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The Goddard memory‐integral expansion for the stress tensor of viscoelastic fluids differs from previous expansions in that it uses a deformation rate referred to a co‐rotating reference frame. It is shown how to get the kernel functions in the Goddard series in terms of the structural parameters occurring in the kinetic theory of macromolecular solutions. That the series has much better convergence properties than previously used expansions is demonstrated by comparison with exact kinetic theory results for steady‐state shear flow and elongational flow.

The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response, in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains.

Several impact problems for nonhomogeneous, four parameter, viscoelastic rods are studied: (i) an oscillatory stress is applied uniformly over the end of a semi-infinite rod, (ii) stresses in the form of Heaviside unit and Dirac delta functions are applied uniformly over the end of a semi-infinite rod, and (iii) the effect of an interface on waves generated by Heaviside impact of a semi-infinite, biviscoelastic rod is examined. In all cases, the rod is initially unstressed and at rest, and the propagation of one-dimensional, longitudinal waves is studied. Formal asymptotic expansions of the solutions are obtained. It is shown how Duhamel's theorem may be used to derive the solution for Dirac delta impact from that for Heaviside impact. For Heaviside impact of the biviscoelastic rod, it is found that reflected waves may be generated at the interface even though the impedance is continuous across it. For Heaviside impact of a rod whose properties satisfy definite functional relationships, the complete expansion is obtained.

The equations of motion for cylindrical and spherical shear waves in nonhomogeneous, isotropic, "standard", viscoelastic media with continuous radial variations are derived. Oscillatory shearing tractions are applied to the boundaries of cylindrical and spherical openings in unlimited viscoelastic media. The propagation of small-amplitude waves is studied, and formal asymptotic expansions of the solutions are obtained. In both cases (cylindrical and spherical), the leading term of the formal asymptotic expansion represents a modulated, oscillating, progressive wave propagating with variable velocity. The modulation depends on the moduli of rigidity and viscosity, whereas the velocity depends on the moduli of rigidity only. Application of our results to the propagation of shear waves in both finite and infinite viscoelastic plates is discussed.

A method for the determination of power consumption of industrial agitators in non-Newtonian fluids on the basis of measurements on a model device is presented. The applicability of this method for purely viscous, time dependent and viscoelastic fluids is discussed. The method was tested experimentally using agitators of two different sizes and results of measurements confirmed the applicability of the present method for engineering calculations.

It is important to clarify how the propagation of a longitudinal wave in a visco-elastic bar is affected by the lateral inertia. This paper purports to clarify the coupling effect of the viscous force and the lateral inertia force on the propagation of the wave, especially in the vicinity of the wave front. It is assumed that the visco-elastic material is represented by a Voigt model. The equation of motion of the bar, in which the effect of the lateral inertia is taken into consideration, is obtained with Hamilton's principle. The solutions of the equation are presented for two cases. One is the case of one end of the bar being impacted at a constant velocity and the other is the case of the stress or strain at the impacted end being kept constant.

The large deformation behavior of a viscoelastic squeeze film being compressed or extended between two parallel circular plates is studied from a Lagrangian viewpoint. A single integro-differential equation is shown to govern the flow of a rubberlike liquid in this geometry, and the solution is compared with results for a Newtonian squeeze film. Instantaneous solid-like response, small amplitude oscillations, and creep-recovery calculations are presented.

Shells are considered for which the constants of viscoelasticity are random functions of curvilinear coordinates of the middle surface. Correlation functions are obtained for the first approximations of the deflection and the stress function, as well as the variance of the critical time.

Constitutive equations are presented for non-linear viscoelastic materials under variable stress loading nd are interpreted in terms of non-linear viscoelastic models. Analytical solutions are obtained for stress, velocity and strain at the wave front in an impulsively loaded semi-infinite rod of material described by these non-linear constitutive equations within the assumptions of small deformation theory.

