“ AN ANALYTIC STUDY OF FRICTIONAL EFFECT ON SLIP PULSES OFEARTHQUAKES „

Seismological observations show the existence of slip pulses with TR/TD<0.3, where TR and TD are, respectively, the rise time at a site on a fault and the duration time of ruptures over the fault. An analytical study of generation of a slip pulse is made based on the continuous form of 1-D spring-slider model, with uniform fault strengths, in the presence of linear, slip-weakening (SW) friction: f=1-u/Δ (u=the displacement and Δ=the characteristic distance) or linear, velocity-weakening (VW) friction: f=1-v/υ (v=the velocity and υ=the characteristic velocity). Let ω0 and tr are, respectively, the predominant angular frequency of the system and the arrival time at a site and define γ=(1-1/Δ)1/2 and σ=(1-1/4υ2)1/2. There are complementary solution (CS), and particular solution (PS) of the equation of motion. The CS shows a slip pulse under some ranges of model parameters for SW friction and for VW friction when υ>>1; while the CS shows a cracklike rupture for VW friction when υ is not too large. For SW friction, TR and TR/TD decrease when the slip pulse propagates in advance along the fault and when ω0 and γ increase. TR and TR/TD also depend on vR (rupture velocity) and increasing L (fault length). For the PS, Tp/TD is a good indication to show the existence of pulse-like oscillations at a site, because Tp (the predominant period of oscillations at a site) is slightly longer than TR. Results show the existence of a pulse-like oscillation at a site for the two types of friction. A pulse-like oscillation is generated when Δ>1.6 for SW friction and when υ>0.6 for VW friction. Tp/TD decreases with increasing Δ. For the two types of friction, To/TD decreases when vR and L increase. Beeler and Tullis, 1996; Cochard and Madariaga, 1996; Perrin et al., 1995; Zheng and Rice, 1998; Nielsen et al., 2000; Lapusta et al., 2000; Ben-Zion and Huang, 2002; Nielsen and Madariaga, 2003; Coker et al., 2005; Rice et al., 2005; Ampuero and Ben-Zion, 2008; Urata et al., 2008; Ando et al., 2010; Garagash, 2012]. Friction used by those authors includes VW friction, SW friction, velocityand state-dependent friction, and thermal-pressurized friction. Results suggest that rupture modes are controlled by several factors, including friction laws, fault strengths, stress conditions on faults, energy and heat generated by faulting, scaling laws of faults, and spatial-temporal complexity of fault slip. In addition, some researchers considered geometrical heterogeneity of slip is a mechanism to stop earthquake rupture. Beroza and Mikumo [1996] suggested that the short TR could yielded by pre-existing stress with heterogeneous fault strengths. The slip pulses have also been studied by some authors [Wu and Chen, 1998; Chen and Wang, 2010; Elbanna and Heaton, 2012] based on the 1-D spring-slider model (abbreviated as the 1-D BK model hereafter) proposed by Burridge and Knopoff [1967]. From analytic studies by using SW friction, Wu and Chen [1998] claimed that SW friction can result in the self-healing slip pulse and the width of a pulse depends on vR and friction strength. From numerical studies by using VW friction Chen and Wang [2010] found the propagation of slip pulses with TR/TD<0.1 along the model. Their simulation results are in agreement with Heaton’s observations. Elbanna and Heaton [2012] pointed out the differences between the BK model and crack models. According to linear elastic fracture mechanics, slip pulses are seldom generated in the continuum models because slipping region inside of a fault cannot release applied stress without continuous slip while rupture is extending. On the other hand, in the BK model slip pulses can be produced due to the following reason. Each slider can completely release stress exerted by the leaf springs by going back to their equilibrium position even while rupture is extending. In other words, each slider does not transfer stress after their stoppage and information on the length of rupture does not feedback. Laboratory experiments also provide significant information on generation of slip pulses. Coker et al. [2005] observed the existence of both pulse-like and crack-like ruptures under certain conditions. Lykotrafitis et al. [2006] observed the pulse-like shear ruptures with self-healing. Lu et al. [2007] found that the rupture modes depend on the level of fault pre-stress and VW friction is important for earthquake dynamics. Biegel et al. [2008] found that off-fault damage can affect the slip-pulse velocity. As mentioned previously, in order to generate slip pulses some authors prefer to VW friction, while other favor SW friction. In this study, I will investigate the frictional effects on the generation of slip pulses using the continuous form of the 1-D BK model with linear SW friction or linear VW friction. Hence, it is significant to examine which friction (SW or VW friction) is more capable than the other for yielding slip pulses. 