Wave dissipation in flexible tubes in the time domain: in vitro model of arterial waves

Earlier work of wave dissipation in flexible tubes and arteries has been carried out predominantly in the frequency domain and most of the studies used the measured pressure waveform for presenting the results. In this work we investigate the pattern of wave dissipation in the time domain using the separated forward and backward travelling waves in flexible tubes. We tested four sizes of latex tubes of 2m in length each, where a single semi-sinusoidal in shape, pressure wave, was produced at the inlet of each tube. Simultaneous measurements of pressure and flow waveforms were recorded every 5cm along the tubes and wave speed was determined using the pressure-velocity loop method (PU-loop). The measured data and wave speed were used to separate the pressure waveform and wave intensity, into their forward and backward directions, using wave intensity analysis (WIA). Also, the energy carried by the wave was calculated by integrating the relevant area under the wave intensity curve. The peak of the measured pressure waveform increased downstream, however, the peak of the separated forward pressure waveform decreased exponentially along the tube. Wave intensity and energy also dissipated exponentially along the travelling distance. The peaks of the separated pressure and wave intensity decreased in the forward in a similar exponential way to that in the backward direction in all four tube sizes. Also, the smaller the size of the tube the greater wave dissipation it caused. We conclude that wave separation is useful in studying wave dissipation in elastic tubes, and WIA provides a convenient method for determining the dissipation of the energy carried by the wave along the travelled distance. The separated pressure waveform, wave intensity and wave energy dissipate exponentially with the travelling distance, and wave dissipation varies conversely with the diameter of elastic tubes.     

Experimental validation of non-invasive and fluid density independent methods for the determination of local wave speed and arrival time of reflected wave

The relationship between the vessel diameter (D) and fluid velocity (U) in arteries and flexible tubes has been recently characterized as linear in the absence of wave reflections. This relationship allowed for determining local wave speed (C(DU)) using the lnDU-loop method. Using C(DU), it was possible to separate U and D waveforms into their forward and backward components. It was also possible to calculate wave intensity (dI(DU)), using D and U, from which the arrival time of reflected wave (Trw(DU)) could be determined. These techniques are fluid density independent and require only non-invasive measurements of D and U. In this work we experimentally validate the relative accuracy of these new techniques in vitro, by comparing their results of C(DU) and Trw(DU) to those determined by the established techniques, PU-loop and wave intensity analysis, C and Trw, respectively. We generated a single semi-sinusoidal wave in long flexible tubes, and simultaneously measured pressure (P), D, and U at the same site. Sequentially in time, we repeated this experiment at three sites along each of the flexible tubes, which were made of different materials and sizes, and three fluids of different densities. C(DU) compared well with that C and likewise Trw(DU) was very similar to Trw. Varying fluid density did not appreciably change the difference between the results of the two techniques. We conclude that the new techniques for determining C(DU) and Trw(DU), although independent of density, provide relatively accurate estimates of wave speed and arrival times of reflected waves in vitro. The new techniques require only non-invasive measurements of D and U, and further in vivo validation is required to establish its advantage in the clinical setting.     

Resolving the time lag between pressure and flow for the determination of local wave speed in elastic tubes and arteries

It is well established that wave speed can be determined using the initial linear part of the pressure–velocity loop (PU-loop). However, the frequency response of most flow measuring devices is usually slower than that of solid-state pressure transducers; making flow waveforms lagging in time behind pressure waveforms. If this lag, which is traditionally determined by eye, is not corrected prior to the analysis, the PU-loop method may provide inaccurate wave speeds. The main aim of this work is therefore to introduce an objective technique to establish the value of this lag.The new technique relies on the linearity between pressure and velocity in the absence of reflections, and determines the highest correlation factor between pressure and velocity in the range of minimum pressure to maximum velocity. We shifted the flow waveform backwards in time steps equal to the sampling interval, and the time shift associated with the highest correlation indicates the correct time lag of the flow waveform. We first tested the new technique in vitro using a uniform latex tube and compared the results to those established using the traditional by eye method, whilst varying the filter setting of the flowmeter. Then we applied the new technique to pressure and flow measured in the ascending aorta of anaesthetised open-chested dogs.We found the time lag between pressure and velocity calculated by the new technique in good agreement with that determined by eye in vitro and that increasing the filtering power generated greater delay between the measured pressure and flow. The results obtained in vivo using the new technique were also in good agreement with those determined by eye. We therefore conclude that the new technique provides a convenient and objective way of correcting the lag and can reliably align pressure and flow.     

