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Mechanical wave in tube – theory

The quantitative description of mechanical wave propagation in tubes is remarkably important for biomechanics (e.g. analysis of pulse wave propagation in cardiovascular system) and also for industry (e.g. analysis of wave propagation in pipelines). Nevertheless, the comprehensive theory of mechanical wave propagation in tube is not currently at hand.

The solution of problems connected with dynamic mechanical behavior of tubes requires firstly suggestion of an adequate mechanical model. Such a model must take into consideration continuously distributed (inertial, elastic and viscose) parameters in entire system. The crucial next step is the derivation of constitutive equations for mechanical wave propagation in the system.

From the mechanical point of view, any elastic or viscoelastic tube filled with viscose liquid is highly complicated system. Mechanical behavior of such structures depends on theirs geometry and on inertial, elastic and viscose forces in wall as well as in liquid inside. Thus we face a considerably complicated task. Helpful may be the sophisticated methodology of derivation of telegraph equations for electric waves in cables applied by W Thompson more than hundred years ago, as the fundamental mechanical and electrical equations are analogical.

Knowledge of constitutive equations of mechanical wave propagation in tubes enables calculations of speed of wave and damping of wave. Further, it enables calculation of wave impedance and reflections of wave. Also, the strains and stresses in tubes walls may be determined.

The constitutive equations can be also used for the inverse problems solutions. Typically, the measurement of the propagation of pulse wave in cardiovascular system may be used for the estimation of viscoelastic parameters changes in arterial walls.

1. Premises, goals and principles
This analysis is concerned with mechanical waves in a cylindrical tube with thin elastic or viscoelastic wall filled with a viscose liquid of defined density.

The tube is considered to be composed of an infinite number of segments (Fig. 1). The geometry and the corresponding symbols are in Figure 2. We will analyze the situation when forces (pressures) and deformations (displacements) are changing in time domain.

The analysis aims to provide formulae for calculation of speed of wave propagation, damping of waves, waves mechanical impedance, reflections of waves and conditions for mechanical matching of tubes and tube endings.


Fig.1. Segmentation of tube

Fig.2. Geometry of tube

2. Force equilibrium in tube (Fig. 3)

Fig.3. Force equilibrium in segment of length x 

2.1. Forces in liquid inside tube (Fig. 4)

Fig.4. Illustration to origin of forces inside tube



Inertial forces

As the wall of tube is thin, mass of wall is negligible with respect to mass of liquid inside.

The geometry of segments is changing during deformation (Fig.4); the position of the center of inertia is thus also changing. Inertial forces obey Newton´s law of inertia.

If the liquid is homogenous, it holds

 and also          

where YM is the position of the center of inertia (radius), m is the mass of segment, ρ is the density, V is the volume of segment.

If the tube is circular, it holds


In case of unit length tube, the inertial force Fm is


The relationships between forces and deformations in rheological models (see e.g. Fig. 10) are as follows


where M is the inertial coefficient.

In case of cylindrical tube of unit geometry, for the inertial coefficient M thus it holds



Radial viscose forces (Fig. 5)

The geometry of segments is changing during deformation. The relative movement of layers in liquid determines radial viscose forces. The tangential stress obeys Newton´s law of viscosity.


where η is the viscosity, vv is the speed of deformation of layer, z is the distance of layers.


Fig.5. Illustration to origin of viscose forces in liquid

Fig.6. Illustration to calculation of viscose forces in liquid


The speed of outer layer (on perimeter of cylinder) is


The speed of layer in the center (in the axis of cylinder) is zero.

The speed gradient is


where yL is the speed of layer.

The gradient is the same for any position of layers (Fig. 6).

Viscose forces FV straining layers


where A is the surface of layer.

The position of center of viscose forces in cylinder is

 The viscose forces in segment are thus


The viscose force FV in cylindrical segment of unit length is


The equation for relationships between forces and deformations in rheological models (see e.g. Fig. 9) is as follows


where N is the Newton (viscose) coefficient.

For the Newton coefficient N thus it holds, in case of unit cylindrical segment


Axial viscose forces

The influx and the outflow of liquid determine axial viscose forces. In short segment these forces are negligible.


2.2. Forces in wall (see Fig. 7 an Fig. 8)



Fig.7. Geometry of deformation of wall - detail


Fig.8. Shear force in deformed segment


Elastic shear stress in the wall


where G is the modulus of  shear stress, A is the area of wall cross section.

Elastic tangent force in the wall


Elastic tangent force in the wall of unit segment



Hooke coefficient in the wall of unit segment


Viscose tangent stress


where ηP is the viscosity of the wall.