2. ONE-DIMENSIONAL SPRING-SLIDER MODEL 2.1 MODEL Burridge and Knopoff [1967] proposed the 1-D BK model (see Figure 1), in which there are N sliders and springs. A slider with mass, m, is connected to its nearest two neighbors by a coil spring of stiffness, Kc. Of course, the two end sliders are only connected to the respective one nearest slider. A moving plate with a constant velocity, Vp, pulls each slider through a leaf spring of stiffness, Kl. Each slider rests in its equilibrium state at time t=0. The position of i-th slider (i=1, ..., N) is denoted by Xi, which is measured from its initial equilibrium position, along the horizontal axis represented by the coordinate y. Hence, Xi is in a function of y and t. Each slider is exerted by a frictional force between it and the moving plate. The frictional force is usually a function of displacement, Xi, and particle velocity, Vi (=dXi/dt), of the slider and shown by the function Fi(Xi;Vi), which has a static frictional force of Fsi=Fi(Xi;0) at rest. The equation of motion is m(∂Xi/∂t )=Kc(Xi+1-2Xi+Xi-1)-Kl(Xi-Vpt)-Fi(Xi;Vi) (1) In Equation (1), there is an implicate parameter ‘a’ which is the space between two sliders in the equilibrium state. The ratio κ=Kc/Kl has been defined by Wang [1995] to be the stiffness ratio of the system. This ratio represents the level of conservation of energy in the system. Larger κ is equivalent to stronger coupling beJeen-Hwa WANG 2 FIGURE 1. An N-degree-of-freedom dynamical spring-slider system. tween two sliders than between a slider and the moving plate, thus leading to a smaller loss of energy through the leaf spring or a higher level of conservation of energy in the system, yet opposite for smaller κ. Since the fault system is dynamically coupling with dissipation, κ must be a non-zero finite value. The Vp is in the order of ~10-12 m/s. The moving plate pulls a slider and thus gradually increases the elastic force, KcVpt, on it. When KcVpt is slightly higher than static frictional force, Fsi, at the i-th slider, the two forces are cancelled out each other and can be ignored during ruptures. After a slider moves, Fsi drops to Fdi (i.e., the dynamic frictional force). 2.2 FRICTION The frictional force between two contact planes is a very complicated physical process. Laboratory experiments have exhibited time-dependent static frictional strength of rocks [Dieterich, 1972] and velocity-dependent dynamic friction [Dieterich, 1979; Shimamoto, 1986]. Dieterich [1979] and Ruina [1983] proposed empirical, rateand state-dependent friction laws. The detailed description of friction laws and the debates concerning the laws and their application to earthquake dynamics can be found in some articles [e.g., Marone, 1998; Wang, 2009; Bizzarri and Cocco, 2006a; Bizzari 2011]. Several simple friction laws have been taken to theoretically and numerically study earthquake dynamics [see Wang, 2016]. The laws are: the velocity-dependent, weakening-hardening friction law [Burridge and Knopoff, 1967]; the slip-dependent friction law [Cao and Aki, 1984/85]; the nonlinear VW friction law [Carlson and Langer, 1989a,b; Carlson, 1991; Carlson et al., 1991; and Beeler et al., 2008]; and the piece-wise, linear velocity-weakening and weakening-hardening friction [Wang, 1995, 1996]. Purely velocity-dependent friction could yield unphysical phenomena and mathematically ill-posed problems as pointed out by Madariaga and Cochard [1994]. Ohnaka [2003] stressed that the pure velocity-dependent friction law is not a one-valued function of velocity. The problem has been deeply discussed by Bizzarri [2011]. Nevertheless, for a purpose of comparison the single-valued linear velocity-dependent friction law is still considered below. Friction is an important factor in controlling earthquake dynamics. Based on the 1-D BK model in the presence of linear VW friction with a decreasing rate, rw, of friction force with velocity, Wang [1996] found three types of rupture propagation: (1) subsonic type with rw>2(Klm) 1/2; (2) sonic type with rw=2(Klm) 1/2; and (3) supersonic type with rw<2(Klm) 1/2. Supersonic-type ruptures are non-causal, because vR is greater than the sound speed. Knopoff et al. [1992] stated that the system is asymptotic to dispersive-free elasticity in the continuum limit when rw=2(Klm) 1/2. They also found that large rw is more capable of generating large events than small rw. Carlson and Langer [1989a,b] used F(v)=1/(v+vc) where vc is the characteristic velocity. The related decreasing rate is 1/vc(1+v/vc) 2 with the values in the range of from 1 to 0 when vc varies from 0 to ∞. Hence, their friction law basically exhibits supersonic behavior with rw<2(Klm) 1/2, and thus is potentially capable of producing very large events. Wang [1997] also stressed the effect of frictional healing on earthquake ruptures. Several authors [Nur, 1978; Carlson and Langer, 1989a,b; Carlson, 1991; Carlson et al., 1991; Knopoff et al.,1992; Rice, 1993; Wang and Hwang, 2001] stressed the influence on earthquake ruptures due to hete


INTRODUCTION
first found the existence of slip pulses of earthquakes.Let T R and T D be the rise time of displacement at a site on a fault and the duration time of ruptures over the entire fault, respectively.He observed T R <<T D and non-uniform T R over the fault plane.This exhibits the complexity of slip over fault plane during an earthquake.Several groups of researchers also observed slip pulses from natural earthquakes [Wald et al., 1991;Wald and Heaton, 1994;Nakayama and Takeo, 1997;Chen et al., 2001;Huang et al., 2001;Nielson and Madariaga, 2003;Lee et al., 2007;Galetzka et al., 2015].Heaton [1990] interpreted the generation of slip pulse-likes using a self-healing assumption based on a crack model in the presence of velocity-weakening (VW) friction.He only considered a small strip which occurs immediately after the rupture front.Johnson [1990] discussed short T R by combining forward propagating waves and backward arresting (or healing) waves from the borders.But, Day et al. [1998] claimed that it is not necessary to consider self-healing as a factor for earthquake dynamics.
