Generalization of Wave Motions and Application to Porous Media

The examination of wave motions is traditionally based on the differential equation of D’Alambert, the solution of which describes the motion along a single dimension, while its bidimensional extension takes on the concept of plane waves. Considering these elements and/or limits, the research is divided into two parts: in the first are written the differential equations relating on the conditions two/three-dimensional for which the exact solutions are found; in the second the concepts are extended to the analysis of the propagation of wave motions in porous media both artificial and natural. In the end the work is completed by a series of tests, which show the high reliability of the physical-mathematical models proposed.


Introduction
The mathematical theory, that analysed the propagation of elastic impulses, has its origin from the mechanics of the wave motions the latter are defined as the integral of a force generated naturally (example: earthquake) or artificially (example: geophysical prospecting; vehicular traffic), time-averaged: Also, the wave nature assumed for the analysis of the phenomenon implies the propagations time of the impulse in accordance with the relationships between the dynamic elastic modules-longitudinal (E d ), tangential (G d ) and volumetric (K d )-through the dynamic Poisson's ratio (ν d ) in turn dependent on the speed of compressions waves (v S ): 0.5 1 1 Known these elements (Figure 1(a)), the analysis of the vibratory phenomenon is attributable to the study of elastic behaviour of the individual waves, that exploit the properties of sine and cosine functions and are repeated periodically in the initial characteristics of the motion, expressed in terms of amplitude A, frequency f and phase costant δ (Figure 1(b)); other fundamental parameters of the motion are: the period: the angular pulsation (or angular frequency): And the number of the wave: In turn the Equations ( 4) and ( 5) can be combined to derive the expression of velocity: Given that the real motions (example: earthquake) can be simulated by the addition of a suitable number of monochromatic waves (Figure 1(b)) each characterized by its own amplitude, frequency and phase (Figure 2).
Under these conditions, a mathematical model of general validity and its extension to the porous media will be described in the continuation of research.

The 1-D Equation of D'Alambert
The analysis of vibratory motions originates from the D'Alembert's differential equation [4,5] which analyze the propagation of monodimensional waves in function of the vectorial component u of the movement (variable between A = 0 and A = 1-Figure 1 The solution of this problem, provided by the author, has been perfected by Euler [6] in the following form expressed in relation to of the elements described by the Equations ( 3)-( 5): In fact, calculated the second derivatives of the Equation (8) as a function of space and time: you can replace the Equations (9b), (9d) and ( 6) in the Equation ( 7) to obtain the identity: Alternatively, can be introduced an The extension of D'Alem mensional case is an unsolved problem that has required the introduction of isotropic homogeneous media in which the motions are propagated as circular waves (Fig- ure 3); at considerable distances from the source, small portions of the circular waves can be approximated by plane waves which propagate in a straight line according to directions normal to the wave fronts represented by the rays; in this way the analysis of the phenomenon is facilitated, as limited to the study of the suns rays having evidently rectilinear trajectories in homogeneous media and curvilinear in those non-homogeneous.
Each monochromatic components of a vib which propagates in a homogeneous medium travels at the same speed, in turn, dependent on the angular fre- q this case, indicated with ρ the density of the medium, it proves [7] that the following equations are valid, for elastic waves compressional (P) and cutting (S): Besides, it is also the relationship v P > v S for the Equatio

General Equation of the Wave Motion
deof the equation of motion will be of the type: n (2a).In inhomogeneous media, for which the speed varies along the path, each frequency component of the vibratory motion is instead equipped with its own speed, such as to arrive to a generic receiver at different times.The described phenomenon, known as the dispersion of the phases, implies that in the means of non-dispersive (homogeneous) the vibratory motion arrives at the receivers with the same initial shape; on the contrary, in dispersive media is possible to identify a group velocity (if you choose to characterize the group through the analysis of the maximum amplitude) and a phase velocity of each component, so that a signal emitted by the source arrives distorted at the receiver.Obviously, means in the two non-dispersive group velocity and phase are.
Denote by s(x,y,z) the displacement vector in space, fined with respect to a reference system coordinated, having components u, v e w; consequently, if the Equation (7) describes the motion of a disturbance which is propagated only in the direction x, the generalized form