The purpose of this paper is to describe the incremental dynamic anisotropic viscoelastic properties of the middle descending thoracic aorta in ten living dogs. A segment of the middle descending thoracic aorta was isolated in situ and connected to a reservoir filled with oxygenated blood. The height of the reservoir could be adjusted to impress a known pressure on the segment. In addition, a sinusoidal fluid displacement pump was connected to this assembly to superimpose small sinusoidal pressures at various frequencies (0-5 Hz); in another phase of the experiment small sinusoidal changes in length were also imposed on the segment. The pressure, the longitudinal force, and the length of the segment were monitored continuously with specially designed transducers. From these data, the incremental elastic ( E ') and the viscous ( E '') moduli in the circumferential (θ), longitudinal ( z ), and radial (γ) directions were calculated around a given state of strain in the physiological range. In general, the viscoelastic moduli were functions of the initial stretches, indicating the nonlinear nature of the arterial wall. Within the physiological range of pressure, E z ' > E θ ' > E γ ' ( P < 0.01) and E z '' > E γ '' > E θ '' (P < 0.01). These results indicated that the vessel wall was anisotropic, having its largest values of elastic and viscous moduli in the longitudinal direction. The ratio of E ' / E ' < 0.123 at 2 Hz, indicating a small viscous component relative to its elastic counterpart. The elastic moduli increased markedly from 0-2 Hz and then settled down to a relatively constant value; this finding was consistent with previously published data. The values of E z ' and E z '' from the in vivo experiments were higher than those from the in vitro experiments, and the difference was attributed to the vascular tethering present in the in vivo state.

In a previous paper the author tried to show that hypothesized mechanical impact and mechanical fatigue processes in the cochlea are consistent with a range of psychophysical data on auditory fatigue and non‐linear effects (e.g., mechanical impact between the hair cells and tectorial membrane, and mechanical fatigue in terms of the viscoelastic or gel properties of the tectorial membrane). The present paper explores the extent to which mechanical processes of this type are also consistent with the recent findings of Goldstein [J. Acoust. Soc. Amer. 41, 676–689 (1967)] regarding combination tones in the cochlea. We develop the idea of a hair vibrating in a pocket within the tectorial membrane, which is formed by the local movement of the hair. It is assumed that this pocket tends to close around the hair with some specified time constant. It is shown that impact between a single hair and the ends of the pocket generates the main features of the combination tone sequence found by Goldstein. Other combinatio...

A formal method that is essentially an extension to viscoelasticity of the Karal‐Keller method for elastodynamics is presented for solving three‐dimensional wave‐propagation problems for a wide class of homogeneous, isotropic, and linear viscoelastic media. As applied to high‐frequency time‐harmonic waves, it consists of asymptotically representing the solutions in a series of inverse powers of the frequency. However, the expansion method is not necessarily limited to time‐harmonic waves. The approximation given by the first term in the expansion is called the geometrical theory of viscoelasticity. Subsequent terms provide corrections to this theory. The method can be employed to solve problems for media that contain boundaries. Applications are given to the reflection of incident waves from smooth obstacles and “asymptotic” reflection laws are derived. As an example, the first two terms are determined in the expansion of the reflected waves that result from a time‐harmonic plane dilatational wave striking a rigid plane.

In this paper a simplified constitutive equation for nonlinear slightly viscoelastic materials is derived by a method similar to that ofColeman andNoll. This equation is also valid for short times, slow motions, or small deformations superposed on a large steady deformation for all viscoelastic materials. An experimental method for determining the material parameters contained in this constitutive equation for dynamic conditions is also presented.

This paper presents the steady‐state response of infinite‐length shells of linear viscoelastic materials to moving axisymmetric ring loads based on the shell theory with corrections for the shear‐deformation and rotatory inertia effects. The complex Fourier transform is used to obtain the exact solution. Numerical results for shells of Maxwell materials with various viscous damping coefficients under various velocities of moving loads are given. The special case, response for elastic shells, is also discussed.

In the last twenty-five years there has been great progress in the development of nonlinear continuum theories in mechanics and other branches of physics, and in the use of such theories to explain and predict phenomena that cannot be explained rationally by the classical linear theories.