The slip pulses have been investigated based on the crack models in the presence of friction [Andrews, 1976[Andrews, , 1985;;Day, 1982;Papageorgiou and Aki, 1983;Boatwright, 1988;Andrews and Ben-Zion, 1996;Beeler and Tullis, 1996;Cochard and Madariaga, 1996;Perrin et al., 1995;Zheng and Rice, 1998;Nielsen et al., 2000;Lapusta et al., 2000;Ben-Zion and Huang, 2002;Nielsen and Madariaga, 2003;Coker et al., 2005;Rice et al., 2005;Ampuero and Ben-Zion, 2008;Urata et al., 2008;Ando et al., 2010;Garagash, 2012].Friction used by those authors includes VW friction, SW friction, velocity-and state-dependent friction, and thermal-pressurized friction.Results suggest that rupture modes are controlled by several factors, including friction laws, fault strengths, stress conditions on faults, energy and heat generated by faulting, scaling laws of faults, and spatial-temporal complexity of fault slip.In addition, some researchers considered geometrical heterogeneity of slip is a mechanism to stop earthquake rupture.Beroza and Mikumo [1996] suggested that the short T R could yielded by pre-existing stress with heterogeneous fault strengths.
The slip pulses have also been studied by some authors [Wu and Chen, 1998;Chen and Wang, 2010;Elbanna and Heaton, 2012] based on the 1-D spring-slider model (abbreviated as the 1-D BK model hereafter) proposed by Burridge and Knopoff [1967].From analytic studies by using SW friction, Wu and Chen [1998] claimed that SW friction can result in the self-healing slip pulse and the width of a pulse depends on v R and friction strength.From numerical studies by using VW friction Chen and Wang [2010] found the propagation of slip pulses with T R /T D <0.1 along the model.Their simulation results are in agreement with Heaton's observations.Elbanna and Heaton [2012] pointed out the differences between the BK model and crack models.According to linear elastic fracture mechanics, slip pulses are seldom generated in the continuum models because slipping region inside of a fault cannot release applied stress without continuous slip while rupture is extending.On the other hand, in the BK model slip pulses can be produced due to the following reason.Each slider can completely release stress exerted by the leaf springs by going back to their equilibrium position even while rupture is extending.In other words, each slider does not transfer stress after their stoppage and information on the length of rupture does not feedback.
Laboratory experiments also provide significant information on generation of slip pulses.Coker et al. [2005] observed the existence of both pulse-like and crack-like ruptures under certain conditions.Lykotrafitis et al. [2006] observed the pulse-like shear ruptures with self-healing.Lu et al. [2007] found that the rupture modes depend on the level of fault pre-stress and VW friction is important for earthquake dynam-ics.Biegel et al. [2008] found that off-fault damage can affect the slip-pulse velocity.
As mentioned previously, in order to generate slip pulses some authors prefer to VW friction, while other favor SW friction.In this study, I will investigate the frictional effects on the generation of slip pulses using the continuous form of the 1-D BK model with linear SW friction or linear VW friction.Hence, it is significant to examine which friction (SW or VW friction) is more capable than the other for yielding slip pulses.

ONE DIMENSIONAL SPRING SLIDER MODEL
2.1 MODEL Burridge and Knopoff [1967] proposed the 1-D BK model (see Figure 1), in which there are N sliders and springs.A slider with mass, m, is connected to its nearest two neighbors by a coil spring of stiffness, K c .Of course, the two end sliders are only connected to the respective one nearest slider.A moving plate with a constant velocity, V p , pulls each slider through a leaf spring of stiffness, K l .Each slider rests in its equilibrium state at time t=0.The position of i-th slider (i=1, …, N) is denoted by X i , which is measured from its initial equilibrium position, along the horizontal axis represented by the coordinate y.Hence, X i is in a function of y and t.Each slider is exerted by a frictional force between it and the moving plate.The frictional force is usually a function of displacement, X i , and particle velocity, V i (=dX i /dt), of the slider and shown by the function F i (X i ;V i ), which has a static frictional force of F si =F i (X i ;0) at rest.The equation of motion is In Equation (1), there is an implicate parameter 'a' which is the space between two sliders in the equilibrium state.The ratio κ=K c /K l has been defined by Wang [1995] to be the stiffness ratio of the system.This ratio represents the level of conservation of energy in the system.Larger κ is equivalent to stronger coupling be-FIGURE 1.An N-degree-of-freedom dynamical spring-slider system.
tween two sliders than between a slider and the moving plate, thus leading to a smaller loss of energy through the leaf spring or a higher level of conservation of energy in the system, yet opposite for smaller κ.Since the fault system is dynamically coupling with dissipation, κ must be a non-zero finite value.The V p is in the order of ~10 -12 m/s.The moving plate pulls a slider and thus gradually increases the elastic force, K c V p t, on it.When K c V p t is slightly higher than static frictional force, Fsi, at the i-th slider, the two forces are cancelled out each other and can be ignored during ruptures.After a slider moves, F si drops to F di (i.e., the dynamic frictional force).

FRICTION
The frictional force between two contact planes is a very complicated physical process.Laboratory experiments have exhibited time-dependent static frictional strength of rocks [Dieterich, 1972] and velocity-dependent dynamic friction [Dieterich, 1979;Shimamoto, 1986].Dieterich [1979] and Ruina [1983] proposed empirical, rate-and state-dependent friction laws.The detailed description of friction laws and the debates concerning the laws and their application to earthquake dynamics can be found in some articles [e.g., Marone, 1998;Wang, 2009;Bizzarri and Cocco, 2006a;Bizzari 2011].
