Åström, Jan; Kellomäki, Markku; Alava, Mikko; Timonen, Juho Propagation and kinetic roughening of wave fronts in disordered lattices

Transcription

1 Powered by TCPDF ( This is an eectronic reprint of the origina artice. This reprint may differ from the origina in pagination and typographic detai. Åström, Jan; Keomäki, Markku; Aava, Mikko; Timonen, Juho Propagation and kinetic roughening of wave fronts in disordered attices Pubished in: PHYSICAL REVIEW E DOI: 1.113/PhysRevE Pubished: 1/11/1997 Document Version Pubisher's PDF, aso known as Version of record Pease cite the origina version: Åström, J., Keomäki, M., Aava, M., & Timonen, J. (1997). Propagation and kinetic roughening of wave fronts in disordered attices. PHYSICAL REVIEW E, 56(5), DOI: 1.113/PhysRevE This materia is protected by copyright and other inteectua property rights, and dupication or sae of a or part of any of the repository coections is not permitted, except that materia may be dupicated by you for your research use or educationa purposes in eectronic or print form. You must obtain permission for any other use. Eectronic or print copies may not be offered, whether for sae or otherwise to anyone who is not an authorised user.

2 PHYSICAL REVIEW E VOLUME 56, NUMBER 5 NOVEMBER 1997 Propagation and kinetic roughening of wave fronts in disordered attices Jan Åström, 1 Markku Keomäki, 1 Mikko Aava, 2,3 and Jussi Timonen 1 1 Department of Physics, University of Jyväskyä, P.O. Box 35, FIN-4351 Jyväskyä, Finand 2 NORDITA, Begdamsvej 17, DK-21 Copenhagen, Denmark 3 Laboratory of Physics, Hesinki University of Technoogy, P.O. Box 11, FIN-215 HUT, Finand Received 13 June 1997 The dynamics of a wave front propagating in diuted square attices of eastic beams is anayzed. We concentrate on the propagation of the first maximum of a semi-infinite wave train. Two different imits are found for the veocity depending on the bending stiffness of the beams. If it vanishes, a one-dimensiona chain mode is derived for the veocity and the ampitude is found to decrease exponentiay. The first maximum is ocaized and the average width of the wave front is aways finite. For very stiff beams an effective-medium mode gives the correct veocity and the ampitude of the first maximum decays according to a power aw. No ocaization of the first maximum is observed in the simuations. In this imit scaing arguments based on Huygen s principe suggest a growth exponent of 1/2, and a roughness exponent of 2/3. The growth exponent fits the simuation data we, but a consideraby ower roughness exponent.5 is obtained. There is a crossover region for the bending stiffness, wherein the wave-front behavior cannot be expained by these imiting cases. S X PACS number s : z, Ct, d I. INTRODUCTION Propagation of a wave front in a homogeneous and continuous medium is rather we understood within easticity theory 1. Wave propagation in a reguar attice is ikewise we understood 2. Amost a rea materias, however, contain some kind of disorder such as random microcracks or structura defects. This compicates the situation and interesting phenomena such as ocaization 3 appear. Disorder aso affects the average wave-propagation veocity and the behavior of the wave s ampitude. We consider here randomy diuted beam attices as discrete modes of eastic soids. As wi be demonstrated beow, both the ampitude decay and the veocity in randomy diuted attices can in many cases be found as a function of the diution parameter. Note that this is made feasibe by considering just the first maximum of the wave front, which is much ess susceptibe to interference. Effects of disorder on ampitude and veocity are, however, just the first characteristics of the wave dynamics. In order to go beyond that, we wi investigate in detai aso the statistica geometry of the wave fronts in terms of their scaing behavior. Consider an initiay straight wave front. After encountering defects in the medium it wi roughen, and the question then is how this happens as measured, e.g., by considering the behavior of the average width of the wave front. The physics behind such changes of shape is best understood by connecting it to the recenty much studied concept of kinetic roughening of growing