Propogation of waves in a linear elastic fluid

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International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

Propogation of waves in a linear elastic #uid E. Momoniat∗ Centre for Di erential Equations, Continuum Mechanics and Applications, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa Received 1 March 2002; received in revised form 11 November 2002

Abstract The non-classical symmetry method is used to determine particular forms of the arbitrary velocity and forcing terms in a linear wave equation used to model the propogation of waves in a linear elastic #uid. The behaviour of solutions derived using the non-classical symmetry method are discussed. Solutions satisfy a given initial pro0le and wave velocity. For some solutions the arbitrary forcing terms and wave velocity can be written in terms of the initial wave pro0le. Relationships between the arbitrary forcing, arbitrary velocity and the solution are derived. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Nonclassical symmetry method; Linear wave equation; Wave propogation; Linear elastic #uid

1. Introduction In this paper we investigate non-classical contact symmetries admitted by the linear wave equation utt − F(t; x)2 uxx = H (t; x);

(1.1)

subject to the initial conditions u(x; 0) = f(x);

ut (x; 0) = g(x);

(1.2)

where F and H are arbitrary functions of t and x, f and g are arbitrary functions of x only and subscripts denote di7erentiation unless otherwise indicated. Functional forms of F and H as well as solutions u(x; t) are obtained from the non-classical contact symmetries. These solutions are discussed. ∗ Centre for Symmetry Analysis, School of Computational and Applied Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, Johannesburg, South Africa. Tel.: +27-11-7176137; fax: +27-11-403-9317. E-mail address: [email protected] (E. Momoniat).

The author’s interest in (1.1) comes from the study of wave propogation in a linear elastic #uid [1] where the velocity of the wave, F, depends on t and x and there is a non-constant forcing term, H , present. Equations of the form (1.1) can arise in the study of small transverse vibrations of a string with variable density [2]. Other applications of (1.1) can be found in [3]. Bluman and Kumei have analysed the linear wave equation utt − c(x)2 uxx = 0 and its equivalent system vt = ux , ut = c(x)2 vx using the Lie group method [4]. They have solved the linear wave equation utt − c2 (x)uxx = 0 for wave speeds c(x) corresponding to two-layered media with smooth transition from layer to layer [5]. Symmetries of non-linear heat and wave equations are presented by Oron and Rosenau [6]. True contact symmetries admitted by a more general form of (1.1) have been investigated by Momoniat [7]. It was found that the wave speeds F had to be quadratic in the variables t and x. Functional forms of H could not be determined. Solutions of (1.1) were found iteratively using the admitted contact

0020-7462/04/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0020-7462(02)00209-3

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E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

transformation. Many books have been written about the application of the classical Lie group method to solve di7erential equations [8–13]. Information on contact transformations can also be found in many books [8,13–16]. The non-classical symmetry method was introduced in 1969 by Bluman and Cole [17] to obtain solutions of the linear heat equation which cannot be found by the classical Lie group method. The non-classical symmetry method has been applied to various equations to obtain solutions which could not be found by the classical Lie group method [18–21]. Nucci has iterated the non-classical method [22] and provided examples to show how some well-known solutions of evolution type equations can be obtained using this method. The non-classical symmetry method is closely related to the conditional symmetry method. Extensions of the conditional symmetry method to higher derivatives are applied to evolution equations by Zhdanov [23]. Nucci has shown that Zhdanov’s conditional Lie-BKacklund symmetries are related to the iterative non-classical symmetries method [24]. The conditional Lie-BKacklund symmetry method is closely related to the generalised conditional symmetry method. Qu has used this generalised conditional symmetry method to obtain solutions of a non-linear di7usion equation with a source [25]. Recently, Qu has applied this method to obtain solutions of some non-linear partial di7erential equations [26]. Mahomed and Qu have introduced the notion of approximate conditional symmetries [27]. Solutions to a class of non-linear heat and wave equations with a small parameter are obtained. The purpose of this paper is to provide a mathematical, rather than physical description of the propogation of waves in a linear elastic #uid. The non-classical symmetry method is employed for this purpose as solutions obtained are not derivable from the classical symmetry methods and may display some interesting physical behaviour [17–21]. Unlike the classical approach in which the initial conditions also have to be invariant under the group admitted by the solution, this does not have to be the case with solutions obtained from the non-classical symmetry method. Hence we are able to impose (1.2) on any solutions we derive. The non-classical contact symmetry method is used as the author has had some success in classifying the arbitrary velocity term in (1.2) using the classical contact symmetry method [7]. The non-classical contact

