Visco-elastic propagation of longitudinal waves in skeletal muscle

/. Biomrchanics.

1972. Vol. 5. pp. I- IO.

Pergamon Press.

Primed in Great Brian

VISCO-ELASTIC PROPAGATION OF LONGITUDINAL WAVES IN SKELETAL MUSCLE* XUAN T. TRUONG Physiometrics Research Laboratory, Veterans Administration Hospital. Houston, Texas 7703 1. and Department of Physical Medicine, Baylor College of Medicine, Houston, Texas 77025, U.S..+.

.ibstract- A method for obtaining the relaxation spectrum of isolated frog skeletal muscle from measurements of phase velocity and attenuation of longitudinal waves, was derived from conventional linear theory of visco-elastic wave propagation. The experimental method for measuring wave velocity and attenuation is described. An approximation method for obtaining the relaxation spectrum from wave velocity alone was developed from theoretical considerations. Experimental results from both the conventional and the approximation methods are presented and compared. Wave propagation measurements appeared to be of value in visco-elastic studies of skeletal muscle, especially when the use of other traditional methods is not practical. ISTRODUCTIOS

tensively used in various fields of engineering and in vascular studies and which consists of measuring the propagation characteristics of visco-elastic waves. Velocity and absorption data have been reported by Goldman and Hueter (1956) among others, for ultra-sound in muscular tissue of various animals. While these data are useful for studying the acoustic impedance and damping of ultra-sound in animal tissues, they have little application to the study of the longitudinal visco-elasticity of muscle. In earlier studies of isolated frog muscle, the author (1963) reported that for a given frequency, the velocity of propagation of longitudinal waves generally increased when the stiffness of the muscle was increased by stretching or activation. There were. however, large variations in wave velocity with changes in frequency. Thus, it is difficult to interpret such findings in terms of the viscoelastic structure of the muscle, unless a method of analysis is first developed on the basis of a specific mechanical model for the muscle. It is the purpose of this paper to present an analysis based on 2 klaxwell model, of the characteristics of propagation of longitudinal sinusoidal strain waves along the longitudinal axis of isolated frog skeletal

viscous and elastic properties of isolated skeletal muscle have been extensively studied in the past with certain well established physiological methods, such as the determination of the lengthtension diagram (Walker, 1960), the recording of stress relaxation and creep (Buchthal et al. 195 1; Long et al. 1964) and the analysis of responses to sinusoidal stress (Buchthal et al. 195 1: Matsumura and Nagai, 1963). These methods have been applied with some success to studies involving skeletal muscles in human subjects (Foley, 1961: Long et al. 1964; 1Moffatt et al. 1969). All of these methods, however, have an inherent limitation when applied to an in ciao situation, namely the inability to make force and displacement measurements that are related to the muscles alone, and not also to such adjacent structures as tendons, ligaments and joint structures. Thus, they can provide valuable rheologic data about the whole muscle-tendon-joint complex, but little selective information about the visco-elastic properties of muscles alone. This problem has prompted the author to consider the application of another method of visco-elastic studies which has been exLONGITUDIN.~L

THE

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muscle, and derive a method for determining the relaxation spectrum of skeletal muscle from wave propagation data. A Maxwell model, composed of a continuous spectrum of Maxwell elements and an auxiliary elastic element, all connected in parallel (Fig. l), was chosen instead of a simpler model because of the inherent structural complexity of muscular tissue. Although simpler models like the Voigt element, the single Maxwell element, and the three or four parameter model were used with success by such authors as Nolle (1947), Lee and Morrison (1956), and Kolsky (1963) in studies of industrial materials, it is unlikely that they can be used to describe the complex mechanical behavior of muscle. On the other hand, a Maxwell model such as the one proposed here was found by Buchthal et al. (195 1) to give a rather good description of the stress relaxation and creep phenomena exhibited by muscle fibers.

modulus, and the longitudinal relaxation time. It has been shown by Kolsky (1963) that these longitudinal constants can define with good approximation the propagation characteristics of longitudinal waves through a visco-elastic rod, if the transverse dimensions of the rod are small compared to its length and compared to the wave length. The ratio of the radius of the specimen to the wave length should be less than O-2 if serious errors are to be avoided. Moreover, if the wave amplitude is small, most non-linear visco-elastic substances can be approximated by linear models such as the one proposed here. These limiting conditions must then be carefully observed if the theoretical method proposed here is to remain valid. If continuous sinusoidal strain waves are propagated through a thin elongated rod of visco-elastic material such as a muscle specimen, the particle displacement at a distance x from the origin and at time t is given by: Vexp(-ax)

sinw(t-x/c),

(1)

where V is the wave amplitude. The following constitutive relations between the propagation constants c (phase velocity) and CY(attenuation coefficient), and the visco-elastic constants for a visco-elastic rod are well known (Sips, 195 1):