interfaces. Interface growth modes usuay give rise to sef-affine interfaces 4,5, confined to a few universaity casses 4 depending on the symmetries of the modes. The scaing of sef-affine interface growth can be characterized by an initia regime, with a roughness or wave-front width r proportiona to r t, and by an asymptotic regime, characterized by r L, where L is the inear system size. In the asymptotic regime the correation ength aong the front is of the order of the system size. One may then ask the question whether wave-front roughening can be characterized by the usua universaity casses, despite being governed at ong time scaes by the Huygens s principe. Consider the simpest growth probems with therma noise: the random deposition RD mode 6, the mode described by the Edwards-Wikinson EW equation 7, and the mode described by the Kardar-Parisi-Zhang KPZ 8 equation. The RD universaity cass describes a set of independent one-dimensiona random waks in the direction of wave propagation. This means that roughness grows indefinitey and 1/2. It is quite obvious that no independent one-dimensiona paths exist in eastic square attices, except in specia cases such as when the bending stiffness of the bonds vanishes, and therefore the RD mode is usuay irreevant in wave-propagation probems. In the EW and KPZ universaity casses there is a surface diffusion term eading to sef-affinity and saturation of the roughness in a finite system. The KPZ equation describes kinetic roughening in cases in which oca growth aways has a finite veocity perpendicuar to the interface. As such, one coud expect it to be reevant for our probem as indeed is the case for scattered, directed cassica waves 9. We wi show beow, however, that neither the KPZ nor the EW mode is appicabe to roughening of wave fronts propagating in a attice. The ony ordinary behavior that can be found beongs, suprisingy enough, to the RD cass, which expains the short-time behavior of the wave-front roughness. For the asymptotic behavior, we empoy a specia argument appicabe to interface propagation obeying the Huygens principe. In the rest of the paper, we begin by describing in detai the attice mode in Sec. II. Section III is devoted to anaytica considerations of veocities and ampitude decay of the waves. In Sec. IV we compare anaytica resuts with numerica simuations, and in Sec. V we anayze the roughening of the wave front. Section VI contains concusions and a discussion X/97/56 5 /642 8 /$ The American Physica Society

3 56 PROPAGATION AND KINETIC ROUGHENING OF WAVE II. LATTICE MODEL We study a numerica attice mode because of two reasons: They are efficient from a numerica point of view and represent a straightforward discretization of a britte soid obeying the Cosserat easticity equation 1,11. A discrete attice can aso be considered as a mode of a granuar materia, in which the attice sites represent the grains and the attice bonds represent the eastic interactions between the grains. We use a square attice where the attice bonds are eastic beams with a square cross section w 2, ength, and Young moduus E. The eastic deformation of a bond is determined by the matrix equation 12 F T T K TU, where the vector F contains the forces and anguar momenta acting on the two ends of the bond, T is a rotation matrix that transforms the oca coordinate system of the beam i.e., the x axis aong the beam axis into the goba coordinate system of the attice, and U is the vector containing the dispacements reated to the forces and momenta in F. The stiffness matrix K in the oca coordinate system is given by K EA EA 12EI 6EI 12EI 6EI EI 4EI 6EI 2EI 2 2 EA EA 12EI 3 6EI 2 6EI 2 4EI 12EI 3 6EI 2 6EI 2 2EI where A w 2 and the moment of inertia is I w 4 /12. This stiffness matrix hods for sma dispacements of a sender beam i.e., shear deformations and noninear effects are negected. The equation governing the eastic response of the entire attice is easiy constructed by summing the stiffness matrices (K ) for a the bonds in the attice. Inertia is introduced in the attice by having masses m on the attice sites, whie the beams are assumed to be massess. Periodic boundary conditions are used in the vertica (y) direction parae to one of the principa