Mathematics Technical Report

It is generally agreed that in the flow of viscoelastic fluids not only tangential stress but also normal stresses may be generated. This fact raises the necessity of the developement of the goniometry of flow. To satisfy this purpose, Roberts1 has developed a series of instruments named“Weissenberg Rheogoniometer.”The instrument reported here is also the one devoted to this purpose. The schematic arrangement and two examples of main assemblies for torque and thrust measurments of this instrument are shown in Figs. (1) to (3), respectively.As is well known, the shear stress and strain velocity relation in laminar shear flow is determined from the data of torque and rotational velocity by the use of Eq. (1) in coaxial cylinders and by Eq. (3) in parallel discs.For the measurements of normal stresses, we adopted an instrument of a parallel plate type, with which the pressure distribution and the total thrust on a stationary disc of radius R are measured under known rotational velocity and gap spacing. Relations between normal stresses and strain velocity are obtained from those data by the use of Eqs. (4) to (9) with the assumptions that the influence of the centrifugal force may be neglected and complexities arising from the edge of bob and the side boundary may be simply replaced by an isotropic term p'.In the sense of phenomenological theory, 1 stresses are correlated to so-called“recoverable strain”and to an over-all rigidity of a fluid in sheared state as shown in Eqs. (12) and (13). The time-dependent features of rigidity may be known by oscillatory laminar shearing tests. The dynamic complex viscosity is determined by Eq. (10) in the coaxial cylinder and by Eq. (11). in the parallel discs. It is of interest to perform the test in which an oscillatory motion is superimposed on the steady shear, for such tests may reveal a fine structure of the rigidity of a fluid in the sheared state. In the present instrument a superimposer of rotational and oscillatory motions is also furnished for this purpose.

SYNOPSIS The stresses and displacements in two layer system having viscoelastic properties of Voigt or Maxwell type are generally treated in this paper by the use of the linear viscoelastic theory. Sixteen types of two-layer systems are considered. Table 1 . Model types of surface layer Model types of lower layer Symbolic expressions elastic elastic E/E viscous viscous V/V elastic viscous E/V viscous elastic V/E elastic Voigt type E/VEV viscous Voigt type V/VEV Voigt type elastic VEV/E Voigt type viscous VEV/V elastic Maxwell type E/VEm viscous Maxwell type V/VEm Maxwell type elastic VEm/E Maxwell type viscous VEm/V Voigt type Voigt type VEV/VEV Maxwell type Maxwell type VEm/VEm Maxwell type Voigt type VEm/VEV Voigt type Maxwell type VEV/VEm Time effect upon stress and displacement is studied by applying the corresponding principle proposed by E. H. Lee for quasistatic viscoelastic problems. The main results obtained herein can be summarized as follows. If relaxation or retardation time of upper layer has the magnitude larger than that of lower layer, the reinforcing and load-spreading effect is more remarkable than is expected from the Burmister’s analysis for elastic system. On the contrary, if material of smaller relaxation or retardation time is deposited in the surface layer, the more concentrated distribution of stress takes place around the axis of load application. Initial and final trends of change in stress and displacement for any other types of systems may be known approximately from the study of initial and final stages of four-elemental models ; E/E, V/V, E/V, and V/E. The degree of approximation of these four-elemental types to other types of systems has been examined with the aid of series expansion in the vicinity of initial and infinite time.

In this paper three-dimensional Maxwell stress-strain relations were deduced phenomenologically. In the first place we applied the Hamilton's principle to the viscoelastic deformation, and obtained the variational equation with respect to the elastic potential and the dissipation function. Then we assumed that the elastic potential is a function only of the stress, and the dissipation function is a function of stress and rate of stress. By the above variational equation of the virtual stress satisfying the equilibrium equation and the boundary conditions, we obtained the relations to be satisfied by the elastic potential and the dissipation function, and the conditions to be satisfied by the dissipation function. From these relations we obtained the required three-dimensional Maxwell stress-strain relations in viscoelasticity. These relations indicate that the strain is the sum of the internal elastic strain and the internal viscous strain. If a given substance is isotropic with respect to stress, the stress-strain relations are expressed by a linear Maxwell model consisting of Hookian spring in series with a Newtonian dashpot. It is the main result of this paper that the three-dimensional Maxwell stress-strain relations in viscoelasticity are deduced from physically appropriate assumptions.

Non-equilibrium thermodynamics with internal variables governed by rate equations is explored as a foundation for non-linear solid mechanics. Rate equations are studied as to type, yielding descriptions of viscoelastic, viscoplastic, and plastic behavior. The special case of uncoupled instantaneous elasticity is considered, as well as materials exhibiting combined behavior.