Wave intensity in the ascending aorta: effects of arterial occlusion

We examine the effects of arterial occlusion on the pressure, velocity and the reflected waves in the ascending aorta using wave intensity analysis. In 11 anaesthetised, open-chested dogs, snares were used to produce total arterial occlusion at 4 sites: the upper descending aorta at the level of the aortic valve (thoracic); the lower thoracic aorta at the level of the diaphragm (diaphragm); the abdominal aorta between the renal arteries (abdominal) and the left iliac artery, 2 cm downstream from the aorta iliac bifurcation (iliac). Pressure and flow in the ascending aorta were measured, and data were collected before and during the occlusion. During thoracic and diaphragm occlusions a significant increase in mean aortic pressure (46% and 23%) and in wave speed (25% and 10%) was observed, while mean flow rate decreased significantly (23% and 17%). Also, the reflected compression wave arrived significantly earlier (45% and 15%) and its peak intensity was significantly greater (257% and 125%), all compared with control. Aortic occlusion distal to the renal arteries, however, caused an indiscernible change in the pressure and velocity waveforms, and in the intensities and timing of the waves in the forward and backward directions. The measured pressure and velocity waveforms are the result of the interaction between the heart and the arterial system. The separated pressure, velocity and wave intensity are required to provide information about arterial hemodynamic such as the timing and magnitude of the forward and backward waves. The net wave intensity is simpler to calculate but provides information only about the predominant direction of the waves and can be misleading when forward and backward waves of comparable magnitudes are present simultaneously.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Determination of wave speed and wave separation in the arteries using diameter and velocity

The determination of arterial wave speed and the separation of the forward and backward waves have been established using simultaneous measurements of pressure (P) and velocity (U). In this work, we present a novel algorithm for the determination of local wave speed and the separation of waves using the simultaneous measurements of diameter (D) and U. The theoretical basis of this work is the solution of the 1D equations of flow in elastic tubes. A relationship between D and U is derived, from which, local wave speed can be determined; C=±0.5(dU_±/d ln D_±). When only unidirectional waves are present, this relationship describes a linear relationship between ln D and U. Therefore, constructing a ln DU-loop should result in a straight line in the early part of the cycle when it is most probable that waves are running in the forward direction. Using this knowledge of wave speed, it is also possible to derive a set of equations to separate the forward and backward waves from the measured D and U waveforms. Once the forward and backward waveforms of D and U are established, we can calculate the energy carried by the forward and backward waves, in a similar way to that of wave intensity analysis. In this paper, we test the new algorithm in vitro and present results from data measured in the carotid artery of human and the ascending aorta of canine. We conclude that the new technique can be reproduced in vitro, and in different vessels of different species, in vivo. The new algorithm is easy to use to determine wave speed and separate D and U waveforms into their forward and backward directions. Using this technique has the merits of utilising noninvasive measurements, which would be useful in the clinical setting.     

Wave intensity analysis of left ventricular filling

Wave intensity analysis (WIA) is a powerful technique to study pressure and flow velocity waves in the time domain in vascular networks. The method is based on the analysis of energy transported by the wave through computation of the wave intensity dI = dPdU, where dP and dU denote pressure and flow velocity changes per time interval, respectively. In this study we propose an analytical modification to the WIA so that it can be used to study waves in conditions of time varying elastic properties, such as the left ventricle (LV) during diastole. The approach is first analytically elaborated for a one-dimensional elastic tube-model of the left ventricle with a time-dependent pressure-area relationship. Data obtained with a validated quasi-three dimensional axi-symmetrical model of the left ventricle are employed to demonstrate this new approach. Along the base-apex axis close to the base wave intensity curves are obtained, both using the standard method and the newly proposed modified method. The main difference between the standard and modified wave intensity pattern occurs immediately after the opening of the mitral valve. Where the standard WIA shows a backward expansion wave, the modified analysis shows a forward compression wave. The proposed modification needs to be taken into account when studying left ventricular relaxation, as it affects the wave type.     