bond directions. The right boundary is free to move without constraints and the sites at the eft boundary are forced to move harmonicay A sin( t) either in the x or in the y direction when the time t. We aways use frequencies that are ower than the owest eigenfrequency of the attice bonds. The entire attice is at rest for t. The dynamics of the attice is cacuated using a discrete form of Newton s equations of motion incuding a sma inear viscous dissipation term, 1, M t 2 2 t U t t C 2M t 2 K U t M t 2 2 t U t t, C where M is the diagona matrix containing the masses, C is the damping matrix, which is aso diagona, and K is the goba stiffness matrix. The time dependent dispacement fied is cacuated by iteration of time steps ( t) starting from equiibrium at t. We foow the first wave front by recording the time when each site in the attice reaches its first dispacement maximum. The ocation of the front is defined for each attice row separatey so that it is at the site in the row that was the ast before a given time to reach its first dispacement maximum. Disorder is introduced by removing randomy a fraction 1 p of the bonds. III. VELOCITY AND ATTENUATION OF ELASTIC WAVE FRONTS A. Anisotropic bond stiffness First we wi derive an approximate soution for the average propagation veocity and the average ampitude decay of the wave front when either the bending or the axia stiffness of the bonds is much smaer than the other. The boundary conditions have been chosen so that the wave front is, in a statistica sense, invariant under transformations in the y direction. Therefore, it is possibe to use a one-dimensiona mode to describe the wave-propagation veocity. The first dispacement maximum traves aong paths of bonds that are oriented in the x direction. These paths are connected via vertica bonds. Consequenty, if the bending stiffness of the bonds is much smaer than their axia stiffness, the vertica bonds wi not affect much the veocity of this wave front i.e., for ongitudina waves induced at the eft boundary of the attice and vice versa for transverse waves. In this case, the veocity can be cacuated by approximating the attice with an ensembe of noninteracting paths of bonds. The equations of motion for such paths are simpe onedimensiona wave equations with zero dissipation m d2 u dt 2 k 2 d2 u dx 2, 2 where u(x,t) are the dispacements from equiibrium of the sites, is the ength of a attice bond, k Ew 2 / for axia dispacements, and k (w/) 2 Ew 2 / for transverse dispacements. The atter is the bending stiffness of a beam with camped ends, which can be used if the moment of inertia of the sites is arge. To cross the attice from one side to the other, the wave front wi have to trave a distance L x in the x direction with a oca veocity v x. At each missing horizonta bond, the wave wi have to trave at east one unit step in the vertica direction. When p is far above its critica vaue at percoation, ony sma connected custers of missing bonds are present. In this case, the probabiity that at any attice site the wave wi have to trave at east a step in the vertica direction is approximatey 1 p. The probabiity for moving at east two steps verticay is approximatey (1 p) 2, etc. Thus we approximate the vertica trave dis-

4 644 ÅSTRÖM, KELLOMÄKI, ALAVA, AND TIMONEN 56 tance by L y L x i 1 (1 p) i. The oca veocity in this direction is v y. The veocities v x and v y are given by v x Ew 2 /m and v y w/ Ew 2 /m for induced ongitudina waves and vice versa for transverse waves. The average wave-front veocity of a ongitudina wave induced at the eft boundary of the attice is then given by v p Ew2 /m 1 1/p 1 /w, and the corresponding veocity of an induced transverse wave by v t p w/ Ew2 /m 1 1/p 1 w/. B. Effective-medium theory for isotropic bond stiffness As aready mentioned, Eq. 3 Eq. 4 is expected to hod when the bending axia stiffness is so sma that ongitudina transverse waves can be considered to propagate aong noninteracting paths. This means that the attice wi not reach a oca eastic equiibrium during the passage of the first dispacement maximum. If, on the other hand, the bending stiffness of the bonds is roughy equa to their axia stiffness, the attice wi ocay remain at an eastic