On the basis of two theorems pertaining to the asymptotic behavior of certain Laplace transforms, the uniqueness of the displacement field in a general linear viscoelastic body (i.e., one with time-variable properties) throughout a time interval is demonstrated, provided the instantaneous elasticity tensor (or, in the case of a generalized Kelvin—Voigt material, the instantaneous viscosity tensor) is positive definite and a continuous function of time, and provided the following information is specified: the displacement field, to within a rigid-body motion, throughout the body and at all times before the given interval; the displacement and velocity fields throughout the body at the beginning of the interval (initial conditions); the body force throughout the body and throughout the interval; and, at each point of the boundary, in each of three orthogonal directions, a component of the traction or of the displacement throughout the time interval. If inertia is neglected, the initial conditions may be dispensed with, but the displacement field is unique only to within a rigid-body motion.

Two three-dimensional models are introduced in order to describe the non-linear behavior and the three-dimensional cross effects occurring in the stress-deformation relations of viscoelastic materials. These models are the analogues of the so-called Maxwell and Voigt models in the classical one­ dimensional case. Non-linearities in the stress-deformation relations are classified into three categories: Elastic, viscous and geometrical ones, which are connected with the energy stored and dissipative mechanisms and the three-dimensional character of the materials, respectively. It is pointed out that we can actually separate these three non-linearities by observing the structural viscosity and the Weissenberg effect. The viscoelasticity of the so-called weakly coupled network model is briefly discussed as an example of the three-dimensional Maxwell model.

The theory of thermomechanical behavior of a linear viscoelastic Cosserat continuum is developed by making use of the fundamental concepts of continuum mechanics and irreversible thermodynamics. It is shown that, under conditions of arbitrary temporal and spatial variation of the temperature field, the stress tensor, the couple stress tensor and entropy density are derivable from a potential which is the free energy density Ψ.

Considered is the problem of thermal buckling of a square plate, simply supported along the edges, of a linear incompressible viscoelasticMaxwell material, and exposed to a temperature field varying randomly with time and across the width of the plate.

The normal modes of fluctuations in an interface, separating two fluid phases, which possess both viscoelasticity and optical anisotropy, are determined. The modes separate into the following four groups if the two surrounding fluids are the same: shear; twist; dilatation and ordering; and transverse vibration and splay modes. In the limit of strong coupling between orientation and transverse vibration, the main effect of molecular alignment is to increase the apparent surface tension by a term proportional to q 2 , where q is the momentum transfer in the surface fluctuations. The intensity and frequency spectrum of coherent light scattered from the thermal fluctuations are calculated. By measuring the depolarization in a suitable geometry and the frequency spectrum, contribution from both the orientation effect and the various viscosity coefficients can be determined. Applications to artificial and biological membranes are briefly discussed.

As is well known, a small compressive disturbance in an elastic gas grows into a shock wave in finite time. If the gas is viscoelastic or has internal degrees of freedom, this result is no longer true unless the initial amplitude of the disturbance is larger than a critical amplitude. This article investigates the effect of diffusion on the growth and decay properties of small disturbances in a mixture of elastic gases. It is shown that a critical amplitude exists for the mixture. Thus, diffusion can be thought of as having a stabilizing effect in that not all compressive disturbances will grow into shock waves.

A dynamical, electrical analogy is used to evaluate criteria for the maximally flat displacement and acceleration responses of a mechanical system equipped with a vibration absorber having a Maxwell-type viscoelastic connection and subjected to harmonic forces.

There have been many studies of the stability of plastically deformable media. Specific problems are solved in [1, 2], etc. In these studies it has been assumed that the process of loss of stability can be investigated in the quasi-static formulation, i.e., an attempt is made to find the values of the external loads at which, together with the unperturbed equilibrium mode, the adjacent perturbed equilibrium state is possible, the transition from the unperturbed to the adjacent perturbed state being assumed to take place without unloading.

We investigate the statistical mechanics of modified rubber-like network structure in order to make a molecular theoretical explanation of viscoelastic behaviors displayed by amorphous high-polymeric substances. In the former part of this paper we discuss a theory in which the relaxation mechanism is ascribed bythe breakage and reformation of junctions, and in the latter part another theory is treated making use of the concept of slipping of chains for the relaxation mechanism. It is shown that the above two theories have the similar results. Owing to this fact, we can conclude that the concept of slipping of chains is essentially equivalent to that of the rearrangement of chains caused by the breakage and reformation of junctions. Furthermore, the stress-strain-time relations of our models are essentially equivalent to that of the so-called generalized Maxwell model in the phenomenological theory of viscoelasticity. §

The propagation of a sinusoidal wave of small amplitude in an initially isotropic viscoelastic solid, which is subjected to a static pure homogeneous deformation, is discussed. The secular equation for a plane wave propagating an arbitrary direction with respect to the principal directions of the static deformation is derived. The complex slownesses corresponding to propagation along the principal axes of the static strain are found. Relations are obtained between these and the principal stresses associated with the static deformation, which generalize earlier formulas obtained by Ericksen for elastic materials.