Determination of wave speed and wave separation in the arteries

Considering waves in the arteries as infinitesimal wave fronts rather than sinusoidal wavetrains, the change in pressure across the wave front, dP, is related to the change in velocity, dU, that it induces by the "water hammer" equation, dP=+/-rhocdU, where rho is the density of blood and c is the local wave speed. When only unidirectional waves are present, this relationship corresponds to a straight line when P is plotted against U with slope rhoc. When both forward and backward waves are present, the PU-loop is no longer linear. Measurements in latex tubes and systemic and pulmonary arteries exhibit a linear range during early systole and this provides a way of determining the local wave speed from the slope of the linear portion of the loop. Once the wave speed is known, it is also possible to separate the measured P and U into their forward and backward components. In cases where reflected waves are prominent, this separation of waves can help clarify the pattern of waves in the arteries throughout the cardiac cycle.     

Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory

Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes' cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the pressure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.     

Differences in cardiac microcirculatory wave patterns between the proximal left mainstem and proximal right coronary artery

Despite having almost identical origins and similar perfusion pressures, the flow-velocity waveforms in the left and right coronary arteries are strikingly different. We hypothesized that pressure differences originating from the distal (microcirculatory) bed would account for the differences in the flow-velocity waveform. We used wave intensity analysis to separate and quantify proximal- and distal-originating pressures to study the differences in velocity waveforms. In 20 subjects with unobstructed coronary arteries, sensor-tipped intra-arterial wires were used to measure simultaneous pressure and Doppler velocity in the proximal left main stem (LMS) and proximal right coronary artery (RCA). Proximal- and distal-originating waves were separated using wave intensity analysis, and differences in waves were examined in relation to structural and anatomic differences between the two arteries. Diastolic flow velocity was lower in the RCA than in the LMS (35.1 +/- 21.4 vs. 56.4 +/- 32.5 cm/s, P < 0.002), and, consequently, the diastolic-to-systolic ratio of peak flow velocity in the RCA was significantly less than in the LMS (1.00 +/- 0.32 vs. 1.79 +/- 0.48, P < 0.001). This was due to a lower distal-originating suction wave (8.2 +/- 6.6 x 10(3) vs. 16.0 +/- 12.2 x 10(3) W.m(-2).s(-1), P < 0.01). The suction wave in the LMS correlated positively with left ventricular pressure (r = 0.6, P < 0.01) and in the RCA with estimated right ventricular systolic pressure (r = 0.7, P = 0.05) but not with the respective diameter in these arteries. In contrast to the LMS, where coronary flow velocity was predominantly diastolic, in the proximal RCA coronary flow velocity was similar in systole and diastole. This difference was due to a smaller distal-originating suction wave in the RCA, which can be explained by differences in elastance and pressure generated between right and left ventricles.     

Numerical assessment of time-domain methods for the estimation of local arterial pulse wave speed

A local estimation of pulse wave speed c, an important predictor of cardiovascular events, can be obtained at arterial locations where simultaneous measurements of blood pressure (P) and velocity (U), arterial diameter (D) and U, flow rate (Q) and cross-sectional area (A), or P and D are available, using the PU-loop, sum-of-squares (∑(2)), lnDU-loop, QA-loop or new D(2)P-loop methods. Here, these methods were applied to estimate c from numerically generated P, U, D, Q and A waveforms using a visco-elastic one-dimensional model of the 55 larger human systemic arteries in normal conditions. Theoretical c were calculated from the parameters of the model. Estimates of c given by the loop methods were closer to theoretical values and more uniform within each arterial segment than those obtained using the ∑(2). The smaller differences between estimates and theoretical values were obtained using the D(2)P-loop method, with root-mean-square errors (RMSE) smaller than 0.18 ms(-1), followed by averaging the two c given by the PU- and lnDU-loops (RMSE <2.99 ms(-1)). In general, the errors of the PU-, lnDU- and QA-loops decreased at locations where visco-elastic effects were small and nearby junctions were well-matched for forward-travelling waves. The ∑(2) performed better at proximal locations.     

