Generalization of Wave Motions and Application to Porous Media

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1 Geomaterials Publishe Online July 03 ( Generaliation o Wave Motions an Application to Porous Meia Romolo Di Francesco Wiar Technology Teramo Italy romolo.irancesco@gmail.com Receive May 03; revise June 03; accepte July 03 Copyright 03 Romolo Di Francesco. This is an open access article istribute uner the Creative Commons Attribution License which permits unrestricte use istribution an reprouction in any meium provie the original work is properly cite. ABSTRACT The eamination o wave motions is traitionally base on the ierential equation o D Alambert the solution o which escribes the motion along a single imension while its biimensional etension takes on the concept o plane waves. Consiering these elements an/or limits the research is ivie into two parts: in the irst are written the ierential equations relating on the conitions two/three-imensional or which the eact solutions are oun; in the secon the concepts are etene to the analysis o the propagation o wave motions in porous meia both artiicial an natural. In the en the work is complete by a series o tests which show the high reliability o the physical-mathematical moels propose. Keywors: D Alambert; Wave Motions; Propagation One/Two/Three-Dimensional; Porous Meia. Introuction The mathematical theory that analyse the propagation o elastic impulses has its origin rom the mechanics o the wave motions the latter are eine as the integral o a orce generate naturally (eample: earthquake) or artiicially (eample: geophysical prospecting; vehicular traic) time-average: t yt t I F t. () Also the wave nature assume or the analysis o the phenomenon implies the propagations time o the impulse in accorance with the relationships between the 0 ynamic elastic moules longituinal (E ) tangential (G ) an volumetric (K ) through the ynamic Poisson s ratio (ν ) in turn epenent on the spee o compressions waves (v S ): E G (a) K E 3 vp (b) v P 0.5. (c) vs vs Known these elements (Figure (a)) the analysis o Figure. (a) The ynamic moules belong to the initial phase o the stress-strain curves relevant to the iel o small eormations the limit o which is set at ε = % []: ientiication o motion parameters o two ientical perioic waves but out o phase between their (Δδ = π/). Copyright 03 SciRes.

2 R. DI FRANCESCO the vibratory phenomenon is attributable to the stuy o elastic behaviour o the iniviual waves that eploit the properties o sine an cosine unctions an are repeate perioically in the initial characteristics o the motion epresse in terms o amplitue A requency an phase costant δ (Figure (b)); other unamental parameters o the motion are: the perio: T (3) the angular pulsation (or angular requency): (4) T An the number o the wave: k. (5) In turn the Equations (4) an (5) can be combine to erive the epression o velocity: v. (6) k T T Given that the real motions (eample: earthquake) can be simulate by the aition o a suitable number o monochromatic waves (Figure (b)) each characterie by its own amplitue requency an phase (Figure ). Uner these conitions a mathematical moel o general valiity an its etension to the porous meia will be escribe in the continuation o research... The -D Equation o D Alambert The analysis o vibratory motions originates rom the D Alembert s ierential equation [45] which analye the propagation o monoimensional waves in unction o the vectorial component u o the movement (variable between A = 0 an A = Figure (b)) in a meia with spee v : u v u. (7) t The solution o this problem provie by the author has been perecte by Euler [6] in the ollowing orm epresse in relation to o the elements escribe by the Equations (3)-(5): t Acosk sint u. (8) In act calculate the secon erivatives o the Equation (8) as a unction o space an time: u A cos k t t cos u Acos k sin t t u ka sin k t sin u k cos sin A k t (9a) (9b) (9c) (9) you can replace the Equations (9b) (9) an (6) in the Equation (7) to obtain the ientity: k k. (0) Alternatively can be introuce only Equations (9b) an (9) to obtain the Equation (7)... The Plane Waves The etension o D Alembert equation to the two-imensional case is an unsolve problem that has require the introuction o isotropic homogeneous meia in which the motions are propagate as circular waves (Figure 3); at consierable istances rom the source small portions o the circular waves can be approimate by plane waves which propagate in a straight line accoring to irections normal to the wave ronts represente by the rays; in this way the analysis o the phenomenon is acilitate as limite to the stuy o the suns rays having eviently rectilinear trajectories in homogeneous meia an curvilinear in those non-homogeneous. Figure. Schematic reprouction o an artiicial impulse an o the siteen monochromatics components [3]..3. The Phenomenon o Dispersion Each monochromatic components o a vibratory motion which propagates in a homogeneous meium travels at the same spee in turn epenent on the angular re- Copyright 03 SciRes.