Several simple friction laws have been taken to theoretically and numerically study earthquake dynamics [see Wang, 2016].The laws are: the velocity-dependent, weakening-hardening friction law [Burridge and Knopoff, 1967]; the slip-dependent friction law [Cao and Aki, 1984/85]; the nonlinear VW friction law [Carlson and Langer, 1989a,b;Carlson, 1991;Carlson et al., 1991;and Beeler et al., 2008]; and the piece-wise, linear velocity-weakening and weakening-hardening friction [Wang, 1995[Wang, , 1996]].Purely velocity-dependent friction could yield unphysical phenomena and mathematically ill-posed problems as pointed out by Madariaga and Cochard [1994].Ohnaka [2003] stressed that the pure velocity-dependent friction law is not a one-valued function of velocity.The problem has been deeply discussed by Bizzarri [2011].Nevertheless, for a purpose of comparison the single-valued linear velocity-dependent friction law is still considered below.
Friction is an important factor in controlling earthquake dynamics.Based on the 1-D BK model in the presence of linear VW friction with a decreasing rate, r w , of friction force with velocity, Wang [1996] found three types of rupture propagation: (1) subsonic type with r w >2(K l m) 1/2 ; (2) sonic type with r w =2(K l m) 1/2 ; and (3) supersonic type with r w <2(K l m) 1/2 .Supersonic-type ruptures are non-causal, because v R is greater than the sound speed.Knopoff et al. [1992] stated that the system is asymptotic to dispersive-free elasticity in the continuum limit when r w =2(K l m) 1/2 .They also found that large r w is more capable of generating large events than small r w .Carlson and Langer [1989a,b] used F(v)=1/(v+v c ) where v c is the characteristic velocity.The related decreasing rate is 1/v c (1+v/v c ) 2 with the values in the range of from 1 to 0 when v c varies from 0 to ∞.Hence, their friction law basically exhibits supersonic behavior with r w <2(K l m) 1/2 , and thus is potentially capable of producing very large events.Wang [1997] also stressed the effect of frictional healing on earthquake ruptures.Several authors [Nur, 1978;Carlson and Langer, 1989a,b;Carlson, 1991;Carlson et al., 1991;Knopoff et al.,1992;Rice, 1993;Wang and Hwang, 2001] stressed the influence on earthquake ruptures due to heterogeneous fault strengths on the fault.Carlson and her co-workers emphasized that de-localized events can be generated when the friction strengths over the fault plane is uniform.
In this study, I will analytically study the frictional effects caused by VW friction or SW friction on the generation of slip pulses by using the 1-D BK model.In order to perform analytic manipulation, only the linear laws are taken into account.The SW friction law (see Figure 2a) is: where X and X c are, respectively, the displacement and the characteristic distance.The VW friction law (see Figure 2b) is: where V=dX/dt is the velocity and V c is the characteristic velocity.The breaking strengths are uniform over the model.This means that only steady travelling waves are taken into account.

EQUATION OF MOTION
Define x i =X i -V p t.This gives X i =x i +V p t and V i =dX i /dt=dx i /dt+V p =v i +V p .Hence, Equation (1) becomes m(∂ 2 x i /∂t 2 )=K c (x i +1 -2x i +x i -1 )-K l x i -F i (x i +V p t;v i +V p ) (4) After a slider moves, V p t and V p can be neglected because of V p t<<x i and V p <<v i during ruptures.This makes Equation (4) be Analytic manipulation will be performed based on the continuous form of Equation ( 5).The details how to set up the continuous form can be found in Carlson and Langer [1989a,b] and Wang [2016], and only a brief description is given below.Equation ( 5) is first normalized and then transformed to its continuous form.In order to normalize the equation, some normalization parameters must be defined below.
The natural angular frequency of oscillation of a single slider attached to a leaf spring in the absence of friction in Equation ( 1) is denoted by ω 0 =(K l / m ) 1/2 , and thus the related natural period is T 0 =2π/ω 0 .Define τ=ω o t as the normalized time.Define D 0 =F 0 /K l as the characteristic slip distance of a slider exerted by a force F 0 through a spring with stiffness of K l .Since the ratio D 0 /V p is the loading time for a leaf spring to stretch enough for overcoming F 0 , V p /D 0 ω 0 is the ratio of the slipping time ω 0 -1 (=T 0 /2π) to the loading time.

SOLUTIONS BASED ON SLIP WEAKENING FRICTION
The normalized SW friction law from Equation ( 2) is φ(u)=1-u/Δ where Δ=X c /D 0 is the dimensionless characteristic distance.Equation (6) becomes Since φ(u) is negative and unreasonable when u>Δ, Equation ( 7) can work only for u<Δ.When the driving force reaches the static strength of the friction whose value is unit in Equation ( 7), stability at the slider is determined by the competition between the rate of friction |∂F strength /∂u|=1/Δ and the rate of stress-relaxation between the slider and the leap spring we have |∂F strength /∂u|> |∂F stress /∂u|.This means that stable motions cannot exist, and is in contrast with the known source time function for dynamic ruptures where an initial acceleration phase should exist.This again makes Equation ( 7) work only when u<Δ.Under SW regime, Equation ( 7) means that max{x p }=D 0 /γ 2 =X c /(Δ-1).This means Δ>1.
