ANALYSE OF PULSE WAVE PROPAGATION IN ARTERIES

Based upon the blood vessel of being regarded as the elasticity tube, and that the tissue restricts the blood vessel wall, the rule of pulse wave propagation in blood vessel was studied. The viscosity of blood, the elastic modulus of blood vessel, the radius of tube that influenced the pulse wave propagation were analyzed. Comparing the result that considered the viscosity of blood with another result that did not consider the viscosity of blood, we finally discover that the viscosity of blood that influences the pulse wave propagation can not be neglected; and with the accretion of the elastic modulus the speed of propagation augments and the press value of blood stream heightens; when diameter of blood vessel reduces, the press of blood stream also heightens and the speed of pulse wave also augments. These results will contribute to making use of the information of pulse wave to analyse and auxiliarily diagnose some causes of human disease.     

The effects of blood viscoelasticity on the pulse wave in arteries

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A computer model of pulse wave and input impedance in human arteries

To predict the propagation of pressure and flow pulses in arterial system and the variation of vascular input impedance, a branched and tapered tube model is studied through one-dimensional transient flow analysis. Coupling the continuity and momentum equations yields a group of quasilinear hyperbolic partial differential equations which can be solved numerically by using the method of characteristics. Several boundary conditions of the arterial system are also simplified suitably. The propagation of the pulses of the arterial system and the vascular input impedance is calculated on computer by using the dimensions and the physiological data of the arterial system. The results point out that the pressure and flow pulses of the arterial system and the vascular input impedance produced by this theoretical model is consistent quite well with the experimental results published.     

Some properties of the solutions of wave equations

Some properties of solutions of initial value problems and mixed initial-boundary value problems of a class of wave equations are discussed. Wave modes are defined and it is shown that for the given class of wave equations there is a one to one correspondence with the roots _i (k) or k _j () of the dispersion relation W(, k)=0. It is shown that solutions of initial value problems cannot consist of single wave modes if the initial values belong to W ₂ ¹ (–, ); generally such solutions must contain all possible modes. Similar results hold for solutions of mixed initial-boundary value problems. It is found that such solutions are stable, even if some of the singularities of the functions k _j () lie in the upper half of the plane. The implications of this result for the Kramers-Kronig relations are discussed.     

Stability of regular shock wave reflection

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 26–33, March–April, 1989.     

Shock wave reflection off coupled surfaces

It has been shown that when a plane shock wave is reflected off a surface consisting of a 75-mm radius circular arc followed by a plane section inclined at 45°, it takes some time for the interaction to reach a pseudosteady reflection configuration. The current study extends this work at a constant Mach number of 1.346, with three compound walls, consisting of leading circular sections of 30, 50 and 75 mm radius, joined to a plane wall section. Testing was done at various wall angles for each of the test pieces. The reflected wave angle was measured and was found to increase along the plane wall section until it reached an asymptotic value, at which time pseudosteady flow was established. The asymptotic values are consistent with reflection off plane wedges and are independent of the leading radius. For lower wall angles which lead to Mach reflection the length required to reach pseudosteady flow increases as the wall angle increases to the pseudosteady transition angle. The reverse occurs when the final pseudosteady reflection is regular, in that as the wall angle increases the distance travelled to reach pseudosteady flow conditions decreases. Additional tests were conducted on a specimen consisting of a plane section at 60° wall angle with 30-mm radius circular arc sections at either end. It is demonstrated how the information from the two slope changes influences the shape of the reflected shock. The trajectories of two perturbations on the reflected shock arising from the joints between the circular sections and the plane wall show that the reflected wave remains linear between these two points, as it received no knowledge from either circular section until the perturbations from the upper and lower joints cross.     