symmetry method reduces (1.1) to four possible cases for which a solution is derived. The arbitrary forcing term H (t; x) is given in terms of the initial wave pro0le f(x). Relationships between the wave velocity and forcing terms are derived. This relationship can be used to determine the wave velocity when the forcing term is known. Another relationship is derived between the wave velocity, the forcing term and the solution. Some of the solutions obtained describe scalings of the initial wave pro0le. Others describe solutions in which both the periodicity and amplitude of the wave change over a period of time. The paper is divided up as follows: in Section 2 we present some mathematical preliminaries. In Section 3 non-classical symmetries admitted by (1.1) are investigated. Solutions obtained from these non-classical contact symmetries are discussed in Section 4. 2. Mathematical preliminaries We 0rst introduce the notion of a group of point transformations. The set of transformations of the independent variables t, x and the dependent variable u, tN = tN(t; x; u; a);

xN = x(t; N x; u; a);

uN = u(t; N x; u; a);

(2.1)

where a is the group parameter, form a one parameter group of point transformations if the group properties hold. The set of transformations (2.1) can be extended to include 0rst derivatives of the dependent variable u such that they form a set tN = tN(t; x; u; ut ; ux ; a);

xN = x(t; N x; u; ut ; ux ; a);

uN = u(t; N x; u; ut ; ux ; a);

ut = ut (t; x; u; ut ; ux ; a);

ux = ux (t; x; u; ut ; ux ; a);

(2.2)

where a is a real parameter. The set of transformations (2.2) form a one parameter group of contact transformations if they satisfy the group properties and @uN @uN (2.3) ut = ; ux = ; @tN @xN holds [8]. The generator of a group of contact transformations (2.2) is given by X = 1 (t; x; u; ut ; ux )@t + 2 (t; x; u; ut ; ux )@x + (t; x; u; ut ; ux )@u ;

(2.4)

E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

449

where @t = @=@t, @x = @=@x; : : : . Prolongations of the contact transformation generator (2.4) are given by

where A is an arbitrary function of t, x, u and ux . The required di7erential consequence of (2.14) is given by

X˜ = X + 1 @t + 2 @x + 11 @utt + 12 @utx

utx = uxx Aux + ux Au + Ax :

+ 22 @uxx + · · · :

(2.5)

W =  − u t 1 − u x 2 :

(2.6)

The coePcients 1 , 2 and  of the contact transformation generator (2.4) can be written in terms of W as follows: 1 = −Wut ;

2 = −Wux ;

 = W − ut W u t − ux W u x :

Secondly, we consider the case W = ux − B(t; x; u; ut )

Lie’s characteristic function is de0ned by

(2.7)

The formulae for the i s can easily be written in terms of W as

(2.15) (2.16)

with corresponding invariant surface condition given by ux = B(t; x; u; ut ):

(2.17)

The required di7erential consequences of (2.17) are given by utx = utt But + ut Bu + Bt ;

(2.18)

uxx = BBu + Bx + But (utt But + ut Bu + Bt ):

(2.19)

Higher-order prolongations can be calculated from the prolongation formulae

By substituting (2.17), (2.18) and (2.19) into the determining equation (2.11) we reintroduce the variable utt into the determining equation. Substituting (1.1) for utt reintroduces uxx . We have a dilemma which can easily be resolved by assuming

i1 i2 :::is = Di1 : : : Dis (W ) − Wuj uji1 :::is ;

W = ux − B(t; x; u)

1 = Wt + ut Wu ;

2 = Wx + ux Wu :

