NOVEMBER 1960 VOL. VIII NO. 6 Circulation Research AN OFFICIAL JOURNAL oftk AMERICAN HEART ASSOCIATION Longitudinal Waves in the Walls of Fluid-Filled Elastic Tubes By ROBERT L. VAN CITTERS, M.D. A PULSE introduced into an elastic sys- tem is propagated in the form of wave motion. In a fluid-filled elastic shell 2 forms of wave motion, pressure and radial disten- tion, are readily recorded and have been studied extensively. While the propagation of pulses was being studied in an elastic model of the aorta, a third mode of vibration, longi- tudinal waves in the wall of the tubing, was observed. Since in the aorta itself these waves may modify the arterial pulse, a method for recording them has been devised and some of their characteristics have been discerned. Methods The basic experimental model (fig. 1) consisted of n 1 M. length of latex Pen rose drain tubing (inside diameter, 1.25 cm.; wall thickness, 0.03 cm.) filled with tap water and fitted at one end to a glass T tube (inside diameter, 0.5 in.). A rubber balloon inflated to 30 nun. Hg was mounted on one arm of the T tube; the pressure in the balloon was transmitted directly to fluid in the system. When the balloon was burst, a step function of pressure was introduced into the sys- tem. A microphone mounted 2 cm. from the balloon recorded the time of! the burst. Gages of pressure, radial distention, and length were mounted adjacent to each other around the circumference of the tubing 1 M. distal to the T tube (fig. 1). The pressure was measured via a 13-gage needle inserted axially into the From the Department of Physiology and Bio- physics, University of Washington School of Medi- cine, Seattle, Wash. Supported in part by Cardiovascular Training Grant HTS 5174 of the National Heart Institute. Received for publication May 12, 1960. Circulation Rctuiarch, Volvmr VIII. Novembrr 196U 1145 center of the fluid column and recorded with a Sanborn model 467B pressure transducer. The radial distention of the tube was measured •with a mutual inductance gage which consisted of a pair of mutual inductance coils, each wound into the shape of a bemicylinder. These coils were mounted opposite each other on the wall of the tubing by means of a tiny drop of cement. One coil was excited with a sine wave, 150 kilocycles 4 volts, from a signal generator. The signal from the secondary coil was rectified by a diode bridge, amplified with a D.C. amplifier, and recorded with the Sanborn polyviso. Changes in the distance separating the coils resulting from variation in the diameter of the tubing altered the degree of mutual coupling and were recorded as a shift in the D.C. level. The apparatus will detect movements of less than 0.05 mm., and the calibra- tion is nearly linear over the range employed. The full-scale response time is less than 20 msec. Changes in the length of the tubing were measured by attaching a variable resistance gage to the wall. This gage consisted of a delicate rubber tubing (inside diameter, 05 mm.) filled with mercury and sealed at both ends by insulated wires. Longitudinal stretching of the tubing in- creases resistance by reducing the cross-sectional area and increasing the length of: the mercury column. When these gages are installed under slight tension, they respond linearly to changes in length up to double the resting length. The experimental model was varied in order to determine some of the characteristics of longi- tudinal waves (figs. 2 and 3). In some experiments, glycerin having specific gravity of 1.249 and vis- cosity of 366 at 25C. was used instead of water. The wall motion was attenuated either by grasp- ing the tube firmly in both hands or by inserting it into a rigidly mounted laboratory clamp. In both instances, the wall of the tubing was in- dented slightly but the continuity of the fluid column was not interrupted. In some experiments by guest on June 6, 2018 http://circres.ahajournals.org/ Downloaded from
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NOVEMBER 1960VOL. VIII NO. 6Circulation Research
AN OFFICIAL JOURNAL oftk AMERICAN HEART ASSOCIATION
Longitudinal Waves in the Walls ofFluid-Filled Elastic Tubes
By ROBERT L. VAN CITTERS, M.D.
A PULSE introduced into an elastic sys-tem is propagated in the form of wave
motion. In a fluid-filled elastic shell 2 formsof wave motion, pressure and radial disten-tion, are readily recorded and have beenstudied extensively. While the propagationof pulses was being studied in an elastic modelof the aorta, a third mode of vibration, longi-tudinal waves in the wall of the tubing, wasobserved. Since in the aorta itself these wavesmay modify the arterial pulse, a method forrecording them has been devised and someof their characteristics have been discerned.
MethodsThe basic experimental model (fig. 1) consisted
of n 1 M. length of latex Pen rose drain tubing(inside diameter, 1.25 cm.; wall thickness, 0.03cm.) filled with tap water and fitted at one endto a glass T tube (inside diameter, 0.5 in.). Arubber balloon inflated to 30 nun. Hg was mountedon one arm of the T tube; the pressure in theballoon was transmitted directly to fluid in thesystem. When the balloon was burst, a stepfunction of pressure was introduced into the sys-tem. A microphone mounted 2 cm. from theballoon recorded the time of! the burst.