equiibrium if the waveength is not very short. In such a case the attice wi behave as an effectivey homogeneous materia and the wave-front veocity is determined by the effective Young moduus and the effective Poisson ratio of the attice. The effective Young moduus can be cacuated within the effective-medium approximation 13. If one assumes that the eastic deformation of a bond is competey determined by a singe constant, the easticity of a attice is formay the same as the eectric conductance of the attice with resistors repacing the eastic beams. If the bending stiffness is different from the axia one, it is required that a bond is deformed through either bending or stretching for the forma simiarity to hod. In such a case, a fiber that is ony bent is considered to have a conductance b, which corresponds to the bending moduus, and a bond that is ony stretched has a conductance a, which corresponds to the axia moduus. Notice that this is the so-caed Born mode 14 of easticity. For a square attice with boundary conditions as described above, it is reasonabe to assume that, for induced ongitudina waves, the vertica bonds are ony bent and the horizonta bonds are ony stretched. The effective-medium approximation is based on the direction symmetry of the current fied caused by a point source in an infinite attice 13. This symmetry hods ony when a b. In the case when a b the situation is somewhat compicated. When p is cose to unity the correct soution is obtained by scaing the y direction by a factor b/a. On the other hand, when p is cose to the percoation threshod p.5 for a square attice, we expect the direction symmetry to be vaid independent of a and b. In our effective-medium soution we use a inear interpoation between these two extremes. Carrying out the effective-medium cacuation as in Ref. 13 gives the effective Young moduus of the attice in the form 3 4 E E p 2 2p 2p 1 a/b 1 p, and the corresponding effective shear moduus in the form p 2 2p 2p 1 b/a 1 p, where E and are the Young moduus and the shear moduus of the perfect attice. Since the Poisson ratio is zero for a square attice, the veocity of induced ongitudina waves is given by v Ew2 m 5 6 p 2 2p 2p 1 w 1 p 7 and the veocity of induced transverse waves by v t w Ew2 m p 2 2p 2p 1 w 1 p. 8 Both Eqs. 3 and 4 and Eqs. 7 and 8 predict the correct veocities for p 1. At the percoation critica point (p.5) the wave-front veocity is zero the shortest connected route in a network is fracta at this point. Atand cose to this point the mode of one-dimensiona paths fais. The effective-medium mode predicts the veocity correcty at the critica point itsef the mean-fied critica point is exact for the two-dimensiona square attice, but it fais within the critica region cose to the critica point. A better resut within this region coud, at east in principe, be obtained using the renormaized effective-medium approximation 15. C. Attenuation In the case of anisotropic bond stiffness the sawtooth chain mode can be appied again for estimating the ampitude decay as a function of p. As ong as the wave front traves aong an unbroken horizonta chain of bonds, the ampitude remains more or ess constant. When the wave front meets a missing bond, it must switch to a neighboring row which takes some time. Meanwhie the wave front propagating aong that neighboring row has propagated past the part of the front that switches row. Thus part of the energy of the front propagating in the origina row is deayed and we can assume that the ampitude is reduced to a fraction of its vaue at each missing bond we assume here that one waveength covers ony one or at most a few missing horizonta bonds. With being constant the ampitude decays exponentiay. The decay as a function of x is then given by A x A exp 1 p 1 x. There aso appears backscattering from missing bonds. In the case of the first maximum of the propagating wave train this effect is negigibe, however. In the case of isotropic bond stiffness the effectivemedium approximation is expected to hod. In this approximation, the network is considered as homogeneous, which impies that there wi be no ampitude decay at a. However, we must take into account that the network is discrete and therefore dispersion of veocities appears. We have 9