To solve Equation ( 7), the Laplace Transformation (LT, denoted by L), which can be seen in numerous textbooks [e.g., Papoulis, 1962], is used to transform it to a different form.The LT of Equation ( 7) is The solution of U includes the complementary solution, U c , and particular solution, U p , that is, U=U c +U p .According to the method given in Johnson and Kiokemeister [1968], the solution of Equation ( 8) is where ψ=(s 2 +1-Δ -1 ) 1/2 .There are two types of waves from Equation ( 9): The first one is the travelling wave represented by the first term along the +ξ direction and the second one along the -ξ direction in its right-handed-side, i.e., U c (ξ,s)=C 1 e -ψξ/h +C 2 e ψξ/h .The second one is the oscillation at a site given by the third term, i.e., U p (ξ,s)=-1/sψ 2 .The second term of the first type with ξ<0 can be rewritten as e -ψ|ξ|/h .The Inverse Laplace Transformation (ILT, denoted by L 1 ) of U c (ξ,s) with |ξ|/h>0 is where C=C 1 or C 2 , γ=(1-Δ -1 ) 1/2 , J 1 […] is the first-order Bessel function, and H(τ-|ξ|/h) is the unit step function (H(z)=0 as z<0 and H(z)=1 as z≥0) representing a travelling plane wave.Since τ=ω 0 t and ξ=y/D 0 are, respec-tively, the normalized time and normalized rupture distance, h=v R /D 0 ω 0 is the normalized rupture velocity.When the rupture propagates from 0 to ξ L , which is the normalized rupture length and equal to L/D 0 (L=the rupture length), the normalized duration time is τ D =ξ L /h, and thus the duration time is Let t r be the arrival time of the travelling wave at a site y, that is, t r =|y|/v R .Substituting u c (ξ,τ)=x c (y,t)/D 0 , ξ=y/D 0 , τ=ω 0 t, and t r =|y|/v R into Equation (10) gives Equation ( 11) shows a propagating wave which is usually represented by a function of the form G(t-|y|/v R ), where t'=t-|y|/v R is known as the retarded time for situations where causality holds [e.g., Perrin et al., 1995;Nielsen et al., 2000].The rise time, T R , is measured from t'=0 or t=t r =|y|/v to larger t'=t* when the wave amplitude and the particle velocity reach their respective peak values.The quantities inside {…} multiplied by CD 0 of Equation ( 11) show the wave amplitude.
It is interesting to investigate the variation in S s (y,t*) with y=v R t r .Inserting t* into S s (y,t) leads to S s (y,t*)=1-  Abramowitz and Stegun, 1972].When S s (y,t*)=0, we have 0.293(γω 0 )2(|y|/v R )=1.This gives |y|=y c =3.413v R /(γω 0 ) 2 and t c =y c /v R =3.413/(γω 0 ) 2 , where t c is t r related to y c .Clearly, the value of y c (>0) is a function of v R , γ, and ω 0 .Hence, S s (y,t*) changes from positive (when |y|<y c or t r <t c ) to negative (when |y|>y c or t r >t c ) by passing through zero (when |y|=y c or t r =t c ). Figure 3a displays an example of a normalized waveform S s (y,t)/S smax , where S smax is the value of S s (y,t) at t* as defined before in the figure, of the rupture wave having a propagation velocity of v R =2 km/sec and the predominant angular frequency of ω 0 =1 Hz at a position on the y-axis when the wave propagates with a travelling time t r =3.607 sec under the action of slipweakening friction force with γ=0.9.This plot shows a FIGURE 3. Figure shows S s (y,t)/S smax (S smax =S s (y,t*)) and S s (y,t)/S svmax (S svmax =S v (y,t*)) at y=v R t r with v R =2 km/sec and t r =3.607 sec: (a) for S s (yy,t)/S smax with γ=0.9 and ω 0 =1.0 Hz; and (b) for S v (y,t)/S svmax with σ=0.9 and ω 0 =1.0 Hz.
pulse-like wave with short T R .Since the value of t c of this case is 4.216 sec, the related value of t*, which is not displayed in Figure 3a, is positive as expected because of t r =3.607 sec<t c .It is necessary to consider a quantitative criterion to confirm the existence of a slip pulse.A simple way is taken based on the value of T R /T D , which is where T D =L/v R .In Equation ( 12), v R t r (denoted by L R ) is the length along which the rupture has propagated on the fault.Obviously, T R /T D increases with L R /L and decreases with increasing γ, or ω 0 , or t r .From the measures made by Heaton [1990], T R /T D is almost between 0.1 and 0.3.Chen and Wang [2010] gave T R /T D <0.1 from numerical simulations.Here, I assume that an acceptable quantitative criterion is T R /T D <0.3.Since, L R /L is always less than 1 during earthquake ruptures, the request of T R /T D <0.3 will be mainly dependent on γ, ω 0 , and t r .Nevertheless, it is easier to generate a slip pulse for long L than for short L.
The inequality T R /T D <0.3 can be represented by [1+3.901/(γω0 t r ) 2 ] 1/2 <1.3.When the rupture proceeds, t r always increases and makes the inequality hold.Hence, t r is less important on the inequality especially for t r >1 sec, which appears very soon after an earthquake starts to rupture.Since the slip pulses have frequencies in a small range of 0.5 to 2 Hz, ω 0 is not the main factor.Since the value of γ varies in a large range from 0 + to 1, γ must be the main factor in controlling the slip pulse.Considering an example of a slip pulse with ω 0 =1 Hz and γ=1, we have (1+3.901/tr 2 ) 1/2 <1.3.When t r >2.378 sec, the inequality T R /T D <0.3 always holds and thus the slip pulse appears.Figure 6 displays the plots of t r versus γ for ω 0 =0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom).The area between the curve and vertical line denotes the solution regime.Clearly, t r decreases with increasing ω 0 .The example shown in Figure 3a can meet the condition.Figure 5 displays the plots of T R /T D versus t r with v R =2 km/sec for various values of γ, ω 0 , and L: (a) for γ=0.1, 0.25, 0.5, and 1.0 (from top to bottom) when L=50 km and ω 0 =1 Hz; (b) for ω 0 =0.1, 0.2, 0.5, 1.0, and 2.0 Hz (from top to bottom) when L=50 km and γ=1; and (c) for L=10, 30, 50, 70, 100 km (from top to bottom) when γ=1 and ω 0 =1 Hz.Obviously, T R /T D decreases very rapidly with increasing t r , thus meaning that T R decreases when the slip pulse propagates along the fault.Meanwhile, T R also decreases when γ, ω 0 , and L increase.
