NOVEMBER
VOL. VIII
Circulation Research
1960
NO. 6
AN OFFICIAL JOURNAL oftk AMERICAN HEART ASSOCIATION
Longitudinal Waves in the Walls of
Fluid-Filled Elastic Tubes
By
ROBERT L. VAN CITTERS,
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PULSE introduced into an elastic system is propagated in the form of wave
motion. In a fluid-filled elastic shell 2 forms
of wave motion, pressure and radial distention, 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, longitudinal 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.
A
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 system. 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 Biophysics, University of Washington School of Medicine, 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
M.D.
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 calibration 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 increases 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 longitudinal waves (figs. 2 and 3). In some experiments,
glycerin having specific gravity of 1.249 and viscosity of 366 at 25C. was used instead of water.
The wall motion was attenuated either by grasping 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 indented slightly but the continuity of the fluid
column was not interrupted. In some experiments
1145
VAN CITTERS
1146
Tube freely suspended
Tube held in hands
6 cm. heavy rigid
tubing in center
Microphone
Length
Diameter
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U
Figure 1
Following introduction of a step function of pressure into the system, waves of pressure, radial
distention, and longitudinal motion of the ivall of
the tubing are recorded. Longitudinal leaves traversed the wall with a velocity of 30 M./sec., tvhile
the velocity of the waves of pressure and radial
distention was 6 M./sec.
a thick-walled rigid tube replaced a 6 cm. segment
of the Penrose tubing. Eecords were obtained when
this segment was suspended freely, and also when
it was clamped rigidly in place. All of these procedures were repeated with various lengths of
Penrose tubing ranging from 10 to 100 em. The
effects of externally applied impacts were also
studied in each experimental condition, the wall
of the tubing being sharply tapped with a light
instrument.
Results
A step function of pressure introduced into
the system is recorded as a series of waves
representing pressure and radial displacement
and longitudinal motion of the wall of the
tubing. The latter 2 traverse the tubing at
independent velocities (fig. 1). The wave of
radial displacement traversed the tubing at
a velocity of about 6 M./sec. and was in phase
with the pressure pulse. At the time of arrival of these waves, the length of the tubing
also changes, the change representing the
local effect of radial displacement. The longitudinal vibration of the wall of the tubing
traveled 1 M. in 35 M./sec, so the velocity
of these waves under the conditions of this
- 1 Second
J•
Figure 2
Attenuation of the wall of the tube damps longitudinal waves but does not interfere with pressure
and radial waves so long as the fluid column is
continuous. Insertion of a section of rigid tubing
does not effect transmission of longitudinal toaves
if the wall is free to move.
experiment is about 30 M./sec Simultaneous
with the arrival of the longitudinal wave a
very small pressure disturbance is recorded.
Attenuation of a segment of the wall,
either by manual grasping or by a rigid
clamp, results in complete damping of the
longitudinal vibrations (fig. 2). Insertion of
a short section of rigid tubing did not effect
longitudinal waves so long as the section was
free to move; when it was clamped rigidly
in place, complete damping resulted (fig. 2).
The differences in density and viscosity of
the fluid column had little effect on the velocity with which longitudinal waves were transmitted, but did modify their amplitude (fig.
3). Successive shortening of the tubing
resulted in proportionate reduction in the
transit time of all components (fig. 3). Impact
applied to the walls of the tubing resulted
in waves of pressure, radial distention, and
longitudinal vibration with characteristics
identical to those of the waves produced by
the step function (fig. 3).
Circulation Research, Volume VIII, November 1960
LONGITUDINAL WAVES
Discussion
Lamb1 has calculated the phase velocity VL
of a plane longitudinal wave propagating
along an empty elastic cylindrical tube of
radius (r), thickness (T), elasticity (E), and
density (P). He found:
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where X is the wavelength. If the tube is
filled with a compressible liquid of density P o ,
the expression for the phase velocity of a
plane longitudinal wave in the liquid is
V D =(TE/2TP 0 )1%
(r<<r<<A).
Wiskind and Talbot2 have calculated the
velocities of longitudinal and dilational waves
in a viscoelastic tube containing an incompressible liquid. Their calculation leads to
CD/CL
S
VD/VL
1147
A
Tube wall tapped
B
C
Tube length reduced
to 3 0 cm.
(L
3 0 cm. tube filled
with glycerin
J
Microphone
Length
Diameter
\
u*-
Pressure
(T < < r < < A).
as the ratio of the dilational to the longitudinal wave velocity. On the basis of the above,
we can calculate the ratio of the time required
by each wave to traverse a given tube length;
we have:
T D /T L = CD/CL = (2 r P/r Po) % .