Gages of pressure, radial distention, and lengthwere mounted adjacent to each other aroundthe circumference of the tubing 1 M. distal tothe T tube (fig. 1). The pressure was measuredvia a 13-gage needle inserted axially into the
From the Department of Physiology and Bio-physics, University of Washington School of Medi-cine, Seattle, Wash.
Supported in part by Cardiovascular TrainingGrant HTS 5174 of the National Heart Institute.
Received for publication May 12, 1960.
Circulation Rctuiarch, Volvmr VIII. Novembrr 196U 1145
center of the fluid column and recorded with aSanborn model 467B pressure transducer. Theradial distention of the tube was measured •witha mutual inductance gage which consisted of apair of mutual inductance coils, each wound intothe shape of a bemicylinder. These coils weremounted opposite each other on the wall of thetubing by means of a tiny drop of cement. Onecoil was excited with a sine wave, 150 kilocycles4 volts, from a signal generator. The signal fromthe secondary coil was rectified by a diode bridge,amplified with a D.C. amplifier, and recorded withthe Sanborn polyviso. Changes in the distanceseparating the coils resulting from variation inthe diameter of the tubing altered the degree ofmutual coupling and were recorded as a shiftin the D.C. level. The apparatus will detectmovements of less than 0.05 mm., and the calibra-tion is nearly linear over the range employed.The full-scale response time is less than 20 msec.
Changes in the length of the tubing weremeasured by attaching a variable resistance gageto the wall. This gage consisted of a delicaterubber tubing (inside diameter, 05 mm.) filledwith mercury and sealed at both ends by insulatedwires. Longitudinal stretching of the tubing in-creases resistance by reducing the cross-sectionalarea and increasing the length of: the mercurycolumn. When these gages are installed underslight tension, they respond linearly to changesin length up to double the resting length.
The experimental model was varied in order todetermine some of the characteristics of longi-tudinal waves (figs. 2 and 3). In some experiments,glycerin having specific gravity of 1.249 and vis-cosity of 366 at 25C. was used instead of water.The wall motion was attenuated either by grasp-ing the tube firmly in both hands or by insertingit into a rigidly mounted laboratory clamp. Inboth instances, the wall of the tubing was in-dented slightly but the continuity of the fluidcolumn was not interrupted. In some experiments
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6 cm. heavy rigidTube freely suspended Tube held in hands tubing in center
UFigure 1
Following introduction of a step function of pres-sure into the system, waves of pressure, radialdistention, and longitudinal motion of the ivall ofthe tubing are recorded. Longitudinal leaves trav-ersed the wall with a velocity of 30 M./sec., tvhilethe velocity of the waves of pressure and radialdistention was 6 M./sec.
a thick-walled rigid tube replaced a 6 cm. segmentof the Penrose tubing. Eecords were obtained whenthis segment was suspended freely, and also whenit was clamped rigidly in place. All of these pro-cedures were repeated with various lengths ofPenrose tubing ranging from 10 to 100 em. Theeffects of externally applied impacts were alsostudied in each experimental condition, the wallof the tubing being sharply tapped with a lightinstrument.
ResultsA step function of pressure introduced into
the system is recorded as a series of wavesrepresenting pressure and radial displacementand longitudinal motion of the wall of thetubing. The latter 2 traverse the tubing atindependent velocities (fig. 1). The wave ofradial displacement traversed the tubing ata velocity of about 6 M./sec. and was in phasewith the pressure pulse. At the time of ar-rival of these waves, the length of the tubingalso changes, the change representing thelocal effect of radial displacement. The longi-tudinal vibration of the wall of the tubingtraveled 1 M. in 35 M./sec, so the velocityof these waves under the conditions of this
Microphone
Length
Diameter
- 1 Second J •
Figure 2Attenuation of the wall of the tube damps longi-tudinal waves but does not interfere with pressureand radial waves so long as the fluid column iscontinuous. Insertion of a section of rigid tubingdoes not effect transmission of longitudinal toavesif the wall is free to move.
experiment is about 30 M./sec Simultaneouswith the arrival of the longitudinal wave avery small pressure disturbance is recorded.
Attenuation of a segment of the wall,either by manual grasping or by a rigidclamp, results in complete damping of thelongitudinal vibrations (fig. 2). Insertion ofa short section of rigid tubing did not effectlongitudinal waves so long as the section wasfree to move; when it was clamped rigidlyin place, complete damping resulted (fig. 2).The differences in density and viscosity ofthe fluid column had little effect on the veloc-ity with which longitudinal waves were trans-mitted, but did modify their amplitude (fig.3). Successive shortening of the tubingresulted in proportionate reduction in thetransit time of all components (fig. 3). Impactapplied to the walls of the tubing resultedin waves of pressure, radial distention, andlongitudinal vibration with characteristicsidentical to those of the waves produced bythe step function (fig. 3).