The second case represents a positive sine-type function and thus can represent a vibration at all sites because it is site-independent from Equation (13).
Figure 8 shows the typical velocity waveform (in the solid plus dotted curve) and the typical displacement waveform (in the solid curve).Since the wave propagates from -y to +y, it stops at the time instant when v=0.In Figure 8, T p is the period of velocity waveform.
Since T R ≈T p /2, the ratio of T p over T D , i.e., T p /T D , is a good indication to show whether a slip pulse can exist or not.The ratio where v R T 0 denotes a characteristic distance of wave propagation in the natural period.For real M≥6 earthquakes v R varies from 1.5 to 4 km/s, L from 30 to 300 km, and T 0 from 10 -1 to 10 sec. Figure 9 shows the plots of T p /T D versus Δ: (a) for v R T 0 =0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km (from bottom to top) when ω 0 L=80 km/s; and (b) for L=20, 40,60,80,100,120,140, and 160 km (from top to bottom) when v R T 0 =2 km.
As mentioned above, the quantitative criterion to confirm the existence of a slip pulse is T R /T D <0.3.Now, T R is almost equal to T p .Hence, a pulse-like oscillation can exist when T p /T D <0.3.In Figure 9 exists only when Δ>2 (as displayed by a vertical line in the figure).Obviously, the major portion of each curve is below the upper bound, thus suggesting the existence of a pulse-like oscillation in the solution regime.
The value of T R of a slip pulse at a site is measured from the arrival time, t r , at a site to the time, t*, when the amplitude of slip pulse reaches its peak value.This gives T R =t*-t r .To obtain t*, we must calculate the time when S v (y,t) reaches its peak value from the necessary condition: dS v (y,t)/d t =0.This can be obtained by taking dS v (y,t)/d t =[dS v (θ)/dθ]dθ/dt=0.Since dθ/dt=t(t 2 -t r 2 ) -1/2 cannot be zero when t>0, we only need to consider if dS s (θ)/dθ equals zero.Mathematical manipulation leads to dS v (θ)/dθ=-(σ 2 ω 0 2 t r )e ϖt {ϖJ 1 (θ)-[J 0 (θ)-J 1 (θ)/θ]}.From the rule of Bessel functions that not any two Bessel functions have the same zero, the function ϖJ 1 (θ)-[J 0 (θ)-J 1 (θ)/θ] cannot be zero.When J 1 (θ)=0 at θ=θ*, J 0 (θ*) is not zero.This results in ϖJ 1 (θ*)-[J 0 (θ*)-J 1 (θ*)/θ*]= -J 0 (θ*)≠0.When J 0 (θ)=0 at θ=θ*, J 1 (θ*) is not zero.This results in ϖJ 1 (θ*)-[J 0 (θ*)-J 1 (θ*)/θ*]=(ϖ+1/θ*)J 1 (θ*)≠0.Meanwhile, e ϖt is equal to 1 at t=0 and larger than 1 when t>0.Hence, dS v (θ)/dθ cannot be zero and there is not an extremum for S v (y,t).This cannot yield T R /T D <0.3 and thus slip pulses do not exist for VW friction.Unlike S s (y,t*), Sv(y,t) does change from positive to negative passing through zero.An example of a normalized waveform S v (y,t)/S vmax , where S vmax is the maximum value of S v in the figure, of the rupture wave having a propagation velocity of v R =2 km/sec and the predominant angular frequency of ω 0 =1 Hz at a position on the y-axis when the wave propagates with a travelling time t r =3.607 sec under the action of velocity-weakening friction force with σ=0.9.This plot which shows an increase in the amplitude with time, thus suggesting a crack-like rupture with long T R .Hence, SV friction can yield pulse-like ruptures only when υ>>1 or V c >>1 and in general produces crack-like ruptures.
The third term of Equation ( 18), i.e., U p (ξ,s)= -1/s(s 2 +1-sυ -1 ), is not a function of locality and thus only represents the oscillations at all sites on the fault.But, it is still significant to examine if it behaves like a pulse-like oscillation or not.The function U p (ξ,s) can be re-written as where e=(1-4υ 2 ) 1/2 .For u c (ξ,τ),υ must be larger than 0.5 as mentioned above.Let q=(4υ 2 -1) 1/2 =ie.Consider a right triangle with three sides: the longest side with a length of R=(1 2 +q 2 ) 1/2 =2υ, and the other two with lengths of q and 1, respectively.The angle between the longest side and the side with L=1 unit is set to be θ.Hence, we have cos(θ)=1/2υ, sin(θ)=q/2υ, and tan(θ)=q.