With the use of a suitable representation for the free energy, explicit forms of the constitutive relations, the dissipation function and related thermodynamical restrictions are derived for infinitesimal non-isothermal deformation of linear viscoelastic solids. Some remarks on further restrictions, consistent with the dissipation inequality, are included for isotropic materials.

The discontinuity of terminal velocity versus bubble volume previously reported as taking place in viscoelastic liquids, was studied experimentally and theoretically. A comparison of terminal velocity data for gas bubbles and glass spheres in aqueous polymer (Separan-AP30) solutions provides strong evidence that the discontinuity results from a change in interfacial conditions from “no-slip” to free shear. A numerical solution of the creeping flow equations using an empirical curve fit for viscosity versus shear-rate indicates that only a fraction of the experimentally observed velocity jump is due to the shear dependence of the viscosity alone (neglecting elastic effects). However, simple qualitative arguments seem to suggest that a relatively modest elastic contribution to the force balance on the bubble would be sufficient to account for the experimental observations. On a etudie experimentalement et theoriquement la relation entre la discontinuite de la vitesse limite et le volume des bulles, laquelle, comme des rapports precedents le pretendent, se produirait dans des liquides visco-elastiques. En comparant les resultats de vitesse limite dans le cas de bulles gazeuses et de spheres de verre dans des solutions aqueuses de polymere (Separan-AP30), on a obtenu une forte preuve que la discontinuite resulte d'un changement dans les conditions interfaciales qui s'exprime par le passage d'une “absence de glissement” a un cisaillement libre. Une solution numerique des equations de l'ecoulement de fluage (creeping), qu'on a obtenue en utilisant une courbe empirique qui s'adaptait a la relation entre la viscosite et la vitesse de cisaillement, indique que seule une fraction du saut de vitesse observe experimentalement est due a la dependance du cisaillement de la viscosite seule (sans consideration des effets elastiques). Toutefois, un simple raisonnement qualitatif semble indiquer qu'il suffirait d'une contribution relativement faible de l'elasticite a l'equilibre de force sur la bulle pour expliquer les observations experimentales.

Equations are constructed for determining the relative strains and stresses in an infinite plate of viscoelastic material exposed to symmetrical heating. Possible methods of solving these equations are examined in relation to various models.

The abnormally high resistance of viscoelastic fluids to sudden deformations and to stretching may be expected to cause the structure of turbulent velocity fields in these systems to differ appreciably from those of Newtonian fluids. An analysis based on these considerations is presented and supported using friction factor data for three concentrations of a water soluble polymer. Pressure losses predicted by use of this correlation are within 15% or less of the experimental values, but as all dimensionless groups which influence the results were not varied independently, the correlation developed may not be universally applicable to all fluids. Suggestions for further and more more specific analyses are made. On peut s'attendre a ce que la resistance extraordinairement elevee des fluides visco-elastiques aux deformations soudaines et a la distension puisse produire dans ces systemes une structure des champs de velocite turbulente bien differente de celle des champs des fluides Newtoniens. On presente une analyse basee sur ces considerations, laquelle est corroboree par les resultats obtenus pour les facteurs de friction dans le cas de trois concentrations d'un polymere soluble dans l'eau. Les pertes de pression predites par l'emploi de cette correlation n'excedent pas 15% des valeurs experimentales, mais, etant donne qu'on n'a pas fait varier independamment tous les groupes sans dimensions qui influencent les resultats, la correlation qu'on a mise au point ne peut s'appliquer a tous les fluides. On fait des suggestions pour faire des analyses plus poussees et plus specifiques.