(2.8)

s = 1; 2; : : : ; (2.9)

with summation on j, where Di is the operator of total di7erentiation given by Di = @xi + ui @u + uij @uj + · · · :

(2.10)

If W is linear in the 0rst derivatives ut and ux , then the contact transformation generator (2.4) reduces to an extended Lie point transformation generator [8]. To calculate the contact symmetry generators of (1.1) we solve the determining equation X˜ (utt − F(t; x)2 uxx − H (t; x))|utt =F(t;x)2 uxx +H (t;x) :

(2.11)

To obtain non-classical contact symmetries the invariant surface condition W (t; x; u; ut ; ux ) = 0;

(2.12)

(2.20)

with corresponding invariant surface condition ux = B(t; x; u):

(2.21)

The required di7erential consequences from (2.21) are given by utx = ut Bu + Bt ;

(2.22)

uxx = BBu + Bx :

(2.23)

3. Non-classical contact symmetries The determining equation to obtain non-classical symmetries of (1.1) with (2.13) can be written as X˜ (utt − F(t; x)2 uxx − H (t; x))|(2:13); utt =F(t;x)2 uxx +H (t;x);(2:14);(2:15) :

(3.1)

also has to be imposed on (2.11). We consider 0rstly the case

Separating the determining equation (3.1) by coeP2 cients of uxx we obtain

W = ut − A(t; x; u; ux )

(F 2 − A2ux )Aux ux = 0:

(2.13)

and its corresponding invariant surface condition

Hence, either

ut = A(t; x; u; ux );

A(t; x; u; ux ) = ux C1 (t; x; u) + C2 (t; x; u);

(2.14)

(3.2) (3.3)

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E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

where C1 and C2 are arbitrary functions of t, x and u or A(t; x; u; ux ) = ux F(t; x) + G(t; x; u);

(3.4)

where G is an arbitrary function of t, x and u. Substituting (3.3) into the determining equation (3.1) separating by uxx ux we obtain 2

(F −

C12 )C1u

=0

(3.5)

and therefore (a) C1 = C1 (t; x)

or

(b) C1 = F(t; x);

(3.6)

where C1 is now an arbitrary function of t and x only. Substituting (3.6(a)) into the determining equation (3.1) and separating by uxx we obtain F 2 C1x + F(−C1 Fx + Ft ) − C1 C1t = 0:

(3.7)

We cannot reduce (3.7) further since both C1 and F are arbitrary functions of t and x. Substitute (3.6(b)) into the determining equation (3.1). We observe from (3.7) that all coePcients of uxx go to zero. We note that by substituting (3.6b) into (3.3) we obtain (3.4). Now substituting (3.3) with (3.6b) into the determining equation (3.1) and separating by coePcients of ux we obtain −2Fx Ft − Ftt − 2Ft C2u + F 2 (Fxx + 2C2xu ) − 2F(C2 C2uu + C2tu ) = 0:

(3.8)

We make the assumption that C2 = uC3 (t; x) + C4 (t; x);

(3.9)

where C3 and C4 are arbitrary functions of t and x. Eq. (3.8) reduces to F 2 (2C3x + Fxx ) − 2C3 Ft − 2Fx Ft − 2FC3t − Ftt = 0:

(3.10)

Substituting (3.9) into the determining equation (3.1) and separating by coePcients of u we obtain F 2 C3xx − 2C3x Ft − 2C3 C3t − C3tt = 0:

(3.11)

The remaining terms from (3.1) are given by

where A1 and A2 are arbitrary constants. Substituting (3.15) into (3.14) and solving the resulting quasi-linear partial di7erential equation by the method of characteristics we obtain A1 −2tC3 H = etC3 h();  = x + tA2 + e : (3.16) 4C32 The invariant surface condition (2.14) can be written ut = ux F + uC3 + C4 :

(3.17)