The plot of T o /T p versus υ is displayed in Figure 7b.Clearly, increasing rate of T o /T p with υ is high when υ<1.744 and becomes low when υ>1.744.Although T p is determined by T 0 (0.1<T 0 <10) here, the value of T 0 for natural faults is not exactly determined.T 0 is defined by the stiffness of a leaf spring and the mass of the slider.As widely known, natural fault systems are usually selfsimilar, so that the unique characteristic period of faulting is T D .Even if natural faults show pulse-like behaviour, their own T p is not necessarily related to T 0 .Elbanna and Heaton [2012] claimed that simultaneous movement of only 10 to 20 sliders in a spring-slider system consisting of 5000 sliders lead to a slip pulse.
From Figure 8, T p /T D =(1-1/4υ 2 ) -1/2 v R T 0 /L is a good indication of the presence of a slip pulse for VW friction.Figure 10 shows the plots of T p /T D versus υ: (a) for v R T 0 =0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, and 4.0 km (from bottom to top) when L=80 km; and (b) for L=20, 40, 60, 80, 100, 120, 160, and 180 km (from top to bottom) when v R T 0 =2.0 km.As mentioned above, the quantitative criterion to confirm the existence of a slip pulse is T R /T D <0.3.Now, T R is almost equal to T p .The upper bound of T p /T D in the figure is 0.3.Meanwhile, the solution exists only when υ>1.744 (as displayed by a vertical line in the figure).Obviously, the major portion of each curve is below the upper bound, thus suggesting the existence of a pulse-like oscillation in the solution regime.

DISCUSSION
The theoretical analyses for the conditions of generating slip pulses are made based on the continuous form of the 1-D BK model in the presence of the linear friction laws: f(u)=1-u/Δ for SW friction as well as f(v)=1v/υ for VW friction.The parameters Δ and υ are, respectively, the characteristic distance and the characteristic velocity of the respective law.First, it is neces-sary to consider physical implications of the results.For SW friction, variable transformation of w=u+γ 2 reduces Equation ( 7) to the Klein-Gordon equation [Polyanin, 2002]: ∂ 2 w/∂τ 2 =h 2 (∂ 2 w/∂ξ 2 )-γ 2 w.This implies that the strength of the leaf spring has a factor of γ 2 due to presence of friction, so that γ 2 per stored energy within the leaf spring is conserved and 1-γ 2 per the energy is dissipated.We can see that 1-γ 2 =D 0 /X c is a ratio of actual frictional work dissipated due to sliding from an area under SW line in Figure 2a.Since the stiffness of a leaf spring multiplied by γ -2 controls energetics, dependence of behavior of the system on k/γ 2 should be discussed.
For VW friction, the restoring force should always have the same sign for stability of the system.Variable transformation of w=u+1 reduces Equation (15) to ∂ 2 w/∂τ 2 =h 2 (∂ 2 w/∂ξ 2 )-w+w'/υ, where w'=dw/dτ.Without loss of generality, the restoring force can be assumed as negative, i.e., -w+w'/υ≤0.Let F[w] be the Fourier transformation of w.Then the following holds: so that should be required, otherwise the friction is not restoring force but repulsive force in high-frequency content.Such a compact supported spectrum in frequency domain is, however, not able to be a compact supported signal in time domain.
There are the complementary solution, x c (y,t) and particular solution, x p (y,t), of the equation of motion.For SW friction, the function x c (y,t) shows a slip pulse when 1>γ=(1-Δ -1 ) 1/2 >0.T R and T R /T D both decrease rapidly with increasing t r , thus meaning that T R and T R /T D are both reduced when the slip pulse propagates along the fault.This means that the slip pulse is generated when the ruptures are far away from the nucleation point.Meanwhile, T R and T R /T D also both decrease when γ, ω 0 , and L increase and when v R is re-duced.Higher γ associated with a low decreasing rate of friction with slip is easier to produce a slip pulse than lower γ related to a higher decreasing rate.A decreases in T R and T R /T D with increasing ωo means that slip pulses should have higher angular frequencies and cannot be long-period rupture waves.The decreases in T R and T R /T D with increasing L mean that a slip pulse can be more easily produced in a long fault than in a short one.Hence, it would be difficult to detect a slip pulse on a short fault.Equation ( 21) shows that lower v R can more easily result in small T R and T R /T D than higher v R .This indicates that the slip pulse has a relatively slow propagation speed.However, v R is less important than other parameters, because the value of v R t r /L in Equation ( 21) is always smaller than 1.
For VW friction when υ>>1; while it shows a cracklike rupture for VW friction when υ is not too big.The present results suggest that unlike Heaton [1990], SW friction is easier to produce a slip pulse than VW friction.Although Perrin et al. [1995] assumed that not all friction laws result in steady travelling pulses, the present result exhibits that the two types of friction in use can generate a slip pulse under their respective ranges of model parameters.
For the particular solution, the ratio of T p over T D of a rupture is a good indication to show the possible existence of a slip pulse, because the T R of slip at a site is shorter than T p .For SW friction with Δ>1, the solution clearly shows oscillations with a predominant period of T p .Computational results in Figure 9 reveal a decrease in T p /T D with increasing Δ.Longer (shorter) Δ means a slower (faster) decay of friction with slip.A slower decay of friction with increasing slip is easier to generating a pulse-like oscillation than a faster decay of friction.Based on the definition of Δ=X c >D 0 , Δ>1 gives X c >D 0 .This suggests that when the characteristic length is longer than the characteristic distance, the pulse-like oscillation can be generated.Figure 9a shows T p /T D <0.4 (or T R /T D <0.2) when Δ>1.2. Figure 9b shows that except for L=20 km, T p /T D <0.4 (or T R /T D <0.2) when Δ>1.5.Theoretical results suggest that Δ>1.5 is a significant condition for generating a pulse-like oscillation at a site under SW friction.