Use of simultaneous pressure and velocity measurements to estimate arterial wave speed at a single site in humans

It has not been possible to measure wave speed in the human coronary artery, because the vessel is too short for the conventional two-point measurement technique used in the aorta. We present a new method derived from wave intensity analysis, which allows derivation of wave speed at a single point. We apply this method in the aorta and then use it to derive wave speed in the human coronary artery for the first time. We measured simultaneous pressure and Doppler velocity with intracoronary wires at the left main stem, left anterior descending and circumflex arteries, and aorta in 14 subjects after a normal coronary arteriogram. Then, in 10 subjects, serial measurements were made along the aorta before and after intracoronary isosorbide dinitrate. Wave speed was derived by two methods in the aorta: 1) the two-site distance/time method (foot-to-foot delay of pressure waveforms) and 2) a new single-point method using simultaneous pressure and velocity measurements. Coronary wave speed was derived by the single-point method. Wave speed derived by the two methods correlated well (r = 0.72, P < 0.05). Coronary wave speed correlated with aortic wave speed (r = 0.72, P = 0.002). After nitrate administration, coronary wave speed fell by 43%: from 16.4 m/s (95% confidence interval 12.6-20.1) to 9.3 m/s (95% confidence interval 6.5-12.0, P < 0.001). This single-point method allows determination of wave speed in the human coronary artery. Aortic wave speed is correlated to coronary wave speed. Finally, this technique detects the prompt fall in coronary artery wave speed with isosorbide dinitrate.     

Wave propagation in a model of the arterial circulation

The propagation of the arterial pulse wave in the large systemic arteries has been calculated using a linearised method of characteristics analysis to follow the waves generated by the heart. The model includes anatomical and physiological data for the 55 largest arteries adjusted so that the bifurcating tree of arteries is well matched for forward travelling waves. The peripheral arteries in the model are terminated by resistance elements which are adjusted to produce a physiologically reasonable distribution of mean blood flow. In the model, the pressure and velocity wave generated by the contraction of the left ventricle propagates to the periphery where it is reflected. These reflected waves are re-reflected by each of the bifurcations that they encounter and a very complex pattern of waves is generated. The results of the calculations exhibit many of the features of the systemic arteries, including the increase of the pulse pressure with distance away from the heart as well as the initial decrease and then the large increase in the magnitude of back flow during late systole going from the ascending aorta to the abdominal aorta to the arteries of the leg. The model is then used to study the effects of the reflection or absorption of waves by the heart and the mechanisms leading to the incisura are investigated. Calculations are carried out with the total occlusion of different arterial segments in order to model experiments in which the effects of the occlusion of different arteries on pressure and flow in the ascending aorta were measured. Finally, the effects of changes in peripheral resistance on pressure and velocity waveforms are also studied. We conclude from these calculations that the complex pattern of wave propagation in the large arteries may be the most important determinant of arterial haemodynamics.     

The role of the surrounding tissue in the propagation of waves through the arterial system.

A theoretical analysis of the flow in arteries is presented, taking into consideration the role played by the surrounding tissues in determining the speed of propagatoion and the damping of the blood pressure pulse. This study was undertaken (a) to exhibit a method of computing the flow properties with a more nearly accurate model, (b) to see if the displacement on the skin would be related to the arterial wall displacement, and hence to pressure, velocity and flow rate of blood in the artery, and if it is likely to be measurable. It was found that the pressure of the 'viscous' part in the surrounding tissue increases the pulse velocity and the damping of the wave over the values found by other models which considered only thick-walled elastic tubes with no surrounding tissue. This study also shows that measurements on the skin can provide information about changes in arterial circulation due to diseases such as: edema, arteriosclerosis and others where the Young's modulus for either the arterial wall or the surrounding tissues is altered.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

Pressure and flow in the systemic arterial system

The dynamic characteristics of the proximal arterial system are studied by solving the nonlinear momentum and mass conservation equations for pressure and flow. The equations are solved for a model systemic arterial system that includes the aorta, common iliacs, and the internal and external iliac arteries. The model includes geometric and elastic taper of the aorta, nonlinearly elastic arteries, side flows, and a complex distal impedance. The model pressure wave shape, inlet and outlet impedance, wave travel, and apparent wave velocity compare favorably with the values measured on humans. Calculations indicate that: (i) reflections are the major factor determining the shape and distal amplification of the pressure wave in the arterial tree; (ii) although important in attenuating the proximal transmission of reflecting waves, geometric taper is not the major cause of the distal pressure wave amplification; (iii) the dicrotic wave is a result of peripheral reflection and is not due to the sudden change in flow at the end of systole; (iv) the elastic taper and nonlinearity of the wall elasticity are of minor significance in determining the flow and pressure profiles; and (v) in spite of numerous nonlinearities, the system behaves in a somewhat linear fashion for the lower frequency components.     