A molecular theoretical investigation of the viscoelastic properties of amorphous polymeric substances in the range of long relaxation time has been made using the temporarily cross-linked network-structure-model. The origin of the energy dissipation is ascribed to the slipping of chains and the change of number of chains induced by deformation.When the network structure is deformed each chain will slip since the network structure does not return to its initial state when the deformation is taken off. As the chains slip the deformations of chains do not coinside with that of the network structure. Then we introduce the two kinds of deformation tensor, one is the observable deformation tensor of the netwok structure and the other is the inner deformation tensor of each chain. Assuming that the deformation of all chains can be characterized by a single average inner deformation tensor, we obtain Eq. (7) which expresses the relation between the observable and the inner deformation tensors.We can also obtain Eq. (31) expressing the change of the number of chains which depends upon the time and the deformation. Eq. (31) has the same form to that of the chemical reaction as expected by the theory of rate process.The strain free energy of each chain is assumed to depend only on the end-to-end distance of each chain and the free energy of whole system is assumed to be given by the sum of the said strain free energy of each chain and the free energy depending on the density of segments which corresponds to the inner pressure of the system.Accordingly the stress is obtained by Eq. (17) as the function of the inner deformation tensor. Using the relation between the observable and the inner deformation tensors, Eq. (7), we obtain the stress- time-deformation relation Eq. (20), which is the natural generalization of equation of Maxwell model. In small deformation the tensors are automatically symmetrized and we obtain Eq. (33) which is just the same formula with that of Maxwell model. As we assume the average inner deformation tensor, namely the average slipping of each chain we have only a single relaxation time which is the average relaxation time in the range of box type relaxation time spectrum. In this formalism, as the average inner deformation is assumed we can not explain that the system must have the box type relaxation time spectrum in the range of the long relaxation time.The energy dissipation expressed by Eq. (37) has two terms, one is due to the slipping of chains and the other for the change of the number of chains.Some applications of the theory are made in last section. We define the stationary state by the state of constant inner deformation tensor. In stationary state the number of chains are kept constant and the origin of the energy dissipation is ascribed to the slipping of chains only. First, we treat the stationary simple elongation, the stress and the energy of dissipation are expressed by Eq. (42) and Eq. (43) respectively and the tensile viscosity coefficient η* is expressed by Eq. (45) in case of no volume change. Secondly, we treat the stationary simple shear, in this case the normal stress effect appears as is shown by Eq. (47) and we have Eq. (49) as the relation between the tensile and the shear viscosity coefficients, which is known as Trouton's law. Lastly we treat the instantaneous simple elongation, as is shown by Eq. (50) and Eq. (51) the stress shows nearly exponential decay since the number of chains changes slowly with time. The inner deformation decreases monotonously and tends to return to its initial state, on the other hand the number of chains decreases at first and passes a minimum value then increases again and tends to approach to the equilibrium value.

The theory of irreversible thermodynamics of continuous media with fading memory is used to formulate general constitutive equations of finite linear viscoelasticity. Specific forms for transversely isotropic and orthotropic media based on both the mechanical and thermodynamic theories are found.

Some current simple integral theories of the Lodge type are compared in simple shearing, small sinusoidal shearing, combined simple and sinusoidal shearing, cessation and start of simple shearing, finite amplitude sinusoidal shearing, and simple elongational motions. Of the simpler models only the recently proposed network‐rupture theory shows a realistic response in elongational flows; in the other flows it gives a reasonable prediction of experimental results from poly‐isobutylene‐cetane solutions.

定常状態におけるBarus効果を山本の3次元粘弾性論 (J. Phys. Soc. Japan, 19, 739 (1964)) によって解析した。山本の理論で導入された内部弾性変位テソソルが回復可能なひずみと同一であることを示した。単純ずり流動において, 内部弾性変位テンソルはずり変形のみでなく伸長変形を受けている。この伸長変形がBarus効果の原因であるとして, Barus効果の大さきを表わす式を導いた。ポリスチレソ溶融体について実験を行ない, この式を検証した。

Closed‐form solutions to the propagation of spherical waves from an internal steady oscillating pressure source as well as to the transient response of a viscoelastic medium to a pressure impulse are presented. The assumption that the loss factors, quantities related to the dissipative property of a viscoelastic medium, are independent of frequency is used here for investigating the wave‐propagation phenomenon due to the action of an impulse. Numerical examples for the displacement and stress functions have been used to illustrate the damping effect of a viscoelastic medium by comparing with its special case—the elastic medium. It is found that this effect shows its significance with increased traveling distance. Also, in the steady‐oscillation problems it is of importance when the vibration frequency is high, while in the transient case it causes the decrease in the peak amplitude and also a time lag in the arrival of the wavefront.