5 56 PROPAGATION AND KINETIC ROUGHENING OF WAVE FIG. 1. Average wave-front propagation veocities of ongitudina waves in attices with 1; m.5,.1; w.3,.6,1.; E 1.; and the frequency.125. The corresponding veocities cacuated from the effective-medium approximation EMA and the one-dimensiona path mode 1DPM are shown by dotted and soid ines, respectivey. shown 16 that, in a perfect square attice, the dispersive widening of the wave front causes a power-aw decay of the form A x x 1/3. 1 In disordered attices with p 1, this type of decay shoud be observed at east for ong waveengths. D. Numerica anaysis of wave-front progation The anaytica resuts Eqs. 3 and 7 are compared with simuation resuts in Fig. 1 and a simiar comparison for Eqs. 4 and 8 is shown in Fig. 2. As can be seen from Fig. 1, the effective-medium approximation EMA foows the FIG. 2. Average wave-front veocities of transverse waves in attices with 1; m.1,.6; w.6,1.,2.5; E 1.; and the frequency.125. The corresponding veocities cacuated from the EMA and the 1DPM are shown by dotted and soid ines, respectivey. FIG. 3. Average veocity of a ongitudina wave front as a function of the driving frequency with 1, m.1, w.3, p.9, and E 1.. The upper and the ower ines are the prediction by the effective-medium approximation and the mode of one-dimensiona paths, respectivey. simuation data we when the bond stiffness is isotropic i.e., w 1.). It is aso evident that the mode of onedimensiona path foows perfecty the simuation resuts for sender bonds (w.3) when p.75. For smaer p the mode fais as expected. In the intermediate case (w.6) none of the modes gives a very good resut. Ony cose to p 1 does the mode of one-dimensiona paths give the correct veocity. In both Figs. 1 and 2 aso the renormaized effective-medium approximation REMA soution is shown. As can be seen, the REMA soutions differ ony a itte from the EMA soutions, which indicate that the scaing regimes are sma. In Fig. 2 simiar resuts are shown for the transverse waves. The effective-medium mode works aso in this case for w 1. The mode of one-dimensiona paths gives the correct veocity for broad beams (w 2.5) when p is cose to unity. We aso expect that there shoud be a crossover between the two mode veocities with a changing driving frequency. This crossover is, however, difficut to observe numericay. This is mainy because of the reativey sma difference between the two soutions. To compete the picture, we show nevertheess in Fig. 3, how the veocity changes with frequency in the case of ongitudina waves for w.3 and p.9. This figure demonstrates a cear trend of increasing veocity at ower frequencies. Notice, however, that the effective-medium mode is not very accurate for these parameter vaues and a 2.5% difference between the simuated veocity and the EMA prediction remains even at the owest frequencies. Next we test numericay the ampitude decay, i.e., Eqs. 9 and 1. In Fig. 4 we show the ampitude A as a function of x for different vaues of 1 p. The parameters used are 1, m.1, w.1, and E 1.. The frequency is.125. A(x) is reasonaby we approximated by an exponentia decay Eq. 9 for sma x. A crossover from the exponentia to a ess rapidy decaying behavior can be seen for arge x and arge 1 p. This crossover phenomenon is an artifact of the dispersion reation 16. As the effective frequency of the first dispacement maximum decreases, the amount of refection aso decreases i.e., increases and

6 646 ÅSTRÖM, KELLOMÄKI, ALAVA, AND TIMONEN 56 FIG. 4. Average ampitude of the wave front as a function of the distance with 1; m.1; w.1,1 p.5,.1,.15,.2,.25,.3; and E 1.. The frequency is.125. The inset shows the fitted exponents as a function of 1 p. eventuay the frequency of the wave front becomes so sma that the attice wi ocay remain at equiibrium and the ampitude wi decrease ony according to Eq. 1. The inset in Fig. 4 proves that the exponent in Eq. 9 is indeed proportiona to the diution parameter 1 p for sma x. The power-aw decay for broader fibers 1, m 1., w 1., and E 1. is demonstrated in Fig. 5. The ampitude decay foows we Eq. 1 for arge enough x. Figures 1 5 support the veocity and ampitude decay predictions by the one-dimensiona