(2)

(3)

Fig. 1. Maxwell model, composed of several Maxwell elements and an auxiliary spring, all connected in parallel. THEORY

The visco-elastic constants that are relevant to longitudinal visco-elasticity of muscle are the extensional elastic modulus or Young’s

where P and Q are normalized expressions of the real and imaginary parts of the complex Young’s modulus and are related to the viscoelastic constants of the medium. E is the normalizing factor and can be obtained from equation (12) shown below. Sips (195 1) gave the following expressions of P and Q for a model composed of discrete parallel Maxwell elements, including an

OF WAVES

PROPAGATION

auxiliary elastic element.

IN SKELETAL

where N(s) is in the normalized satisfies the condition:

I0=N(s)

(4) (3 with the normalizing condition: i Ai= i=l

dAds s) dAds

,

where N(A, s) is the number of Maxwell elements as a function of A and s, so that the tota number of elements is: I

II0

x

0

iV(A, s) dAds.

If we now introduce the relaxation frequency spectrum function: V(s) = Iox AN(A,

s) dA,

we have: P=AoU(w)+

Q=

lo= m

(6)

ds,

form and

ds = 1.

There are several formulae (Gross, 1953; Nolle, 1950) for inverting the integral in (6) or (7) to obtain the relaxation frequency spectrum function N(s). Nolle’s formula is the simplest to use:

1,

where A i = modulus of elastic component in Maxwell element; w = cyclic frequency; Ji = reciprocal of relaxation time; A,U(w) = step function representing the auxiliary elastic element of normalized modulus A”. If instead of a number of discrete Maxwell elements as in Sips’ model, we have a continuous spectrum of Maxwell elements, we can write expressions (4) and (5) as:

x osAN(A, s’+o?

3

MUSCLE

R;(o)

=

dP dw’

(8)

From this, the relaxation time spectrum function is obtained according to Gross (195 3): F(T) = lV( 1IT)/?.

(9)

Thus, the conventional method for obtaining relaxation spectra requires measurements of both the phase velocity c and the attenuation coefficient (Y which are necessary for the calculation of P for various frequencies. An analytical frequency function can then be fitted to the experimental P vs. frequency curve, and the relaxation spectrum obtained by differentiation according to equation (8). It should be noted here that, as a result of differentiation of the step function expressing the contribution of the auxiliary elastic element to P, this element will be represented by a spectral line of ‘magnitude’ A0 at s = 0. Under experimental conditions it is often difficult to measure (Ywith reasonable accuracy because of certain experimental problems that will be discussed later. In such cases, a procedure for obtaining the relaxation spectrum without the need for attenuation measurements would be advantageous. With some additional algebra, we can eliminate CY from relations (2) and (3) and derive an expression relating the relaxation spectrum to the phase velocity alone:

”

2E =~‘P+(P”+Q’)W

P’+Q’

(10)

4

X. T. TRUONG

where c is real, and P and Q are related to the relaxation frequency spectrum by expressions (6) and (7). There are no easy rigorous methods to derive N(S) from the velocity equation [equation (lo)]. However, if we accept certain approximations, equation (15) could be simplified. In order to define these approximations, let us consider one of the elements in the Maxwell model, with parameters Ai and SC. According to expressions (4) and (5), the contributions of that element to P and Q are Pi =a

L.L 2

w

t

Fig. 2. Pi and Qi vs. cyclic frequency. Pi and Qi in units of Ai. si = reciprocal of relaxation the.

and Qi=B. If we plot Pi and Qi against the cyclic frequency (Fig. 2), we can see that Pi approaches a value Ai for o > Si, and zero for o < si, whereas Qr approaches zero for values of o greater and smaller than Si and reaches a maximum value AJ2 at w = si. Therefore, in the whole model, for any given frequency o,, the magnitude of P results mainly from the contributions of all elements whose si are less than unr whereas the magnitude of Q depends on the relatively few elements whose s i have values around w,. This suggests that the magnitude of Q is generally much smaller than that of P, and could be approximately neglected in comparison with P. The relative errors caused by this approximation would be greatest at the extreme range of low frequencies when few elements would contribute to either P or Q. Thus if Qz is neglected in equation (lo), the velocity equation reduces to: c’ = P . Elp,

4

or

P = c?.p/E.