The Equation of Wave Motion in Space nded to
The Equation ( 12), expressed in matrix and exte form, in the case of the means non-dispersive, reduces the form: in which v j identifies the column vector of th Then, the solution can still be expressed using the reedi Equation ( 8) on condition to take account of the th mensionality of the phenomenon: By repeating the procedure previously seen wit tion (7) of the motion 1D, you get in sequence: In Equation ( 16), replace the components of ity, expressed according to Equation ( 6), finally to prove the validity of the proposed solution: the veloc- The latter also demonstrates that the fundamental period of the wave (and consequently the frequency fundamental) corresponds to the period (a the individual components.ition v z > v y = v x we are seein ng: nd frequency) of In conclusion, for orthotropic media and from the structure of Equation ( 14), which is easily seen that the waves propagate according ellipsoids scalene (Figure 4 Place also: we come to the equation of an ellipsoid: e Laplacian content in Equation general (13a)- 2 s-is the quadratic form of the divergence of the vector functions that, for positives values in a generic point P denotes the existence of an flow in the neighborhood of P; so, to the condition imposed v z > v y > v x corresponds w > v > u which describes dent of temspects the law of conservation of energy.

(24)
On the other hand, th outgoing an ellipsoid.The discussed model is indepen perature of the system assumed adiabatic, that re

The Equation of Wave Motion on a Plane
In case of the plane x z the Equation (13b) is reduced on the form: whose solution is a special case of the Equation ( 14): Again, the procedure is applied now known we arr at the identity relation: which proves the accuracy of the solution according to l or circular forms.

The Propagation of Waves in
propagation in porous media has been addressed previously in [8] from the law of mass balance for two-phase media now extended to three-the propagation of the waves on a plane with elliptica

Porous Media
The problem of waves Copyright © 2013 SciRes.GM phase drives: aseous (ρ a ) capable of describing a structerstitial interonnected with the speed, through In other words, must be considered as a porou consists of three continuous elastic means that occ sa Th same impulse, through the solid skeleton and phases fluid and gases contained in the in this way we obtain a system of three equations for the pr In the Equation (28) appear porosity (n), the degree of saturation (S) and the density of the solid phase (ρ s ), fluida (ρ w ) and g ture consisting of a skeleton solid with in connected pores between them (Figure 5).
Since the mass is c the Equations ( 11), (2a) and (2b) shows a dependence of the type:

 
, , , , , , , s solid upy the me region of space and which interact between them carving up the propagation of the same elastic pulse.

Speed of Compression Waves in
Porous Media e problem outlined can be mathematically simplified if one analyzes the propagation of three waves decoupled from the pores (Figure 6); opagation of compression waves in anisotropic porous media:

   
, , 1 1 Analyzing the Equation (30) analyzing the equation it turns out that the first bracket identifies the anisotropic sional aves; the following brackets indicate instead the components related to the isotropic liquid and vapor phases through the corresponding elastic moduli volumetric (K w e K a ).
The symbol E * d,i identifies a law of variation longitudinal elastic modulus dynamic, determined from [11] and [12] and based on the introduction of factors of co component, relative to the propagation through the solid skeleton, linked with only the elastic modulus being able to neglect the effect of the transverse contraction [10] for having brought the problem to three one-dimen w of the ntraction seen in [13]: In the Equation (31) appear the parameterwhich represents a material constant and must be determinated experimentally, meanwhile the value of ρ, presents in the Equations (30) changes as a function of the porosity according to the law (28) of the mass balance.
Once defined the general structure of the equations, the problem can be further simplified if one considers that the speed of propagation of the impulses in water and air are approximately v w ≈ 1500 m/s and v a ≈ 340 m/s which would alter the Equations (30): 1 5 0   1 1500 Using the Figure 7 as a refer co l symmetry and, accordingly, the related formulations can be obtained from Equations (32) purified from the isotropic components .ence it is seen that the mponents of the shear waves are 6, of which three independent mathematica 1 1500 340 1 In conclusion, the same way as seen with the tions ( 13) ÷ ( 19) related to the generalized waves, the Equations ( 30) and (32) describe scalene ellipsoidal waves, ellipsoid of revolution or spherical with respect to the relationships between the elastic modulus of the soil skeleton that affect the dynamic response the anisotropic component.