and the effective medium modes in their respective regions of vaidity. The simuations reveaed, somewhat suprisingy, aso a third wave-front veocity. This veocity does not depend on the average properties of the attice, but is instead a transient that propagates ony a short distance and takes advantage of the fastest routes that exist. This is demonstrated in Fig. 6. The distances of the attice sites from the eft edge of the attice are potted in this figure as a function of the arriva time of the first dispacement maximum at these sites. The attice size is FIG. 6. Distance of the sites in a network of size as a function of the time when the site reaches its first dispacement maximum with 1, m.1, w.1, E 1., p.9, and the frequency.125. The ines are given by Eq. 3 with p.9 and p in attice units. The network parameters are 1, m.1, w.1, p.9, E 1., and frequency.125. Two veocity branches appear. The sower veocity is we predicted by Eq. 3 the ower ine in the figure, whie the faster veocity the upper ine in the figure is the veocity that woud appear in the perfect attice i.e., Eq. 3 or 7 with p 1. The faster signa dies out before the opposite end of the system is reached. This can aso be seen in Fig. 7, where the ampitudes are potted as a function of the distances from the eft edge of the attice for the same data as in Fig. 6. This figure shows that the ampitude of the faster signa decreases exponentiay, whie the ampitude of the sower signa decreases much sower for arge x i.e., according to Eq. 1. The fast transients are essentiay signas that trave aong short pieces of unbroken straight chains of beams. Their exponentia decay is caused by coupings to the surroundings via the vertica bonds. The dynamics of this transient signa wi be reported in more detai in Ref. 16. IV. WAVE-FRONT ROUGHENING So far we have ony considered the mean veocity of the wave front. This does not, however, describe the dynamics FIG. 5. Ampitude of the wave front as a function of distance with 1; m 1.; w 1.; p.7,.8; and E 1.. The frequency is.125. The fitted ines are given by Eq. 1. FIG. 7. Variation of the ampitude with the distance for the data of Fig. 6.

7 56 PROPAGATION AND KINETIC ROUGHENING OF WAVE FIG. 9. Roughness in attices with 1; m 1 6 ; w.1; E 1.; p.7,.8,.9, and the frequency.125. The ines are given by Eq. 12. of the front competey. Caused by the disorder, the initiay straight front wi get rough as it propagates. This is demonstrated for three exampes in Fig. 8. The wave fronts propagate from eft to right; as time evoves the initiay fat fronts become more and more compicated. Notice the difference in roughness between the ast two cases, arising from a much higher bending stiffness in the atter. It woud be reasonabe to expect that the roughening of the wave front woud beong to one of the usua universaity casses. Simuations show, however, that this is not the case. Instead, if one considers in a simiar fashion the eary time and the asymptotic interface widths, a very compicated behavior is encountered. We beieve that this is due to the fact that the wave-front veocity behaves differenty in the two imits, as expained above. The situation is further compicated by the appearance of the transient veocity cf. Fig. 6. In the imit of a vanishing axia or bending stiffness, it is possibe to cacuate the dynamics of the roughening exacty. In the case of ongitudina waves and vanishing bending stiffness, the wave wi trave poory in the y direction aong the vertica bonds and the wave front can be considered to trave aong independent straight paths unti a missing bond is encountered and the propagation stops. This means that the average veocity wi decrease exponentiay with time (t). The average ocation of the wave front x m (t) is then given by x m t p 1 pvt, 11 1 p where v is the wave veocity in the case when p 1 given by Eqs. 3 and 7. When t, x m wi approach the vaue p/(1 p). The roughness r(t) of the wave front is given by r 2 t 1 p 2 p 2 1 p vt 2 1 p vt 1 p p 2 2p vt 1 vt 1 p vt vtp vt 1 p 2 1 p vt vt 1 p vt p vt FIG. 8. Wave-front ocation thick ines at different times in attices with top w.1 and p.98, midde w 1., and p.7, and bottom w 2. and p.7. When t the roughness r(t) approaches the vaue