Classification of pseudo-steady shock wave reflection types

Classification of various types of the reflections of a shock wave over a straight wedge is proposed. The idea about entire reflection phenomenon as a result of interaction of two processes—the shock wave reflection process and the flow deflection process—serves as a basis for the classification. To recognize the types of reflection, changes in the shapes of the reflected wave, Mach stem, and contact surface (slipstream) are taken into account. The boundaries and domains of existence for various types of reflection configuration are reported. New terms for some types of reflection are proposed. The domain of irregular non-Mach reflection is analyzed carefully. It is shown that the von Neumann reflection pattern can result from not only the weak shock reflection but also the strong shock reflection over thin wedges. Shadowgraph images of different types of irregular reflection that illustrate the suggested classification are presented. Emphasis is placed on near-wall behavior of the contact discontinuity in the Mach configuration.     

The reflection of an ionized shock wave

In a previous paper, we studied the thermodynamic and kinetic theory for an ionized gas, in one space dimension; in this paper, we provide an application of those results to the reflection of a shock wave in an electromagnetic shock tube. Under some reasonable limitations, which fully agree with experimental data, we prove that both the incident and the reflected shock waves satisfy the Lax entropy conditions; this result holds even outside genuinely nonlinear regions, which are present in the model. We show that the temperature increases in a significant way behind the incident shock front but the degree of ionization does not undergo a similar growth. On the contrary, the degree of ionization increases substantially behind the reflected shock front. We explain these phenomena by means of the concavity of the Hugoniot loci. Therefore, our results not only fit perfectly but explain what was remarked in experiments.     

Estimating pressure in nonsteady reflection of a shock wave

A comparative analysis of the results of physical [1–4] and mathematical experiments [5–8] is used to elucidate the mechanism of additional pressure lift at a Mach wavefront. The possibility and range of application of simplified flow models for the estimation of the pressure at certain characteristic points of the reflection surface (for example, a prism, a cylinder, or a sphere) was investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 100–106, March–April, 1977.     

A note on weak shock wave reflection

This work discusses the possibility of reconstructing, both numerically and experimentally, the steady state flow field and shock reflection pattern close to the triple point of von Neumann, Guderley and Vasilev reflections. First, a criterion for the orientation of shock wave fronts, even in the case of subcritical/subsonic flow downstream the front, is introduced and formalized. Then, a technique for obtaining a close view of the above reflection patterns centered about the triple point is described and a numerical example, within the framework of shallow water flow, is presented and discussed.     

A weak shock wave reflection over wedges

When a weak shock wave reflects from wedges its reflection pattern does not appear to be a simple Mach reflection. This reflection pattern is known to be von Neumann Mach reflection in which a Mach stem can not necessarily be straight. In this paper the local change of the Mach stem curvature was experimentally and numerically investigated. A distinct triple point, at which the curvature becomes infinite as appears in a simple Mach reflection, was not observed but the Mach stem curvature became a maximum between foot of the Mach stem and a point, P₁, at which an incident shock met with a reflected shock. Maximum curvature point P₂ and P₁ do not coincide for small wedge angles and tend to merge over a certain wedge angle. Experimental results agreed with numerical results. The trajectory angle of P₂ was found to be expressed well by Whitham's shock-shock angle.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1.0 and the AMS fonts, developed by the American Mathematical Society.     

Stability and analogy of shock wave reflection in steady flow

Numerical simulations have been performed to investigate the stability of shock wave reflection in supersonic steady flow. Wall deflection control has been applied just downstream of the reflection point in the regular reflection configuration. The results provide the magnitude of the disturbance required to cause transition from one configuration to the other throughout the range of incident shock angle. An argument focusing on the subsonic region generated behind the Mach stem in the Mach reflection configuration explains the mechanism of the transition. Numerical results show that both regular and Mach reflections are possible in the dual-solution domain, and also indicate the presence of the hysteresis effect. The transition processes and the stability of the possible states are shown to be described consistently by an analogy based on the potential energy of a particle on a surface. The necessity of more sophisticated experimental investigations is emphasized to verify the argument about the stability of shock reflections and proposed analogy. Received 17 March 1997 / Accepted 26 February 1998     

The total reflection of a sound pulse of arbitrary form

The problem of a sound pulse of arbitrary form incident on a half space with an angle of incidence greater than the critical angle is discussed. Formulae are obtained for the transmitted and reflected pressure fields. An expression is obtained for the energy flux across the interface, and it is shown that the net energy flow per unit area over all time is zero.     