Velocity of the Shear Wave on Porous Media
In the determination of the speed of shear waves is necessary to consider: 1) the physical impossibili to propagate in fluids; 2) their polarization on mutually orthogonal planes (waves S V e S H ); 3) their development in Equatheory of of ty of waves all directions with respect to the source; 4) to Equation (11b); 5) a law of variation of Poisson's ratio: Finally, the Equations (2a), ( 31) and (34) can be combined with each other: .
Equations ( 35), compared with th demonstrate the existence of time dif of the shear wave-compared to compression-which must necessarily increase with the increase of the distance from the source.

Tests of Model Validation
(35c) e Equations (32), ferences of arrival The graph of Figure 8 illustrates the experimental results, conducted on both dry rocks that saturated water, expressed in terms of rate of change of P-waves as a 3 3 3  function of porosity.In detail, as reported in [15], it is possible to note that the increase due to saturation is very evident both in the lava issued from Etna (ET; near Catania, Italy) than in the Campi Flegrei (CF, Italy near Naples) both characterized by a low overall porosity of approximately in the range n = 5% -20%, on the contrary, in tuffs and ignimbrites (having a porosity in the range n = 30% -60%), the increase of speed found in the saturated state is significantly higher than the lava rocks.At the same graph have been superimposed the theoretical laws of variation of P-waves velocity, and their tabulated data, calculated using Equations (28), (31) and (32c); as can be noted from the same theoretical model fails to approximate, for  = 2.126 introduced in the calculation of , the speed difference between the dry and saturated ro s with the increase of the porosity.
Final interpretation of the results with the theoretical model must be considered that the theoretical behavior of lava rocks (lavas of Etna and Campi Flegrei) and pyroclastics (tuffs and ignimbrites) was standardized in terms composition (Basalts and Trachytes alkaline in the first case, the variable in the second case [16]) induces to assume the existence of different starting values of the dynamic elastic modulus and density; in other words, the same experimental results are not comparable to each other if not qualitatively, as indeed demonstrated by the dispersion of its employee data from the variation of the physical properties quoted.
The second test was performed using the experimental data contained in [17], relating to the correlations v P -n and v S -n measured in samples of alumina ceramic (Al 2 O 3 ) known for the acid resistance and low thermal conductivity so as to be used as a catalyst in the chemical industry and as a graft material in the biomedical; therefore, with reference to Figure 9(a), it turns out that the theoretical model describes with high accuracy the performance of the P-waves in the ceramic (per  = 1.46) while S waves are sufficiently approximated by n < 15% and approximated for n > 15%.
The Copyright © 2013 SciRes.GM in the test: 1) the P-waves velocity for saturated samples is slightly lower than that responsible for the dry sampl respectively are linear and non-linea odulus.The last test was conducted using the experimental results contained in [18] and related to the propagation of P-waves in a generic group of rocks not necessarily related to each other (Figure 10); Also in this case, the theoretical model is able to approximate (per  = 2.381) the dynamic behavior generally using the same starting values of the density and the dynamic elastic modulus.