3 R. DI FRANCESCO 3 s s t v v v v vy y v y (a) s v u v t. (b) v v v w vy y v y y Figure 3. In homogeneous isotropic meia the P an S waves propagate accoring to concentric spheres (circles in the plane) at a great istance can be approimate by the wave ronts plans generically einite plane wave. quency an the wave number through Equation (6); in this case inicate with ρ the ensity o the meium it proves [7] that the ollowing equations are vali or elastic waves compressional (P) an cutting (S): v P P P k T P P (a) K 43G v S S S G. (b) k T S S Besies it is also the relationship v P > v S or the Equation (a). In inhomogeneous meia or which the spee varies along the path each requency component o the vibratory motion is instea equippe with its own spee such as to arrive to a generic receiver at ierent times. The escribe phenomenon known as the ispersion o the phases implies that in the means o non-ispersive (homogeneous) the vibratory motion arrives at the receivers with the same initial shape; on the contrary in ispersive meia is possible to ientiy a group velocity (i you choose to characterie the group through the analysis o the maimum amplitue) an a phase velocity o each component so that a signal emitte by the source arrives istorte at the receiver. Obviously means in the two non-ispersive group velocity an phase are.. General Equation o the Wave Motion Denote by s(y) the isplacement vector in space eine with respect to a reerence system coorinate having components u v e w; consequently i the Equation (7) escribes the motion o a isturbance which is propagate only in the irection the generalie orm o the equation o motion will be o the type:.. The Equation o Wave Motion in Space The Equation () epresse in matri an etene orm in the case o the means non-ispersive reuces to the orm: s s v j (3a) t w t y s u v v v v y in which v j ientiies the column vector o the spee. Then the solution can still be epresse using the Equation (8) on conition to take account o the three- imensionality o the phenomenon: s yt Acosk cosky y cos k sin 3t (3b). (4) By repeating the proceure previously seen with Equation (7) o the motion D you get in sequence: s 3 A cosk cosky y t (5a) cos k cos 3t s 9 Acos k cosky y t (5b) cos sin 3 k t u ka sin k ky y cos cos k sin t u k cos cos A k ky y cos k sin t v kya cos k ky y y cos sin k sin t (5c) (5) (5e) Copyright 03 SciRes.

4 4 R. DI FRANCESCO v k cos cos ya k ky y y k sin t cos w ka cos k ky y sin cos k sin t k cos cos A k ky y cos k sin t (5) (5g) (5h) 3 v k vy ky v k. (6) In Equation (6) replace the components o the velocity epresse accoring to Equation (6) inally to prove the valiity o the propose solution: k k k. (7) 3 y k ky k The latter also emonstrates that the unamental perio o the wave (an consequently the requency unamental) correspons to the perio (an requency) o the iniviual components. In conclusion or orthotropic meia an rom the structure o Equation (4) which is easily seen that the waves propagate accoring ellipsois scalene (Figure 4(a)); similarly or the conition v > v y = v we are seeing waves with the orm o ellipsois o revolution (Figure 4(b)) while the conition v = v y = v leas to the evelopment o spherical waves (Figure 4(c)). In this regar we can rewrite Equation (6) creating the conitions: ky mk mn ; mn (8) k nk obtaining: Place also: 3 y v k mv k nv k (9a) v mv nv. (9b) 3 y k Figure 4. Three-imentional geometry o waves: (a) Scalene ellipsoi (v > v y > v ); (b) Ellipsoi o revolution (v > v y = v ); (c) Sphere (v = v y = v ). you get: 3 a (0) k y a v m v n v (a) v mv y nv 0. (b) a a a Finally putting too: m a b n m: a b c (3) n a c we come to the equation o an ellipsoi: v v y v 0. (4) a b c On the other han the Laplacian content in Equation general (3a) s is the quaratic orm o the ivergence o the vector unctions that or positives values in a generic point P enotes the eistence o an outgoing low in the neighborhoo o P; so to the conition impose v > v y > v correspons w > v > u which escribes an ellipsoi. The iscusse moel is inepenent o temperature o the system assume aiabatic that respects the law o conservation o energy... The Equation o Wave Motion on a Plane In case o the plane the Equation (3b) is reuce on the orm: s v u v w (5) t whose solution is a special case o the Equation (4): s t Acosk cosk. (6) sin t Again the proceure is applie now known we arrive at the ientity relation: k k (7) k k which proves the accuracy o the solution accoring to the propagation o the waves on a plane with elliptical or circular orms. 3. The Propagation o Waves in Porous Meia The problem o waves propagation in porous meia has been aresse previously in [8] rom the law o mass balance or two-phase meia now etene to three- Copyright 03 SciRes.

