linear waves does not take place. The inverse problem of determining materials properties according to pulse shape changes is discussed.     

Parametric resonance in non-linear elastodynamics

We show that finite amplitude shearing motions superimposed on an unsteady simple extension are admissible in any incompressible isotropic elastic material. We show that the determining equations for these shearing motions admit a general reduction to a system of ordinary differential equations (ODEs) in the remarkable case of generalized circularly polarized transverse waves. When these waves are standing and the underlying unsteady simple extension is composed of a harmonic perturbation of a static stretch it is possible to reduce the determining ODEs to linear or non-linear Mathieu equations. We use this property for a detailed study of the phenomenon of parametric resonance in non-linear elastodynamics.     

A geometric approach for establishing dynamic buckling loads of autonomous potential N-degree-of-freedom systems

Non-linear dynamic buckling of autonomous non-dissipative N-degree-of-freedom systems whose static instability is governed either by a limit point or by an unstable symmetric bifurcation is thoroughly discussed using energy and geometric considerations. Characteristic distances associated with the geometry of the zero level total potential energy “hypersurface” in connection with total energy-balance equation lead to dynamic (global) instability criteria. These criteria allow the determination of “exact” dynamic buckling loads without solving the non-linear initial-value problem. The reliability and efficiency of the proposed geometric approach is demonstrated via several dynamic buckling analyses of 3-degree-of-freedom systems which subsequently are compared with corresponding numerical analyses based on the Verner–Runge–Kutta scheme.     

Multiple solutions generated by statistical linearization and their physical significance

Three cases are examined where the statistical linearization (SL) procedure can yield multiple solutions for the first and second moments of the response. The first is an oscillator with a hardening non-linear stiffness excited by a narrow-band random excitation, the second is an oscillator with two potential wells excited by wide-band random excitation, and the third is an oscillator where the non-linear features present in the first two problems are combined. The results of an SL analysis are quantitatively compared with the behaviour of digitally simulated sample functions of the displacement response. In all cases a definite correspondence is found between the occurrence of multiple solutions generated by the SL method and the appearance of noticeable jumps in sample functions of the response. In some cases a quantitative agreement exists between the first and second moment values of the multiple solutions and the magnitude of “local” moments of the response.     

Numerical simulation of the onset of slug initiation in laminar horizontal channel flow

Results are presented for the initiation of slug-type structures from stratified 2D, two-layer pressure-driven channel flow. Good agreement is obtained with an Orr–Sommerfeld-type stability analysis for the growth rate and wave speed of very small disturbances. The numerical results elucidate the non-linear evolution of the interface shape once small disturbances have grown substantially. It is shown that relatively short waves (which are the most unstable according to linear theory) saturate when the length of the periodic domain is equally short. In longer domains, coalescence of short waves of small-amplitude is shown to lead to large-amplitude long waves, which subsequently exhibit a tendency towards slug formation. The non-uniform distribution of the interfacial shear stress is shown to be a significant mechanism for wave growth in the non-linear regime.     

Diagnostics of the medium structure by long wave of finite amplitude

The averaged systems of hydrodynamic equations for a structured medium in the Lagrangian and the Eulerian coordinates are discussed. In the general case, the equations cannot be reduced to the average hydrodynamic terms. Under propagation of long waves in media with structure, the non-linear effects appear and they are analyzed in the framework of the asymptotic averaged model. The heterogeneity in a medium structure always increases the non-linear effects for the long-wave perturbations. A new method for diagnostics of the properties of medium components by long non-linear waves is suggested (inverse problem). The mass contents of components in the media can be determined by this diagnostic method.     

Non-linear longitudinal waves in bars

The finite amplitude longitudinal waves along a uniform bar are examined by using the method or multiple scales. The evolution of the complex amplitude of a quasi-monochromatic progressive wave is shown to be governed by a non-linear Schrödinger equation. The analysis reveals that the constant amplitude progressive waves are stable against modulation.