Circulation Research, Volume VIII, November 1960
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DiscussionLamb1 has calculated the phase velocity VL
of a plane longitudinal wave propagatingalong an empty elastic cylindrical tube ofradius (r), thickness (T), elasticity (E), anddensity (P). He found:
A B C
Tube length reduced 30 cm. tube filledTube wall tapped to 30 cm. with glycerin
where X is the wavelength. If the tube isfilled with a compressible liquid of density Po,the expression for the phase velocity of aplane longitudinal wave in the liquid is
VD=(TE/2TP0)1% ( r < < r < < A ) .Wiskind and Talbot2 have calculated the
velocities of longitudinal and dilational wavesin a viscoelastic tube containing an incom-pressible liquid. Their calculation leads to
CD/CL S V D / V L (T < < r < < A).as the ratio of the dilational to the longitu-dinal wave velocity. On the basis of the above,we can calculate the ratio of the time requiredby each wave to traverse a given tube length;we have:
TD/TL = CD/CL = (2 r P/r Po)%.For the tube used in the present experiment
P/Po s l, 2 T/T S 50which leads to
TD/TL s 7whereas the data indicates a ratio
TD/TL e* 5.The concept of longitudinal waves may be
significant in the interpretation of phenom-ena observed in propagation of the aorticpulse. Rushmer3 suggested the presence oflength changes which are out of phase withcircumference changes as an explanation forthe phase difference he observed in the pres-sure-volume relations of the aorta.
The velocity of impact waves in the arter-ies of normal men has been observed to beabout 18 M./sec.4 This value is intermediatebetween the velocity of the pulse wave andthe velocity of sound in blood. A similar re-lationship has also been noted for the velocityof longitudinal waves in the elastic model.Although no completely satisfactory set ofphysical constants is available for arteriesunder dynamic conditions, a value forYoung's modulus has been calculated forhuman arteries in the frequency range of im-
Circulation Research, Volume VIII, November 1960
(L J
Microphone
Length
Diameter
Pressure
\ u*-
I — 1 Second '
Figure 3A. When a sharp impact is applied to the tubewall, waroes of pressure, radial distention, and lon-gitudinal motion are recorded. Wave velocities aresimilar to those observed following introduction ofa step function. B. Reduction of tube length re-sults in proportionate decrease in the time requiredfor all components to traverse the system. C. Slightdifferences in wave characteristics occur when theviscosity of the fluid within the tubing is altered.
pact waves.5 Substitution of this value alongwith appropriate physical measurements intoLamb's equations results in a value for thevelocity of longitudinal waves which is prac-tically identical with that measured for im-pact waves in arteries. It is suggested, there-fore, that impact waves which have beenstudied in human arteries are manifestationsof longitudinal waves in the arterial walls.
It is apparent that the role played by longi-tudinal waves in formation of the arterialpulse contour requires further investigation.Determination of the damping characteristicsof longitudinal waves is of primary impor-tance. While it can be shown from theoreticalconsiderations that the damping coefficientfor these waves is identical with that of radialwaves, the limitations of the recorder and theinertial effects of the gages on the wall aresuch that quantitative treatment could notbe attempted.
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the aorta have demonstrated the presence oflongitudinal waves in the wall of the tubing.This mode of vibration was predicted byLamb from the theoretical considerations, buthas not previously been described. The char-acteristics of "impact waves" in arteries andof the longitudinal mural waves in the modelappear to be simi'ar. Longitudinal waves un-doubtedly influence the arterial pulse con-tour. Theoretically, the damping coefficientfor them is the same as that for radial waves.
AcknowledgmentThe author wishes to acknowledge the assistance
of Harold Falk, Graduate Assistant, Department ofPhysics, University of Washington, in the theoreticalaspects of this paper.
Summario in InterlinguaExperimentos con un modello aortic de tubage de
ennchu ha demonstrate le presentia de undas longi-tudinal in le pariete. Iste inodo de vibration essevapredicite per Lamb super le base de eonsi(leratione<
theoric sed ha non previemente essite describite. Lecharacteristicas de undas de impacto in arterias e leillos del undas parietal longitudinal pare esser simile.Undas longitudinal influe sin dubita super le contornodel pulso arterial. Thcorieamente, le coefficiente (leamortimento de illos es identic con le correspondentecoefficiente de undas radial.
References1. LAMB, H.: On the velocity of sound in a tube,
as effected by the elasticity of the walls.Memoirs Manchester Literary and Philosophi-cal Society, 1898, Vol. 42.
2. WISKIND, H. K., AND TALBOT, S. A.: PhysicalBasis of Cardiovascular Sound: An AnalyticalSurvey. AFOSR Technical Report No. T.R-58-160; ASTIA Document No. AD 207 459;December, 1958.
M. KUSHMER, R. F.: Pressure-circumference relationsin the aorta. Am. J. Physiol. 183: 545, 1955.
4. LANDOWNE, M.: Characteristics of impact andpulse wave propagation in brachi.il and radialarteries. J. April. Physiol. 12: 9.1, 1958.
•5. —: Pulse wave velocity as an index of arterialelastic characteristics. In Tissue Elasticity,Edited by J. W. Remington. Washington,D. C, Waverly Press, 1957, pp. 1168-1176.
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