For the tube used in the present experiment
P/Po s l, 2 T/T S 50
which leads to
TD/TL s 7
whereas the data indicates a ratio
T D /T L e* 5.
The concept of longitudinal waves may be
significant in the interpretation of phenomena observed in propagation of the aortic
pulse. Rushmer3 suggested the presence of
length changes which are out of phase with
circumference changes as an explanation for
the phase difference he observed in the pressure-volume relations of the aorta.
The velocity of impact waves in the arteries of normal men has been observed to be
about 18 M./sec.4 This value is intermediate
between the velocity of the pulse wave and
the velocity of sound in blood. A similar relationship has also been noted for the velocity
of longitudinal waves in the elastic model.
Although no completely satisfactory set of
physical constants is available for arteries
under dynamic conditions, a value for
Young's modulus has been calculated for
human arteries in the frequency range of imCirculation Research, Volume VIII, November 1960
I
— 1 Second
'
Figure 3
A. When a sharp impact is applied to the tube
wall, waroes of pressure, radial distention, and longitudinal motion are recorded. Wave velocities are
similar to those observed following introduction of
a step function. B. Reduction of tube length results in proportionate decrease in the time required
for all components to traverse the system. C. Slight
differences in wave characteristics occur when the
viscosity of the fluid within the tubing is altered.
pact waves.5 Substitution of this value along
with appropriate physical measurements into
Lamb's equations results in a value for the
velocity of longitudinal waves which is practically identical with that measured for impact waves in arteries. It is suggested, therefore, that impact waves which have been
studied in human arteries are manifestations
of longitudinal waves in the arterial walls.
It is apparent that the role played by longitudinal waves in formation of the arterial
pulse contour requires further investigation.
Determination of the damping characteristics
of longitudinal waves is of primary importance. While it can be shown from theoretical
considerations that the damping coefficient
for these waves is identical with that of radial
waves, the limitations of the recorder and the
inertial effects of the gages on the wall are
such that quantitative treatment could not
be attempted.
VAN CITTERS
1148
Summary
Experiments with a rubber tube model of
the aorta have demonstrated the presence of
longitudinal waves in the wall of the tubing.
This mode of vibration was predicted by
Lamb from the theoretical considerations, but
has not previously been described. The characteristics of "impact waves" in arteries and
of the longitudinal mural waves in the model
appear to be simi'ar. Longitudinal waves undoubtedly influence the arterial pulse contour. Theoretically, the damping coefficient
for them is the same as that for radial waves.
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Acknowledgment
The author wishes to acknowledge the assistance
of Harold Falk, Graduate Assistant, Department of
Physics, University of Washington, in the theoretical
aspects of this paper.
Summario in Interlingua
Experimentos con un modello aortic de tubage de
ennchu ha demonstrate le presentia de undas longitudinal in le pariete. Iste inodo de vibration esseva
predicite per Lamb super le base de eonsi(leratione<
theoric sed ha non previemente essite describite. Le
characteristicas de undas de impacto in arterias e le
illos del undas parietal longitudinal pare esser simile.
Undas longitudinal influe sin dubita super le contorno
del pulso arterial. Thcorieamente, le coefficiente (le
amortimento de illos es identic con le correspondente
coefficiente de undas radial.
References
1. LAMB, H.: On the velocity of sound in a tube,
as effected by the elasticity of the walls.
Memoirs Manchester Literary and Philosophical Society, 1898, Vol. 42.
2. WISKIND, H. K., AND TALBOT, S. A.: Physical
Basis of Cardiovascular Sound: An Analytical
Survey. AFOSR Technical Report No. T.R-58160; ASTIA Document No. AD 207 459;
December, 1958.
M. KUSHMER, R. F.: Pressure-circumference relations
in the aorta. Am. J. Physiol. 183: 545, 1955.
4. LANDOWNE, M.: Characteristics of impact and
pulse wave propagation in brachi.il and radial
arteries. J. April. Physiol. 12: 9.1, 1958.
•5. —: Pulse wave velocity as an index of arterial
elastic characteristics. In Tissue Elasticity,
Edited by J. W. Remington. Washington,
D. C, Waverly Press, 1957, pp. 1168-1176.
Circulation Research, Volume VIII, November I960
Longitudinal Waves in the Walls of Fluid-Filled Elastic Tubes
ROBERT L. VAN CITTERS
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Circ Res. 1960;8:1145-1148
doi: 10.1161/01.RES.8.6.1145
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Copyright © 1960 American Heart Association, Inc. All rights reserved.
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