Optimal modeling of an arterial system

The aortic blood flow is described by a set of nonlinear hyperbolic partial differential equations that account for mass and momentum conservation, and nonlinear models for the mechanical properties of the artery. Identification is used for determining the wave speed, arterial taper, and cross section: these parameters reflect the elastic characteristics of the aorta wall and control the pulsatile response. The differential equations were numerically integrated by the Lax-Wendroff scheme of Abarbanel and Goldberg [J. Comput. Phys. 10:1–21 (1972)] that avoids nonlinear oscillations. The Gauss-Newton technique was used for the parameter identification. By reference to reported elocity and pressure input-output pairs, a parameter vector is found such that the distance in the L₂ norm between the predicted outputs and the measured functions is minimal. Calculations of the velocity and pressure waves show excellent compatibility of the model with reported experimental data: starting from arbitrary parameter estimates, which yield grossly distorted waveforms, the error is typically reduced to 7–8%. Introduction of viscoelastic behaviour for the arterial wall in the form of a Volterra integral for the cross section does not lead to significant improvement. Numerical examples are presented which prove the convergence, accuracy, and stability of the algorithm. Emphasis is placed on the computational feasibility of the proposed system identification.     

<Emphasis Type="Italic">In silico</Emphasis> coronary wave intensity analysis: application of an integrated one-dimensional and poromechanical model of cardiac perfusion

Coronary wave intensity analysis (cWIA) is a diagnostic technique based on invasive measurement of coronary pressure and velocity waveforms. The theory of WIA allows the forward- and backward-propagating coronary waves to be separated and attributed to their origin and timing, thus serving as a sensitive and specific cardiac functional indicator. In recent years, an increasing number of clinical studies have begun to establish associations between changes in specific waves and various diseases of myocardium and perfusion. These studies are, however, currently confined to a trial-and-error approach and are subject to technological limitations which may confound accurate interpretations. In this work, we have developed a biophysically based cardiac perfusion model which incorporates full ventricular–aortic–coronary coupling. This was achieved by integrating our previous work on one-dimensional modelling of vascular flow and poroelastic perfusion within an active myocardial mechanics framework. Extensive parameterisation was performed, yielding a close agreement with physiological levels of global coronary and myocardial function as well as experimentally observed cumulative wave intensity magnitudes. Results indicate a strong dependence of the backward suction wave on QRS duration and vascular resistance, the forward pushing wave on the rate of myocyte tension development, and the late forward pushing wave on the aortic valve dynamics. These findings are not only consistent with experimental observations, but offer a greater specificity to the wave-originating mechanisms, thus demonstrating the value of the integrated model as a tool for clinical investigation.     

Pulse Wave Propagation in Elastic Tubes Having Longitudinal Changes in Area and Stiffness

The behavior of both step waves and sinusoidal waves in fluid-filled elastic vessels whose area and distensibility vary with distance is explored theoretically. It is shown that the behavior of these waves may be explained, to a large extent, by considering the effect of the continuous stream of infinitesimal reflections that is set up whenever any wave travels in a region of vessel where the local impedance, (that is, the ratio of elastic wavespeed to tube area) is not constant. It is found that in such vessels the behavior of sinusoidal waves over distances which are a fraction of a wavelength can be quite different from their average behavior over several wavelengths. Both behaviors are described analytically. The results are applied to the mammalian circulatory system, one of the most interesting results being that a longitudinal variation in the pressure and velocity amplitudes which has a wavelength roughly one-half that of standing waves is predicted. The treatment is essentially a linearized quasi-one-dimensional one, the major assumptions being that the fluid is inviscid, the mean flow is zero, and the vessel is perfectly elastic and constrained from motion in the longitudinal direction. As in the physiological situation, the ratio of fluid velocity to pulse propagation speed is assumed small. For comparison with the analytical results, the linearized equations are also solved numerically by computer.