Conclusions
The analysis of wave motions takes origin by the differential equation of D'Alembert [4,5], the solution of which, based on the properties of the trigonometric sine and cosine functions, simulates the propagation of an elastic wave in a continuous medium 1-dimensional, and its ex-ts that make up th these equations are valid for continuous media, w e existence of three continuous with interparticle voids paths from different types of rocks and in samples of alumina ceramics.
Ultimately, the tests showed that the proposed models fail to accurately simulate the behavior of a particular material or group of materials that share the same origin while further applications may be later developed: 1) in es n < 60%; 2) for n > 60% the P-waves velocity of saturated samples becomes greater than that of the dry samples while the model manifests a marked non-linearity; 3) for n = 100% the speed of the saturated samples is reduced to that of water (point A) while that of the samples dried in the air (point B).Finally, as expected, n = 100% for the S-waves velocity vanishes.
The trends of the P-waves velocity associated to the fields 0 < n < 0.6 and 0.6 < n < 1depend on the laws of variation adopted for ρ and E * d that, described by Equations (28) and (31), r; which means that, while the density varies linearly for 0 < n < 1 the elastic modulus assumes a-heating an approximately linear for n < 0.6 and not linear for n > 0.6.Consequently, the increase of speed setting attributable to fluid phases fails to compensate-for n < 0.6the effects adducts by variations in density and elastic m tension to the floor is brought back to the simplified concept of circular waves reduced to plane waves.
After a brief review of the main elemen e rolling waves of D'Alembert, was written the differential equation governing the wave motion in space (generalized theory) to follow, it was found that the exact solution shows that the waves take the form of ellipsoids scalene in orthotropic means which, in turn, reduce to the ellipsoids of revolution in the means transversely isotropic and spherical waves in isotropic media; in the same way, which step consequent, the 3D equation has been reduced to the 2D field whose solution leads to waves having elliptical shapes that are reduced to circular waves.But be aware that hich can be described by assigning them appropriate scalar and vector fields defined by means of functions regular and continuous over the entire domain configuration.
Which next step, given the nature of the particle actually real media (with particular reference to geomaterial), the equations have been applied to porous media, th description of which is based on concepts known in Geo technical that predict the media (solid skeleton water and air) that interact between them carving up the propagation of the same impulse elastic divided into the components of compression and shear.Finally, the search has been completed with some tests based on known experimental data relating to the propagation of waves in Copyright © 2013 SciRes.GM the field seismological, considered that the Equations (30) reduce to Equations (11a) for means infinitely porous, as in

Figure 1 .
Figure 1.(a) The dynamic modules belong to the initial phase of the stress-strain curves, relevant to the field of small deformations, the limit of which is set at ε = 1% [1,2]: identification of motion parameters of two identical periodic waves, but out of phase between their (Δδ = π/2).

Figure 2 .
Figure 2. Schematic reproduction of an artificial impulse and of the sixteen monochromatics components [3].

Figure 3 .
Figure 3.In homogeneous isotropic media the P and (a)); similarly, for the cond g waves with the form of ellipsoids of revolution (Figure 4(b)) while the condition v z = v y = v x leads to the development of spherical waves (Figure 4(c)).In this regard, we can rewrite Equation (16) creating the conditions:

Figure 6 .
Figure 6.The basic hypothesis involves the schematization of porous media in three continuous elastic means and in this way an impulse can be decoupled into three components which propagate without mutual interaction.

Figure 5 .
Figure 5.Typical structure of soils and identification of the basic parameters of porous media in general [9].

Figure 8 .
Figure 8.Comparison between the experimental data of the speed of samples of dry rocks and waterlogged [15] and the theoretical results predicted by the model.

Figure 9 (Figure 9 .
Figure 9. (a) Comparison between the experimental data of the speed of samples of alumina ceramics [17] and the theoretical results predicted by the model; (b) Extension of the model to the entire field of porosity.

Figure 10 .
Figure 10.Comparison between the experimental velocity of P diction of the proposed model is extended to the field of poros -waves of a generic group of rocks [18] and the theoretical preity.