(11)

We have then an approximate expression of P in terms of the velocity alone without terms in LYin contrast to the rigorous constitutive

expression both c and Hence, equation spectrum alone:

for P [equation (2)], which has (Yterms. by differentiation according to (8), an approximate relaxation can be derived from velocity data

N(w)=plE-2

= p/E*=.

d(P)

(12)

For each particular material the validity of this approximation depends on how small experimental Qz values are in comparison with the experimental P’ values [as calculated from equation (2) and equation (3) for the range of frequencies of interest] so that neglecting Qz in equation (10) can be justified. This will be shown in the experimental results for the particular case of frog muscle. The ratio p/E in equations (2), (3) and (11) can be obtained from equations (6), (7) and ( 10) by making w approach infinity: c?+

E/p

as

w-x.

(13)

Thus, E/p can be obtained by extrapolating the experimental velocity vs. frequency curve to infinite frequency. If w in equations (6) and (7) is made to approach zero, equation (10) reduces to: c”-+ Elp.A,,

as

o-,5.

(14)

PROPAGATION

OF WAVES

Hence, A0 can be estimated by extrapolating the experimental velocity-frequency curve to zero frequency.

IN SKELETAL

MUSCLE

5

experimental components residing outside of the muscle segment located between the two pick-up sites, remained constant during measurements at both proximal and distal EXPERIMENTAL METHOD sites, they were cancelled out by the subtracting and dividing processes involved in Adult winter frogs (Rana pipiens) were used without regard to sex. The sartorius, the velocity and attenuation calculations. The a thin and elongated muscle, was excised results of these calculations therefore were whole from the right leg of each animal. related to the muscle properties alone. The instrumentation for measuring the Wave reflections from the end of the propagation constants of continuous sinuspecimen and at the contact between pick-up soidal strain waves was similar to the one and specimen presented no problems at fredescribed by Hillier (1961) and Kolsky (1963). quencies above 1000 Hz, because of the The driving system consisted of an electrorelatively high attenuation in the specimen. mechanical vibrator (AGAC-Derritron VP-2) Below 1000 Hz, terminal wave reflections driven by a sine-wave variable frequency could be minimized by carefully attaching the oscillator. The receiving pick-up, which distal end of the muscle to a thin and long consisted of two similar semi-conductor strip of foam rubber, impregnated with a highstrain gages (Pixie 8 101) joined in series and viscosity silicone fluid. On the other hand, bonded to the opposite sides of a thin steel below this frequency, reflections at the pickblade with a pointed tip, was so arranged as up contact significantly affected the velocity to be much more sensitive to longitudinal and attenuation measurements which would vibrations in the specimen than to transverse then require rather cumbersome mathematical vibrations. The electrical signals from the corrections (Ballou and Smith, 1949: Kolsky, oscillator, which served as reference, and the 1963). However, according to Ballou and output from the pick-up were displayed on a Smith (1949), if the product of the attenuation dual-trace oscilloscope. For any given fre- coefficient by the distance between driving quency, the pick-up was applied to a proximal source and pick-up is greater than 2.3 nepers. site on the muscle and the apparent time lag then the error in wave amplitude measurebetween pick-up and reference signals was ments is less than 1 per cent. Also the effects read as T,. The amplitude of the pick-up of this type of reflection on velocity measureproximal signals was also measured as G’,. ments become negligible as the distance The pick-up was then moved to a distal site between source and pick-up becomes large. on the muscle, and similarly, a ‘distal’ time One way to extend such distance between lag T, and a ‘distal’ wave amplitude L’?were source and pick-up is to interpose between measured. The wave propagation time be- source and muscle specimen a long strip of tween proximal and distal muscle sites was visco-elastic material with high attenuation given by 1, - T,, and the phase velocity was properties and with such propagation characobtained by dividing the distance .r between teristics that wave reflections at the materialproximal and distal sites by the wave propagamuscle boundary are minimal for the range tion time. The attenuation coefficient was of frequencies of interest. According to calculated as: Rayleigh (1894) and Kolsky (1963). wave reflections at the boundary of two viscoelastic rods are minimized when the rods (Y= i In (VI/l’,). have similar acoustic impedance (product of cross-section area, density and wave propagaSince all time lags and loss factors, due to tion velocity) and similar attenuation proper-