p/(1 p), which hods for a p except for p 1. A comparison of Eq. 12 with numerica resuts is shown in Fig. 9. In this figure 1, m 1 6, w.1, E 1., and the frequency is.125. It is evident that Eq. 12 fits the simuation resuts very we. Two further observations can aso be made based on the Fig. 9: There is, in terms of the usua interface growth modes, a trivia pinning transition that takes pace in the infinite time imit and the interface width does not depend on the system size. Interface dynamics is thus simiar to that of the random deposition mode, in which oca fuctuations set the time dependence. With a nonzero bending stiffness the situation becomes immediatey more compicated. At eary enough times the front behavior may in some cases be of the Edwards- Wikison type ( 1/4), i.e., there is a parae correation ength aong the interface dictated by diffusive dynamics. At

8 648 ÅSTRÖM, KELLOMÄKI, ALAVA, AND TIMONEN 56 FIG. 1. Wave front at ten different times in a network with bonds ony diuted in the shaded square cose to the eft boundary, 1, m.1, and w 1.. onger time scaes, however, there is no typica saturation behavior manifested by a size-dependent interface width. We next consider the case when the bending and axia stiffnesses are equa. In this case we are not abe to cacuate the roughness exacty. We expect, however, that the attice wi behave more or ess ike a continuous medium with randomy ocated hoes. Then the wave-front roughness shoud be governed by the Huygens principe. This is demonstrated in Fig. 1. The wave-front patterns in this figure are not exacty spherica shapes as predicted by the Huygens principe, but a sight distortion of the shape is expected because the effective stiffness of the network is ower than the average cose to a hoe. Furthermore, the veocity is not quite isotropic, which aso causes a distortion of the norma spherica shape. The roughness of an interface governed by the Huygens principe has been anayzed earier as a mode of sputter deposition for amorphous fims 2. A rough interface wi be subject to a smoothing effect caused by the atera, synchronous growth of peaks on the interface. This wi cause height differences h, a distance x apart, to be smoothened out in a time t such that FIG. 11. Averaged wave-front roughness as a function of time in attices with p.7 and vertica heights L y 3,6,9,12; 1, m 1., w 1., E 1., and the frequency is.125. The fitted ine is given by r(t) t 1/2. The inset shows the same data on a semiogarithmic scae, with the y axis rescaed by L.5. In a attice of inear size L the maximum height difference is therefore proportiona to L 2/3, which means that the roughness exponent is 2/3. The corresponding crossover time scae is t crossover x 4/3 /v. 16 What was not taken into account above, however, is the effective decrease in the frequency of the first dispacement maximum. For ower frequencies, the detais of the attice are not as easiy fet by the wave front. Intuitivey, we woud expect this effect to have a decreasing effect on the roughness. In Fig. 11 we show the roughness obtained by simuations for networks of sizes 3 3, 6 3, 9 3, and 12 3, with m w E 1., p.7, and the frequency.125. The first wave front eaves the eft edge of the network at t 2. As can be seen from the figure, the transient signa affects the roughness of the front for t 1. For 1 t 5 the roughness grows diffusivey according to t 1/2, as predicted by Eq. 14. For ate times the roughness decreases, which demonstrates that the decreasing frequency h x 2 /vt, 13 where v is the interface propagation veocity. Roughening of the interface wi be induced by the uncorreated random vacancies in the attice. The average height fuctuations, resuting from this uncorreated noise, wi increase ike h vt 1/2. 14 The roughness of an initiay fat interface, induced by missing bonds, wi therefore increase ike r t 1/2. Roughness wi then, however, reach a state when there is a baance between the two opposite mechanisms described by Eqs. 13 and 14. This happens when h x 2 / h 2 h x 2/3. 15 FIG. 12. Averaged wave-front roughness as a function of time in TLM attices with p.85 and vertica heights L y 3,6,12,24; the driving frequency is.314. The dashed ine is given by r(t) t 1/2.