For VW friction, when υ>0.5 this solution shows a sine-function oscillation at a site.Because of υ=V c /D 0 ω 0 , the inequality υ>0.5 leads to V c >D 0 ω 0 /2.This means that a pulse-like oscillation at a site can be generated when V c >D 0 ω 0 /2.Since V c is the characteristic velocity of VW friction law, higher V c indicates a slower decay of friction with velocity.The present result suggests that a slower decay of friction is more capable of generating the pulse-like oscillation than a faster one.Unlike the pulse-like oscillation produced by SW friction, the amplitude of pulse-like oscillation generated by VW friction could increase with time due to an extra term e τ/2υ .This suggests that VW friction can cause higher wave energy than SW friction.Smaller υ will cause a faster increase in wave amplitude than larger υ. Figure 10 reveals a decrease in T p /T D with increasing υ.Higher (lower) V c means a slower (faster) decay of friction with velocity.Clearly, a slower decay of friction with velocity is easier to generating a slip pulse than a faster decay of friction.
Because of υ=V c /D 0 ω 0 , υ>1 means V c >D 0 ω 0 /2.When the characteristic velocity of the model is higher than the rate of a slider in the characteristic distance, a pulse-like oscillation can be generated.Figure 10 shows T R /T D <0.4 (or T R /T D <0.2) when υ>0.6.Hence, υ>0.6 could be a significant condition for generating a pulse-like oscillation at a site under VW friction.
In addition, Figures 9a and 10a both show a decrease in T p /T D with decreasing v R T 0 , which is the characteristic rupture distance.Lower v R is more capable of generating a pulse-like oscillation than higher v R .Meanwhile, shorter T 0 is more capable of generating a pulse-like oscillation than longer T 0 .Because of T 0 = (m/K l ) 1/2 /2π=(ρ A /κ l ) 1/2 /2π, strong coupling (denoted by κ l ) between the moving plate and a fault system is capable of producing a pulse-like oscillation than weak coupling.On the other hand, lighter or lower-density fault rocks are easier to producing a pulse-like oscillation than heavier or higher-density fault rocks.Figures 9b and 10b exhibit a decrease in T p /T D with increasing L, that is, longer L is more capable of generating a pulse-like oscillation than shorter L. However, the difference in the values of T p /T D between two sequential rupture lengths decreases with increasing L. This means that when L is longer than a certain value, the effect of rupture length on T p /T D is reduced.Of course, the L-effect also depends on v R and T 0 .
A comparison between Figure 9 and Figure 10 shows that T p /T D is in general smaller for VW friction than for SW friction, thus suggesting that the former is more capable of producing a pulse-like oscillation at site than the latter.In addition, under VW friction a pulse-like oscillation can be produced even though L is short.
Based on Equation (7), Wang [2016] obtained that the complementary solution exhibits ω -1 scaling in the whole range of ω for SW friction.But, for the particular solution SW friction results in spectral amplitudes only at three values of ω.Based on Equation (15), Wang [2016] obtained that for VW friction with υ>0.5, the spectral amplitude versus ω exhibits almost ω 0 scaling when ω is lower than the corner angular fre-quency, ω c , which is independent on υ and increases with ω 0 .When ω>ω c , the spectral amplitude monotonically decreases with ω following a line with a slope value of -1, which is the scaling exponent.This again confirms that crack-like rupture is generated by VW friction.Hence, it is easier to yield slip pulses from SW friction than from VW friction.

CONCLUSIONS
Seismological observations show the existence of slip pulses with T R /T D <0.3.For the present model, there are complementary and particular solutions of the equation of motion.For SW friction, the complementary solution shows a slip pulse when γ>1. T R and T R /T D both decrease rapidly with increasing t r , thus meaning that T R and T R /T D both decrease when the slip pulse propagates along the fault.T R and T R /T D also both decrease when γ, ω 0 , v R , and L increase.For VW friction, a slip pulse is yielded when υ>>1 or V c >>1 and the crack-like ruptures is generated when υ is not too big.When υ>>1 or V c >>1, the results produced by VW friction are essentially similar with those by SW friction.For the two types of friction, a slower decay of friction is more capable of generating slip pulses than a faster one.Lower v R is more capable of generating a slip pulse than higher v R .Longer L is more capable of generating slip pulses than shorter L. Of course, the importance of v R and L are lower than γ and ω 0 .
For the particular solution, the ratio of predominant period (T p ) of oscillations at a site is slightly longer than T R , and thus T p /T D , is a good indication to show the existence of pulse-like oscillations at a site because of T R <T p .Results show the existence of pulse-like oscillations for SW friction when Δ>1.6 (or X c >1.6D 0 ) and for VW friction when υ>0.5 (or V c >0.5D 0 ω 0 ).T p /T D decreases with increasing Δ for SW friction and with increasing υ for VW friction.For the two types of friction, T 0 /T D and T p /T D both decreases when v R and L increase.For the two types of friction, a slower decay of friction is more capable of triggering pulse-like oscillations at a site than a faster one.Shorter T 0 is more capable of generating pulse-like oscillations than longer T 0 .In other words, strong coupling between the moving plate and a fault system is capable of producing a slip pulse than weak coupling.Lighter or lowerdensity fault rocks are easier to producing pulse-like oscillations than heavier or higher-density fault rocks.Longer L is more capable of generating pulse-like oscillations than shorter L.