5 R. DI FRANCESCO 5 phase rives: n n S n S s w a. (8) In the Equation (8) appear porosity (n) the egree o saturation (S) an the ensity o the soli phase (ρ s ) luia (ρ w ) an gaseous (ρ a ) capable o escribing a structure consisting o a skeleton soli with interstitial interconnecte pores between them (Figure 5). Since the mass is connecte with the spee through the Equations () (a) an (b) shows a epenence o the type: v v y n S E G t. (9) In other wors must be consiere as a porous soli consists o three continuous elastic means that occupy the same region o space an which interact between them carving up the propagation o the same elastic pulse. 3.. Spee o Compression Waves in Porous Meia The problem outline can be mathematically simpliie i one analyes the propagation o three waves ecouple rom the same impulse through the soli skeleton an phases lui an gases containe in the pores (Figure 6); in this way we obtain a system o three equations or the propagation o compression waves in anisotropic porous meia: * E Kw v n n S P w K n S a a (30a) E y K w vpy n ns w (30b) K a ns a Figure 5. Typical structure o soils an ientiication o the Figure 6. The basic hypothesis involves the schematiation o porous meia in three continuous elastic means an in this way an impulse can be ecouple into three components which propagate without mutual interaction. E K w vp n ns w. (30c) K a ns a Analying the Equation (30) analying the equation it turns out that the irst bracket ientiies the anisotropic component relative to the propagation through the soli skeleton linke with only the elastic moulus being able to neglect the eect o the transverse contraction [0] or having brought the problem to three one-imensional waves; the ollowing brackets inicate instea the components relate to the isotropic liqui an vapor phases through the corresponing elastic mouli volumetric (K w e K a ). The symbol E * i ientiies a law o variation o the longituinal elastic moulus ynamic etermine rom [] an [] an base on the introuction o actors o contraction seen in [3]: E E n. (3) i 0 i In the Equation (3) appear the parameterwhich represents a material constant an must be eterminate eperimentally meanwhile the value o ρ presents in the Equations (30) changes as a unction o the porosity accoring to the law (8) o the mass balance. Once eine the general structure o the equations the problem can be urther simpliie i one consiers that the spee o propagation o the impulses in water an air are approimately v w 500 m/s an v a 340 m/s which woul alter the Equations (30): E vp n S n 0nS basic parameters o porous meia in general [9]. (3a) Copyright 03 SciRes.

6 6 R. DI FRANCESCO E y E 0 n G (3b) 0 n vpy n 500 ns 340nS E vp n 500nS. (3c) 340n S In conclusion the same way as seen with the Equations (3) (9) relate to the generalie theory o waves the Equations (30) an (3) escribe scalene ellipsoial waves ellipsoi o revolution or spherical with respect to the relationships between the elastic moulus o the soil skeleton that aect the ynamic response o the anisotropic component. 3.. Velocity o the Shear Wave on Porous Meia In the etermination o the spee o shear waves is necessary to consier: ) the physical impossibility o waves to propagate in luis; ) their polariation on mutually orthogonal planes (waves S V e S H ); 3) their evelopment in all irections with respect to the source; 4) to Equation (b); 5) a law o variation o Poisson s ratio: 0 n. (33) Finally the Equations (a) (3) an (34) can be combine with each other:. (34) Using the Figure 7 as a reerence it is seen that the components o the shear waves are 6 o which three inepenent mathematical symmetry an accoringly the relate ormulations can be obtaine rom Equations (3) puriie rom the isotropic components: v v G y n (35a) Sy n S G (35b) G y vs y n. (35c) Equations (35) compare with the Equations (3) emonstrate the eistence o time ierences o arrival o the shear wave compare to compression which must necessarily increase with the increase o the istance rom the source Tests o Moel Valiation () () (3) The graph o Figure 8 illustrates the eperimental results conucte on both ry rocks that saturate water epresse in terms o rate o change o P-waves as a () () (3) (3) (3) (33) Figure 7. Components o the seismic moment tensor [4]. Copyright 03 SciRes.