Viscose tangent force


The viscose tangent force in unit segment


Newton coefficient NP in unit segment



3. Constitutive wave equations for Maxwell model of the wall

3.1. Rheological scheme of tube for Maxwell model of wall

 Fig.9. Rheological scheme for Maxwell model of wall 


Applied formulae
For simplicity, we use the speed of deformation (v) instead the deformation (y) in following formulae.



Derivation of constitutive equations

The movement of liquid leads to decline of force on wall as follows


Thus it holds


The movement of wall leads to decline of speed of deformation


Thus it holds



3. 2. Wave equations

From equations (9) and (10) the constitutive (wave) equations follows



3. 3. Speed of wave propagation
The solution of equations (11) and (12) leads to the formula for calculation of speed of wave







Example 1 – Speed of wave propagation on frequency


Tab. E1.The variable and parameters of tube


Fig. E1. Dependency of speed of wave propagation on frequency for viscoelastic tube 


Example 2 – Speed of wave propagation on radius


Tab. E2.The variable and parameters of tube

Fig.E2. Dependency of speed of wave propagation on wall radius for viscoelastic tube 


Example 3 – Speed of wave propagation on wall thickness

Tab. E3.The variable and parameters of tube


Fig.E3. Dependency of speed of wave propagation on wall thickness for viscoelastic tube


Example 4 – Speed of wave propagation on wall shear modulus

Tab. E4.The variable and parameters of tube

Fig.E4. Dependency of speed of wave propagation on shear modulus of wall for viscoelastic tube 


3.4. Damping of wave








Example 5 –Damping of wave on tube length

Tab. E5.Parameters of tube


Fig.E5. Dependency of damping on tube length for viscoelastic tube


3. 5. Wave mechanical impedance

3.6. Wavefront of non-harmonic wave

Harmonic analysis

The equations (13 - 17) hold for simple harmonic (sinusoidal) waves. The harmonic analysis enables extend theirs applicability on all periodic functions.

Methodology of harmonic analysis consists in the representation of periodic functions as the superposition of harmonic functions (harmonics). As superposition principle holds for solutions of a linear homogeneous ordinary differential equation, any periodic function may be approximate as a Fourier series. A Fourier series is an approximation of a periodic function in terms of an infinite sum of sines and cosines. Then it is possible to recombine harmonics to obtain the solution to the original problem.

Fourier series

According definition, for the Fourier series the following holds


where f(x) is the periodic function, T is the period, n is the integer.

For coefficients a resp. b in equation (18) it holds:

Thus in time domain it holds:


where t is the time, ω is the angular speed.

Example 6 –Propagation of non-harmonic wave


Suppose the viscoelastic tube of following parameters:


Tab. E6.Parameters of tube


Suppose further that wavefront of pulse wave at point A (see Fig. E6) is according Fig. E7.

Fig.E6. Propagation of wave in tube


The curve in Figure E7 is the sum of 3 harmonics.

In general, for harmonics it holds: where f is the frequency of the first harmonic.

For the curve in Figure E7 it holds

For the second harmonic it holds:

For the third harmonic it holds:



Wavefront of pulse wave at point A (x = 0) is the sum of the harmonics at point A (see Fig. E7)

According formulae (13 and (14) is the speed of wave propagation for all harmonics. The phase shifts of harmonics at point B are thus possible to calculate.

According formulae (15 and (16), damping of the harmonics at point B may be calculated.

Consequently, parameters of harmonics at point B are possible to be obtained.

Wavefront of pulse wave at point B (x = 0.5 m) is the sum of the harmonics at point B (see Fig. E8)


               Fig. E7. Wavefront at point A                                 Fig. E8. Wavefront at point B


The comparison of wavefronts at A and at B is shown in Figure E9.

Fig. E9. Comparison of wavefronts (full line wavefront at A, dotted line at B)


Comment: In reality, for calculation computer with adequate software is necessary to use.


4. Constitutive wave equations for Voigt model of wall

4. 1. Rheological scheme of tube for Voigt model of wall


Fig.10. Rheological scheme for Voigt model of wall


4.2. Transformation of Voigt model to Maxwell model


In many situations, rheological parameters (HV, NPV) of wall are available for Voigt model (e.g. if measurement is based on torsion oscillations). In this case, the Voigt model is appropriate. Nevertheless, the constitutive equations derived for Maxwell model are also applicable, provided rheological parameters of Voigt model are transformed into Maxwell ones (H, NP). The transforming equations are as follows:

The previous formulae are applicable if this holds: k > 10 which is usually fulfilled.


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Mechanical wave in tube –theory



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