Substituting (3.15) into (3.17) and solving the resulting quasi-linear partial di7erential equation by the method of characteristics we obtain C4 u(x; t) = − + etC3 K(): (3.18) C3 Imposing the initial conditions (1.2) on (3.18) we obtain that C4 = 0. We obtain transformed initial curves where A1 u(x; 0) = f(x); N xN = x + : (3.19) 4C32 The transformed initial velocity is given by N g(x) N = C3 f(x) ut (x; 0) = g(x); N 
A1 f (x); + A2 − N 2C3

(3.20)

N x. N The solution (3.18) can be writwhere f (x)=df=d ten as u(x; t) = etC3 f():

(3.21)

Substituting (3.21) with (3.15) and (3.16) into (1.1) we obtain h() = C32 f() + 2A2 C3 f ():

(3.22)

We now make the assumption that F=F(x). Solving (3.13) we get that

−HC3 − FHx + F 2 C4xx − 2C4x Ft + Ht − 2C4 C3t − C4tt = 0:

We assume that F = F(t). Solving the resulting ordinary di7erential equation from (3.13) we obtain A1 −2tC3 e ; (3.15) F(t) = A2 − 2C3

(3.12)

We assume C3 and C4 are arbitrary constants. Then (3.11) is satis0ed and (3.10) and (3.12) reduce to F 2 Fxx − 2(C3 + Fx )Ft − Ftt = 0;

(3.13)

Ht − FHx − C3 H = 0:

(3.14)

F = A 4 x + A5 ;

(3.23)

where A4 and A5 are arbitrary constants. Substituting (3.23) into (3.14) and solving by the method of characteristics we obtain 1 tA4 H = etC3 h();  = e (A4 x + A5 ): (3.24) A4

E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

Substituting (3.23) into (3.17) and solving by the method of characteristics we obtain C4 u(x; t) = − + etC3 K(): (3.25) C3 Imposing the initial conditions (1.2) on (3.25) we obtain again that C4 = 0. We again obtain transformed initial curves where 1 u(x; 0) = f(x); N xN = (xA4 + A5 ): (3.26) A4 The transformed initial velocity is given by ut (x; 0) = g(x); N

g(x) N = C3 f(x) N + A4 xf N  (x): N (3.27)

The solution (3.25) can be written as u(x; t) = etC3 f():

(3.28)

Substituting (3.28) with (3.23) and (3.24) into (1.1) we obtain h() = C32 f() + A4 (A4 + 2C3 )f ():

(3.29)

Solving (3.13) for F(t; x) is not a trivial task. Instead of continuing with this analysis, we make the assumption in (2.13) that W = ut − G(t; x; u):

(3.30)

The invariant surface condition corresponding to (3.30) is given by ut = G(t; x; u);

(3.31)

where G is an arbitrary function of t, x and u. The required di7erential consequence from (3.31) is given by utx = ux Gu + Gx :

(3.32)

By imposing (3.30) with (3.31) and (3.32) the required determining equation becomes X˜ (utt − F(t; x)2 uxx −H (t; x))|(3:30); utt =F(t;x)2 uxx +H (t;x);(3:31);(3:32) : (3.33)

451

From (3.34) we observe that F = F(x). Integrating (3.35) and (3.36) we obtain G = uD1 (t) + D2 (t; x);

(3.37)

where D1 is an arbitrary function of t only and D2 is an arbitrary function of t and x. Substituting (3.37) into (3.33) and separating by coePcients of u we obtain D1tt + 2D1 D1t = 0:

(3.38)

Solving the ordinary di7erential equation (3.38) we obtain D1 (t) = −D3 tan(tD3 );

(3.39)

where D3 is a constant. Substituting (3.39) into the determining equation (3.33) we obtain D3 H tan(tD3 ) + 2D32 sec(tD3 )2 C2 + F 2 C2xx + Ht − C2tt = 0:

(3.40)

Since C2 is arbitrary we make the assumption C2 = 0 then (3.40) reduces to the quasi-linear partial di7erential equation D3 H tan(tD3 ) + Ht = 0;

(3.41)

which can be solved by the method of characteristics to give H (t; x) = D4 (x) cos(tD3 );

(3.42)

where D4 is an arbitrary function of x. The invariant surface condition (3.31) becomes ut = −uD3 tan(tD3 );