6

X. T. TRUONG

ties. Coupling the muscle specimen to the vibrator through a long thin strip of foam rubber impregnated with a high viscosity silicone fluid was found to approach these conditions and reduce effects of wave reflections at the pick-up contact to a reasonable degree for frequencies above 25 Hz. The lower the frequency, the longer the coupling strip needs to be, on account of the lesser attenuation. The lack of wave reflections was ascertained in a few specimens by obtaining complete plots of wave propagation time and amplitude vs. gradually increasing distance between distal and proximal pick-up sites, and by observing the lack of oscillations in the propagation time vs. distance, and amplitude vs. distance curves (Ballou and Smith, 1949; Hillier, 1961; Kolsky, 1963). The same tension on the specimen was maintained by a 7 g weight attached to the end of the distal foam rubber strip through a pulley. After relative stablization of creep, this tension resulted approximately in a 25 per cent lengthening of the muscle beyond resting length. The muscle specimen, which was suspended in free air, was kept moist with a Frog’s Ringer Solution drip. Muscle temperature was maintained at 25.0°C by controlling the temperatures of the room and of the Ringer solution. The condition of the

muscle was considered viable throughout each experiment if it produced similar twitch tension upon electrical stimulation before and after the experiment. The tip of the pick-up was pressed against the muscle lightly but sufficiently to produce consistent amplitude readings. The amplitude of vibrations at the driving source was kept at a minimum compatible with consistency and accuracy of measurements and the various frequencies were applied in a random sequence. Figure 3 illustrates the instrument system. With this system, measurements were possible between the frequencies of 25 and 10,000 Hz. Above this range, measurements were limited by excessive attenuation, while below 25 Hz they were limited by excessive wave reflections and lack of measurable attenuation. RESULTS

The root mean square error of measurements in the same specimen was about 5 per cent for velocity, and 10 per cent for attenuation coefficient. R.m.s. variation among specimens from the different animals was about 15 per cent for velocity and 25 per cent for attenuation coefficient. The effect of frequency on phase velocity is shown in Fig. 4 (curve with closed circles), where the data points represent average values

Odlator

Foam rubber strips

Weight

Fig. 3. Instrumentation for measuring phase velocity and attenuation coefficient.

PROPAGATION

I

0

2

OF WAVE

1

3

IN SKELETAL

I

I

4

5

Frequency,

MUSCLE

,

6

7

I

8

9

kc/set

Fig. 4. Phase velocity (closed circles) and attenuation (open circles) vs. wave frequency.

from 10 different specimens. The velocityfrequency curve shows a plateau at the high frequency end. A definitive plateau at the low frequency end was difficult to ascertain because of increasing error due to wave reff ection effects. The average attenuation-frequency curve is shown in Fig. 4 (curve with open circles). As frequency is increased toward 5000 Hz, the increase in attenuation seems to level off to a plateau. Above 5000 Hz, attenuation seems to unexpectedly rise again rapidly. There is strong suspicion that this attenuation rise at the higher frequencies was due to some type of experimental error. Indeed, there was an abnormal distribution of wave amplitude near the driving source, which added errors to amplitude measurements in this region. Since, at the higher frequencies, measurements had to be made closer to the driving source because of the high attenuation, they were affected by these errors. A similar problem was reported by Nolle (1947). Therefore, instead of using the experimental attenuation data for frequencies above 5000 Hz, values were extrapolated from the attenuation curve at lower frequencies (dotted line in Fig. 4) and were used in the determination of the relaxation spectrum.

IO IO

coefficient

The curve representing the average values of P for various frequencies as calculated by equation (2) is shown in Fig. 5 (solid curve with closed circles). To this curve was fitted a function of w of the general form: P(w) =S+d.

The parameters n, b and d of P(o) were determined by non-linear regression analysis using a Taylor Series reiterative method. The resulting regression equation was: 1.2ow + 0.02. p(w)=w+(1.95 x 10d) The relaxation time spectrum, according to equations (8) and (9) is then: 2.34 x lo4 F(r) = [( 1.95 x 104) + 11” For illustration purpose it is convenient to plot the spectrum on logarithmic scales, and it is customary to transform the spectral function according to the condition: L(ln T) X dln r = F(T) dr (Buchthal et al. 195 1; Nolle, 1950). The resulting function L(ln r) = I . F(T) is shown in Fig. 6, solid curve.

8

X. T. TRUONG

r

Cyclic frequency.

~10’ rod./sec

Fig. 5. Q2(a. c). !‘“(a. c), P(a, c) and approximated P(c) vs. frequency. Ordinate in dimensionless units.