Normal shock wave reflection on deformable solid walls

Summary A theoretical-experimental study has been performed on the interaction between normal shock waves and deformable solid materials, with particular reference to the characteristics of the reflected shock wave.The present study is of importance in all those cases in which a shock wave impings on solid walls (i. e. explosions, sonic booms etc.).The first part presents a theoretical analysis which gives the solution for the shock wave intensity reflected into the gas and for the transmitted shock wave propagating into the elastic solid material for both the cases of linear (Hooke type) and non linear solids. For the first kind of materials a closed form solution is found. For the non-linear materials the problem is solved through a numerical analysis.The experiments concern with two kinds of expanded foams (closed cell and open cell foams). These materials have been selected because of the good deformation and density characteristics which both are necessary to evidentiate the shock wave interaction phenomena.The tests have been performed putting a cylindrical model of the material under study at the closed end of a shock tube and creating shock waves of variable strength which impinge on the material. Slightly upstream of the free end wall of the model, the pressures history is recorded by means of a pressure pick up and of an oscilloscope. The range of the shock wave Mach number has been1,2M_s2,2 with initial pressures ranging between50p ₁ 760 mmHg.The theoretical analysis seems to be in a very good agreement with the experiments performed, especially when due account is given to the gap existing between shock tube and model walls.

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Sommario È stata eseguita una indagine teorico-sperimentale sulla interazione tra onde d'urto normali e materiali solidi de formabili con particolare riferimento alla determinazione delle caratteristiche dell'onda riflessa.Tale studio é di importanza per tutti i casi in cui si verifica la interazione tra onde d'urto e superfici solide (esplosioni, sonic booms, etc.).L'analisi teorica porta alla determinazione dell'intensità dell'onda d'urto riflessa nel gas e di quella trasmessa nell'interno dei solidi elastici, lineari (tipo Hooke) e non. Per i primi si arriva ad una soluzione in forma chiusa per le caratteristiche della riflessione. Per i materiali solidi che esibiscono una elasticità non lineare entro il campo di tensioni-deformazioni determinate dalle sovrapressioni causate dall'onda d'urto, il problema é risolto a mezzo di un'analisi numerica.Sperimentalmente sono state esaminate due categorie di materiali: resine espanse a cellula aperta (tipo gomma piuma) e a cellula chiusa (tipo isolante termico). Tali materiali sono stati selezionati essenzialmente per le loro buone caratteristiche di deformabilità e di densità che hanno entrambe consentito di evidenziare molto bene il fenomeno.Le prove sono state eseguite ponendo un modello cilindrico del materiale in esame all'estremità chiusa di un tubo d'urto e facendo incidere su di esso onde d'urto di diversa intensità. Poco a monte della parete del modello esposta all'urto viene rilevata la storia delle pressioni a mezzo di un piezometro e di un oscillografo. Il campo del numero di Mach delle onde d'urto indagato é stato1,2M_s2,2 con pressioni iniziali50p ₁ 760 mmHg.L'analisi teorica é in ottimo accordo con gli esperimenti specie se si tiene conto della correzione dovuta al gioco esistente tra modello e pareti del tubo d'urto.

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This work has been partly supported by the C.N.R.     

The two-dimensional transmission and reflection of a solitary wave

In the present paper, limited to the discussion of weak non-linear shallow water waves, the transmission and reflection of a planar soliton on a two-dimensional structure are considered. The whole flow field is divided mainly into two subfields. One is in the vicinity of the structure, called the inner field; the other is far from the structure, called the outer field. In the outer field, according to its definition, the influence of the structure on the flow is negligible; to the order O(α, β) the governing equation for the flow is replaced by the Boussinesq equation. In the inner field the effect of the structure on the flow is significant, so the full Laplace equation is adopted as the governing equation for the flow field. Then the matched asymptotic expansion method is employed to connect smoothly the inner and outer solutions. Owing to the irregularity of the bottom of the structure, the boundary element method is incorporated. As an example, the case in which the incoming wave is a solitary wave is calculated and the time histories of transmitted and reflected waves are plotted.