(3.43)

which can be solved by the method of characteristics to give u = D5 (x) cos(tD3 ):

(3.44)

Substituting (3.44) with (3.42) and F =F(x) into (1.1) we obtain F(x)2 D5xx (x) + D32 D5 (x) + D4 (x) = 0:

(3.45)

uxx : FFt = 0;

(3.34)

ux2 : Guu = 0;

(3.35)

Thus, given the wave velocity F(x) and the coePcient D4 (x) of the forcing term H , we can calculate D5 (x) from (3.45) and hence the solution (3.44) of (1.1). However, if we impose the initial conditions (1.2) on (3.44) we obtain D5 (x) = f(x) and g(x) = 0. Thus (3.44) can be written

ux : Gxu = 0:

(3.36)

u(x; t) = f(x) cos(tD3 );

Separating (3.33) by coePcients of uxx and powers of ux we obtain

(3.46)

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E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

where f(x) is the initial curve. The di7erential equation (3.45) can now be written

Solving (3.57) by the method of characteristics we obtain

F(x)2 f (x) + D32 f(x) + D4 (x) = 0;

H (t; x) = E4 (t)e



2

(3.47)

2

where f (x) = d f=d x . Thus, if we know the initial curve f(x) and the wave velocity F(x) the coePcient D4 (x) of the forcing term H is given by 2



D4 (x) = −(F(x) f (x) +

D32 f(x)):

(3.48)

Similarly, if the coePcient D4 (x) of the forcing is known, the wave velocity can be obtained from F(x)2 =

D2 f(x) + D4 (x) − 3 : f (x)

(3.49)

Finally, we consider the case (2.20). The required determining equation is given by X˜ (utt − F(t; x)2 uxx − H (t; x))|(2:20); utt =F(t;x)2 uxx +H (t;x);(2:21);(2:22);(2:23) : (3.50) Separating the determining equation by coePcients of ut2 and ut we obtain ut2 : Buu = 0;

(3.51)

ut : Btu = 0:

(3.52)

Therefore B = uE1 (x) + E2 (t; x);

(3.53)

where E1 is an arbitrary function of x and E2 is an arbitrary function of t and x. Substituting (3.53) into (3.50) and separating by coePcients of u we obtain F(2E1 E1x + E1xx ) + 2Fx (E12 + E1x ) = 0: Writing (3.54) as 2E1 E1x + E1xx Fx =− F 2(E12 + E1x ) and integrating with respect to x we obtain E3 (t) ; F(t; x) =  E1 (x)2 + E1x (x)

(3.54)

E1 (x)

;

(3.58)

where E4 is an arbitrary function of t. The invariant surface condition (2.21) can be written ux = uE1 (x);

(3.59)

which can be solved by the method of characteristics to give u(x; t) = E5 (t)e



E1 (x)

;

(3.60)

where E5 is an arbitrary function of t. Substituting (3.60) with (3.58) and (3.56) into (1.1) we observe that E3 (t), E4 (t) and E5 (t) have to satisfy the relation E5tt (t) − E3 (t)2 E5 (t) − E4 (t) = 0:

(3.61)

For some cases of E3 (t) and E4 (t) (3.61) has to be solved numerically. Imposing the boundary conditions (1.2) on (3.60) we obtain f (x) E1 (x) = ; f(x) = g(x); (3.62) f(x) where we have chosen E5 (0) = 1;

E5t (0) = 1:

(3.63)

Eqs. (3.56) and (3.58) reduce to  f(x) F(t; x) = E3 (t) ; f (x) H (t; x) = E4 (t)f(x)

(3.64)

and (3.60) becomes u = E5 (t)f(x):

(3.65)

4. Discussion (3.55)

(3.56)

where E3 is an arbitrary function of t. Since E2 in (3.53) is arbitrary, we set E2 = 0. Substituting (3.56) into (3.50) we obtain Hx − HE1 (x) = 0:



(3.57)