7 R. DI FRANCESCO 7 composition (Basalts an Trachytes alkaline in the irst case the variable in the secon case [6]) inuces to assume the eistence o ierent starting values o the ynamic elastic moulus an ensity; in other wors the same eperimental results are not comparable to each other i not qualitatively as inee emonstrate by the ispersion o its employee ata rom the variation o the physical properties quote. The secon test was perorme using the eperimental ata containe in [7] relating to the correlations v P -n an v S -n measure in samples o alumina ceramic (Al O 3 ) known or the aci resistance an low thermal conuctivity so as to be use as a catalyst in the chemical inustry an as a grat material in the biomeical; thereore with reerence to Figure 9(a) it turns out that the theoretical moel escribes with high accuracy the perormance o the P-waves in the ceramic (per =.46) while S waves are suiciently approimate by n < 5% an approimate or n > 5%. The Figure 9(b) in turn illustrates the etension o the theoretical moel to the entire range o variability o porosity between a material ininitely compact (n = 0%) an ininitely porous (n = 00% ) epresse in terms o P-waves velocity or ry samples an saturate an S- waves velocity. In this case the elements are key inings Figure 8. Comparison between the eperimental ata o the spee o samples o ry rocks an waterlogge [5] an the theoretical results preicte by the moel. unction o porosity. In etail as reporte in [5] it is possible to note that the increase ue to saturation is very evient both in the lava issue rom Etna (ET; near Catania Italy) than in the Campi Flegrei (CF Italy near Naples) both characterie by a low overall porosity o approimately in the range n = 5% - 0% on the contrary in tus an ignimbrites (having a porosity in the range n = 30% - 60%) the increase o spee oun in the saturate state is signiicantly higher than the lava rocks. At the same graph have been superimpose the theoretical laws o variation o P-waves velocity an their tabulate ata calculate using Equations (8) (3) an (3c); as can be note rom the same theoretical moel ails to approimate or =.6 introuce in the calculation o E the spee ierence between the ry an saturate rocks with the increase o the porosity. Final interpretation o the results with the theoretical moel must be consiere that the theoretical behavior o lava rocks (lavas o Etna an Campi Flegrei) an pyroclastics (tus an ignimbrites) was stanarie in terms o ρ an o E although the ierent mineralogical Figure 9. (a) Comparison between the eperimental ata o the spee o samples o alumina ceramics [7] an the theoretical results preicte by the moel; (b) Etension o the moel to the entire iel o porosity. Copyright 03 SciRes.

8 8 R. DI FRANCESCO in the test: ) the P-waves velocity or saturate samples is slightly lower than that responsible or the ry samples n < 60%; ) or n > 60% the P-waves velocity o saturate samples becomes greater than that o the ry samples while the moel maniests a marke non-linearity; 3) or n = 00% the spee o the saturate samples is reuce to that o water (point A) while that o the samples rie in the air (point B). Finally as epecte n = 00% or the S-waves velocity vanishes. The trens o the P-waves velocity associate to the iels 0 < n < 0.6 an 0.6 < n < epen on the laws o variation aopte or ρ an E * that escribe by Equations (8) an (3) respectively are linear an non-linear; which means that while the ensity varies linearly or 0 < n < the elastic moulus assumes a-heating an approimately linear or n < 0.6 an not linear or n > 0.6. Consequently the increase o spee setting attributable to lui phases ails to compensate or n < 0.6 the eects aucts by variations in ensity an elastic moulus. The last test was conucte using the eperimental results containe in [8] an relate to the propagation o P-waves in a generic group o rocks not necessarily relate to each other (Figure 0); Also in this case the theoretical moel is able to approimate (per =.38) the ynamic behavior generally using the same starting values o the ensity an the ynamic elastic moulus. 4. Conclusions The analysis o wave motions takes origin by the ierential equation o D Alembert [45] the solution o which base on the properties o the trigonometric sine an cosine unctions simulates the propagation o an elastic wave in a continuous meium -imensional an its e- tension to the loor is brought back to the simpliie concept o circular waves reuce to plane waves. Ater a brie review o the main elements that make up the rolling waves o D Alembert was written the ierential equation governing the wave motion in space (generalie theory) to ollow it was oun that the eact solution shows that the waves take the orm o ellipsois scalene in orthotropic means which in turn reuce to the ellipsois o revolution in the means transversely isotropic an spherical waves in isotropic meia; in the same way which step consequent the 3D equation has been reuce to the D iel whose solution leas to waves having elliptical shapes that are reuce to circular waves. But be aware that these equations are vali or continuous meia which can be escribe by assigning them appropriate scalar an vector iels eine by means o unctions regular an continuous over the entire omain coniguration. Which net step given the nature o the particle actually real meia (with particular reerence to geomaterial) the equations have been applie to porous meia the escription o which is base on concepts known in Geo- technical that preict the eistence o three continuous meia (soli skeleton with interparticle vois paths rom water an air) that interact between them carving up the propagation o the same impulse elastic ivie into the components o compression an shear. Finally the search has been complete with some tests base on known eperimental ata relating to the propagation o waves in ierent types o rocks an in samples o alumina ceramics. Ultimately the tests showe that the propose moels ail to accurately simulate the behavior o a particular material or group o materials that share the same origin while urther applications may be later evelope: ) in Figure 0. Comparison between the eperimental velocity o P-waves o a generic group o rocks [8] an the theoretical preiction o the propose moel is etene to the iel o porosity. Copyright 03 SciRes.

