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(b)) in a media with speed v_x:

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 only Equations (9b) and (9d) to obtain the Equation (7).

The extension of D’Alembert equation to the two-dimensional case is an unsolved problem that has required the introduction of isotropic homogeneous media in which the motions are propagated as circular waves (Figure 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 vibratory motion which propagates in a homogeneous medium travels at the same speed, in turn, dependent on the angular fre-

quency and the wave number through Equation (6); in 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 Equation (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.

The Equation (12), expressed in matrix and extended form, in the case of the means non-dispersive, reduces to the form:

in which v_j identifies the column vector of the speed.

Then, the solution can still be expressed using the Equation (8) on condition to take account of the threedimensionality of the phenomenon:

By repeating the procedure previously seen with Equation (7) of the motion 1D, you get in sequence:

In Equation (16), replace the components of the velocity, expressed according to Equation (6), finally to prove the validity of the proposed solution:

The latter also demonstrates that the fundamental period of the wave (and consequently the frequency fundamental) corresponds to the period (and frequency) of the individual components.

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(a)); similarly, for the condition v_z > v_y = v_x we are seeing 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:

obtaining:

Place also:

you get:

Finally, putting too:

we come to the equation of an ellipsoid:

On the other hand, the Laplacian content in Equation general (13a)—Ѳs—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 outgoing flow in the neighborhood of P; so, to the condition imposed v_z > v_y > v_x corresponds w > v > u which describes an ellipsoid. The discussed model is independent of temperature of the system assumed adiabatic, that respects the law of conservation of energy.

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 arrive at the identity relation:

which proves the accuracy of the solution according to the propagation of the waves on a plane with elliptical or circular forms.

The problem outlined can be mathematically simplified if one analyzes the propagation of three waves decoupled from the same impulse, through the solid skeleton and phases fluid and gases contained in the pores (Figure 6); in this way we obtain a system of three equations for the propagation of compression waves in anisotropic porous media:

Analyzing the Equation (30) analyzing the equation it turns out that the first bracket identifies the anisotropic 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-dimensional waves; 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 of the longitudinal elastic modulus dynamic, determined from [11] and [12] and based on the introduction of factors of contraction seen in [13]:

In the Equation (31) appear the parameterawhich 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):

In conclusion, the same way as seen with the Equations (13) ÷ (19) related to the generalized theory of 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 of the anisotropic component.

In the determination of the speed of shear waves is necessary to consider: 1) the physical impossibility of waves to propagate in fluids; 2) their polarization on mutually orthogonal planes (waves S_V e S_H); 3) their development in 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:

Using the Figure 7 as a reference it is seen that the components of the shear waves are 6, of which three independent mathematical symmetry and, accordingly, the related formulations can be obtained from Equations (32) purified from the isotropic components:

Equations (35), compared with the Equations (32), demonstrate the existence of time differences of arrival of the shear wave—compared to compression—which must necessarily increase with the increase of the distance from the source.

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

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 a = 2.126 introduced in the calculation of, the speed difference between the dry and saturated rocks 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 of ρ and of, although the different mineralogical 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 (Al₂O₃) 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 a = 1.46) while S waves are sufficiently approximated by n < 15% and approximated for n > 15%.

The Figure 9(b), in turn, illustrates the extension of the theoretical model to the entire range of variability of porosity, between a material infinitely compact (n = 0%) and infinitely porous (n = 100%), expressed in terms of P-waves velocity for dry samples and saturated and Swaves velocity. In this case the elements are key findings

in the test: 1) the P-waves velocity for saturated samples is slightly lower than that responsible for the dry samples 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), respectively are linear and non-linear; 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 < 0.6— the effects adducts by variations in density and elastic modulus.

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 a = 2.381) the dynamic behavior generally using the same starting values of the density and the dynamic elastic modulus.