Relaxmix

Me,

ICC

Fig. 6. Normalized relaxation time spectrum of skeletal muscle. in dimensionless units (solid curve). Approximated relaxation time spectrum (dashed curve).

The dashed curve with open circles in Fig. 5. represent average values of Qz for various frequencies. It can be seen that for frog muscle QZ values were notably smaller than P2 values (dashed curve with closed circles) at most frequencies, thus suggesting the possibility of neglecting QZ in the velocity equation (10) to give an approximated P according to equation (1 l), from which an approximate relaxation spectrum can be derived according to equations (12) and (9). The average values for P thus approximated

are represented by the soiid curve with open circles in Fig. 5. It can be seen that this approximate P(c) curve is not widely different from the solid curve with closed circle representing the exact P(a, c). Although the relative difference is considerably greater at the lower frequencies, the absolute difference remains rather small for all frequencies and the shapes of the two curves are basically similar. Since, according to equations (8) and (12), the spectrum functions are mainly dependent on the slopes or shapes of the P curves, it can be expected that the spectrum based on the exact P would not differ too greatly from the one based on the approximate P. This is shown in Fig. 6 where the dashed curve represents the approximate spectrum. The relaxation spectrum alone, if given in the normalized form, gives information only about the relative distribution of relaxation times, and no indication of the absolute magnitude of the elastic components involved. To have a fuller characterization of the viscoelastic properties of muscle, there must be also a measurement of the normalizing factor E, which by definition is J,“~(T) dr, wheref(r) is the non-normalized relaxation tune spectrum function. Since J,“f(f> dT is equal to the total sum of the elastic moduli of all elements in the

PROPAGATION

OF WAVES

Maxwell model, E is then a measurement of the ‘total’ elastic component in the muscle. By using a frog muscle density of 1.035 g/cm3, as determined by the buoyancy method, the normalizing factor is 1.84 X lOa average dyn/cm’, as calculated from equation (13). If under certain circumstances the density of the muscle is not expected to change a great deal in comparison with visco-elastic changes, the value of cx alone could be used as an estimate of the absolute total strength of the spectrum, and together with the normalized spectrum could give us a fairly complete characterization of the visco-elastic properties of skeletal muscle. The modulus A0 of the elastic component that corresponds to the auxiliary elastic element of the model can be estimated from equation (14). The lack of data at frequencies near zero, however, makes any extrapolation in this region rather questionable. DISCUSSION

The velocity-frequency curve for frog muscle appeared compatible with the general behaviour of a Maxwell model, since the velocity equation, equation (12) calls for similar plateaus as the frequency tends to zero and infinity. At the low frequency end, the wave velocity is primarily dependent on the auxiliary pure elastic element of the model, while at the high frequency end, the velocity depends primarily on the sum of the elastic moduli of all the elements of the model. The attenuation-frequency curve for frog skeletal muscle tended toward zero at the low frequency end and toward a plateau around 5000 Hz, and thus appeared also to be compatible with a Maxwell model, whose attenuation equation, as derived from equations (2) and (3), is: cy3_

&

1E

. (p’+

Q’)“”

p’+Q’

_

p

’

where P and Q are given by equations (6) and (7). It can be seen that the attenuation

IN SKELETAL

9

MUSCLE

tends to zero as the cyclic frequency approaches zero, and to a definite limiting value as the frequency approaches infinity. Few visco-elastic spectra have been derived from wave propagation data. An approximate relaxation time spectrum for Buna Rubber was reported by Nolle (1950). Buchthal er al. (195 1) have derived the following relaxation time spectrum for isolated frog muscle fiber from stress relaxation studies: F(T) = 7

K

7,;-1

X-+1nm

“&__? ’

1

’ ”

1

where K, X-, T,, and rZ are constants. This spectral function shows a general shape that is similar to ours, except for small relaxation times, when it tends to infinity as the relaxation time approaches zero, and the spectrum integral fr F(r) dr is infinite. This would mean that the mnial stress, in response to a sudden finite strain, is infinite according to the stress equation (Buchthal, 195 1):

cr,,=,,E[,,+j+; F(r)dr]. where ~~ is the initial stress; eO, the finite strain; A 0, the modulus of the auxiliary elastic element; E, the normalizing factor. Obviously, such a response is unnatural and some artificial cut-off limit was needed at the lower end of Buchthal’s spectrum, in order to adapt it to experimental results. In contrast, our relaxation time spectral functions have finite values at zero relaxation time, and their infinite integrals are convergent, thus allowing for a finite initial stress in response to a sudden strain. Buchthal et cl. (1951) reported that half of the relaxation times in their spectrum for resting muscle fibers were less than 1 msec. Our spectrum for whole frog muscle showed approximately the same proportion of relaxation times below 1 msec. In stress-relaxation