9 56 PROPAGATION AND KINETIC ROUGHENING OF WAVE of the wave front has a strong effect on roughness. The roughness exponent 2/3 fits the data for the two argest systems of Fig. 11, but the two smaer systems 3 3, and 6 3 have an asymptotic roughness that is too high to foow Eq. 15. We expect that these two sma systems are not yet in the scaing region. The roughness exponent that fits best a the simuation data is therefore smaer than 2/3 i.e., around.5, which is demonstrated by the inset in Fig. 11. To further test Eqs. 14 and 15 we aso used a numerica agorithm transmission ine method TLM wave automaton, that soves the cassica wave equation by directy appying Huygens s principe 21. Using this mode, we again found that roughness grows diffusivey (t 1/2 ), but the roughness exponent is ower than the 2/3 predicted by Eq. 15. A best fit to the data gave a roughness exponent around.5 Fig. 12. V. DISCUSSION AND CONCLUSIONS In summary, we have demonstrated that the propagation veocity and the ampitude decay of the first dispacement maximum in randomy diuted square attices of eastic beams can be argey understood within two simpe modes. In the imit of vanishing axia or bending moduus, a onedimensiona mode correcty describes the dynamics of the wave front. When the bending and the axia modui are roughy equa, an effective-medium approximation combined with continuum easticity theory is sufficient for describing the wave-front propagation. The roughness of the wave front can be exacty cacuated in the imit of a vanishing axia or bending moduus. In this imit the first wave front is aways ocaized and the average wave-front width is finite. As the time evoution is governed by Poissonian fuctuations, this is a random-deposition equivaent phenomenon for wave fronts. For beams that have a nonvanishing bending stiffness, the two-dimensiona character of wave propagation makes the roughening process resembe standard kinetic roughening phenomena. However, the dynamic behavior cannot be mapped to the standard modes, except perhaps at eary times. For roughy equa bending and axia modui, the wavefront roughness grows initiay ike t 1/2. For ate times, Huygens s principe suggests a roughness exponent 2/3, but simuations gave an exponent cose to.5. This discrepancy is sti not fuy understood but is probaby due to finite-size effects. 1 L. D. Landau and E. M. Lifshitz, Theory of Easticity Pergamon, Oxford, A. L. Fetter and J. D. Waecka, Theoretica Mechanics of Partices and Continua McGraw-Hi, New York, Scattering and Locaization of Cassica Waves in Random Media, edited by P. Sheng Word Scientific, Singapore, A.-L. Barabasi and H. E. Staney, Fracta Concepts in Surface Growth Cambridge University Press, Cambridge, J. Krug and H. Spohn, in Soids Far From Equiibrium, edited by C. Godereche Cambridge University Press, Cambridge, F. Famiy, Physica A 19, L S. F. Edwards and D. R. Wikinson, Proc. R. Soc. London, Ser. A 381, M. Kardar, G. Parisi, and Y.-C Zhang, Phys. Rev. Lett. 56, S. Feng, L. Goubovic, and Y.-Z. Zhang, Phys. Rev. Lett. 65, W. Nowacki, Theory of Micropoar Easticity Springer- Verag, Udine, H. J. Herrmann and S. Roux, in Statistica Modes for the Fracture and Easticity of Disordered Media, edited by H. J. Herrmann and S. Roux North-Hoand, Amsterdam, This is the same formaism as that used in the dynamic finite eement method. 13 S. Kirkpatrick, Rev. Mod. Phys. 45, M. Born and K. Huang, Dynamica Theory of Crysta Lattices Oxford University Press, New York, M. Sahimi, Appications of Percoation Theory Tayor & Francis, London, M. Keomäki, J. Åström, and J. Timonen unpubished. 17 L. Briouin, Wave Propagation and Group Veocity Academic, New York, I. Tostoy, Wave Propagation McGraw-Hi, New York, A. Sege and G.H. Handeman, Mathematics Appied to Continuum Mechanics Macmian, New York, C. Tang, S. Aexander, and R. Bruinsma, Phys. Rev. Lett. 64, P. O. Luthi, B. Chopard, and J.-F. Wagen, Lecture Notes in Computer Science Springer, Berin, 1996.