9 R. DI FRANCESCO 9 the iel seismological consiere that the Equations (30) reuce to Equations (a) or means ininitely porous as in the case in which the seismic compressional waves are reuce to acoustic waves when intercept the earth s surace an propagate into the atmosphere; ) still in the iel seismological or in the analysis o vibrational motions artiicial through the introuction o amping unction o the signal; 3) or the etermination o the porosity o geomaterials coarse loose that can not be sample an measure in the laboratory. 5. Acknowlegements The author wishes to thank Luca Lussari Department o Mathematics an Physics o University o Sacro Cuore o Brescia Italy or his suggestions an observations. REFERENCES [] G. Lano an F. Silvestri Sismic Local Responce Hevelius Eiioni Benevento 999. [] D. Lo Presti Behaviour o Soils in Dynamic an Ciclic Conitions International Center or Mechanical Sciences CISM Uine Italy 999. [3] E. Carrara A. Rapolla an N. Roberti Geophysical Survey or the Stuy o the Subsoil: Geoelectric an Seismic Methos Liguori Eitore Napoli 99. [4] J. D Alembert Research on the Curve Forme by a Rope Set in Vibration History o the Royal Acaemy o Sciences an Belles Letters in Berlin Vol pp [5] J. D Alambert Aition to the Memory on the Curve Forme by a Rope Set in Vibration History o the Royal Acaemy o Sciences an Belles Letters in Berlin Vol pp [6] C. B. Boyer A History o Mathematics John Wiley & Sons New York 968. [7] L. Lanau an E. Lisits Theory o Elasticity Eitori Riuniti Roma 979. [8] R. Di Francesco an M. Siena The Contribution o Geophysics in Geotechnical Design an Control Work in Progress XXIII National Conerence o Geotechnical Engineering Paova 6-8 May 007 pp. -8. [9] R. Di Francesco Introuction to Soil s Mechanics Part I Dario Flaccovio Eitore Palermo 03. [0] O. Bellui Science o Construction Vol. IV Zanichelli Eitori Bologna 963. [] G. Onracek The Quantitative Microstructure Fiel Property Correlation o Multiphase an Porous Materials Review on Power Metallurgy an Physical Ceramics Vol. 3 No pp [] D. N. Boccaccini an A. R. Boccaccini Depenence o Ultrasonic Velocity on Porosity an Pore Shape in Seintere Materials Journal o Nonestructive Evaluation Vol. 6 No pp [3] R. Di Francesco Introuction to Continum s Mechanics Dario Flaccovio Eitore Palermo 0. [4] K. Aky an P. G. Richars Quantitative Seismology: Theory an Methos University Science Books Sausalito 00. [5] A. Zollo an A. Emolo Earthquakes an Waves. Methos an Practice o Moern Seismology Liguori Eitori Napoli 0. [6] G. Negretti an B. Di Sabatino Course o Petrography CISU Rome 983. [7] M. Asmani C. Kermel A. Leriche an M. Ourak Inluence o Porosity on Young s Moule an Poisson s Ratio in Alumina Ceramics Journal o the European Ceramic Society 00 Vol. No. 8 pp [8] V. Rhevsky an G. Novik The Physics o Rocks Mir Publichers Moscow 97. Copyright 03 SciRes.