10

X. T. TRUONG

studies of human forearm muscles in tiivo, Long et al. (1964) reported a range of relaxation times between 0.1 and 6.0 sec. The iack of representation from smaller relaxation times in Long’s results, was obviously due to the relatively long duration of the manually produced transient stretch, which would have caused relaxation of most of the components with small relaxation times before the total stress relaxation curve was obtained. Moreover results from such in ciao studies included parallel joint and ligament factors as well as muscle properties. The wave propagation method of studying visco-elasticity in skeletal muscle appears, then, to be of value in providing dynamic visco-elastic information, especially in the higher frequency ranges. Its primary limitation, when continuous waves are used, is the lower frequency limit imposed by wave reflection problems in specimens of limited length. REFERENCES Ballou, J. W. and Smith, J. C. (1949) Dynamic measurements of polymer physical properties. /. appf. Phys. 20,493-502. Buchthal, F., Kaiser, E. and Rosenfalck, P. (1951) Rheology of cross-striated muscle fiber with particular reference to isotonic conditions. Dan. Biol. Med. 21, 1-3 18. Foley, J. (196 1) The stiffness of spastic muscle. J. Neural. Neurosurg. Psychiat. X12513 1. Goldman. D. E. and Hueter, T. F. (1956) Tabular data of the velocity and absorption of high-frequency sound in mammalian tissues. J. Acous. Sot. Am. 28,35-57. Gross, B. (1953) Marhemaricnl Stracture of the Theories of Visco-elasticity. Herman and Cie, Paris. Hillier, K. W. (1961) The measurement of dynamic elastic properties. In Progress in Solid Mechanics, Vol. II, Ed. by I. N. Sneddon and R. Hill. NorthHolland, Amsterdam. Koisky, H. (1963) Stress Waves in Solids. Dover, New York. Lee, E. H. and Morrison, J. A. (1956) A comparison of

the propagation of Ioagitudiial waves in rods of visco-elastic materials. J. Polymer Sci. 19,93-l 10. Long, C., Thomas, D. and Crochetiere, W. J. (1964) Objective measurement of muscle tone in the hand. Clin. Pharmacol.

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17.

Matsumura, M. and Nagai. T. (1963) Dynamic viscoelastic properties of glycerol-extracted muscle fibers. Jap. J. Physiof. 13,246-259.

Moffatt, C. A., Harris, E. H. and Haslam, E. T. (1969) An experimental and analytic study of the dynamic properties of the human leg. J. Eiomechanics 2,373-387. Nolle, A. W. (1947) Acoustic determination of the physical constants of rubber-like materials. J. Acoos. Sot. Am. 19,194-201. Nolle, A. W. (1950) Dynamic mechanical properties of rubber-like materials. J. Polymer Sci. 5, l-54. Rayleigh, J. W. S. (I 894) The Theory of Sound, Reprint (1945). Dover, New York. Sips, R. (1951) Propagation phenomena in elasticviscous media. J. PolymerSci. 3,285-293, Truong, X. T., Walker, S. M. and Wall, B. J. (1963) The use of velocity of elastic waves in the determination of elastic constants in frog muscle. The Physiologist. 6,289.

Walker, S. M. (1960) The relation of stretch and of temperature to contraction of skeletal muscle. Am. J. Phys. Med. 39,234-258.

A* C CO Cr

E 07)

Im N(s) Q’ Re S t x a

P r w

NOMENCLATURE modulus of auxiliary elastic element in Maxwell model (dynlcm’) modulus of elastic component of each Maxwell element in model (dynlcm?) phase velocity (cm/set) phase velocity limit at zero frequency (cmlsec) phase velocity limit at infinite frequency (cmlsec) sum of moduli of all elastic components in Maxwell model (dynlcm’) normalized relaxation time spectrum function imaginary part relaxation frequency spectrum function normalized real part of complex Young’s Modulus normalized imaginary part of complex Young’s Modulus real part reciprocal of relaxation time (set-‘) time (set) distance along longitudinal axis of specimen (cm) attenuation coefficient (neperslcm) density of propagating medium (g/cm? relaxation time (set) cyclic frequency or angular velocity (rad./sec)