In this paper we have derived solutions for the linear wave equations A–D listed in Table 1. The corresponding non-classical contact symmetry generators are listed in Table 2. We observe from Table 2 that the Lie characteristic functions, W , are linear in the derivatives ut and ux . Hence the non-classical contact symmetry generators are just extended non-classical point symmetry generators [8]. The solutions of the linear wave equations A and B represent waves in which both the amplitude and periodicity are a7ected

E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

453

Table 1 Linear wave equations and their corresponding solutions

Wave Equation


Solution

A1 −2tC3 e 2C3

2

 = x + tA2 +

utt − (A4 x + A5 )2 uxx = etC3 (C32 f() + A4 (A4 + 2C3 )f ())

u(x; t) = etC3 f()

=

C

utt − F(x)2 uxx = D4 (x) cos(tD3 )

u(x; t) = cos(tD3 )f(x)

F(x)2 = −

D

utt − E3 (t)2

u(x; t) = E5 (t)f(x)

E5tt (t) − E3 (t)2 E5 (t) − E4 (t) = 0

utt − A2 −

B

uxx = etC3 (C32 f() + 2A2 C3 f ())

A1 −2tC3 e 4C32

u(x; t) = etC3 f()

A

f(x) uxx = E4 (t)f(x) f (x)

1 tA4 e (xA4 + A5 ) A4 D32 f(x) + D4 (x) f (x)

Table 2 Contact symmetry generators of the linear wave equations listed in Table 1

A

Non-classical contact symmetry generators
 A1 −2tC 3 − uC3 − C4 W = ut − ux A2 − e 2C3

B

W = ut − ux (A4 x + A5 ) − uC3 − C4

C

W = ut + uD3 tan(tD3 )

D

W = ux −

f (x) u f(x) Fig. 1. u(x; t) = et sin( 14 e−2t + t + x) in the interval x ∈ [0; 2 ] and t ∈ [0; 10].

with changing time. The solutions of the linear wave equations C and D represent changes in amplitude of the initial curve f(x) over time. The solution of A is plotted in Fig. 1 for A1 = A2 = C3 = 1 and f() = sin . The solution of B is plotted in Fig. 2 for A4 = A5 = C3 = 1 for f() = sin . Note in both 0gures how the initial sin pro0le changes its amplitude and periodicity. The solution of C represents a change in amplitude of the initial curve f(x) from an amplitude of one to zero following the periodicity of the cos(tD3 ) function. The solution of C is plotted in Fig. 3 for D3 = 3. The solution of D depends on E5 (t) which in turn depends on the coePcient E4 (t) of the forcing term and the coePcient E3 (t) of the velocity. Let us consider an initial curve f(x) = sin x. The forcing term is then given by E4 (t)sin x. The quantity f(x)=f (x)

Fig. 2. u(x; t) = et sin(et (1 + x)) in the interval x ∈ [0; 2 ] and t ∈ [0; 10].

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E. Momoniat / International Journal of Non-Linear Mechanics 39 (2004) 447 – 455

From Fig. 4 we observe that the solution of D retains the shape of the initial pro0le f(x) = sin x while the amplitude increases at a rate of E5t (t). Acknowledgements I would like to thank Prof. C. Qu and Prof F.M. Mahomed for valuable discussion. I also thank the National Research Foundation for funding. References Fig. 3. u(x; t) = cos(3t) sin x in the interval x ∈ [0; 2 ] and t ∈ [0; 10].

Fig. 4. u(x; t) = 12 (et + cos t + sin t) sin x in the interval x ∈ [0; 2 ] and t ∈ [0; 10].

evaluates to −1. Hence the wave velocity is given by iE3 (t). The simplest case occurs when we choose the wave velocity to cancel the imaginary constant i.e. E3 (t) = i. We let the coePcient E4 (t) of the forcing term be et . The resulting ordinary di7erential equation for E5 (t) is given by E5tt (t) + E5 (t) − et = 0:

(4.1)

Solving (4.1) with respect to (3.63) we obtain 1 E5 (t) = (et + cos t + sin t): (4.2) 2 The solution of D subject to (4.2) and f(x) = sin x is plotted in Fig. 4.

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