Hervé É Riou
email: riou@lmt.ens-cachan.fr

Pierre Ladevèze
email: ladeveze@lmt.ens-cachan.fr

Benjamin Sourcis
email: sourcis@lmt.ens-cachan.fr

Béatrice Faverjon
email: beatrice.faverjon@insa-lyon.fr

Louis Kovalevsky
email: kovalevsky@lmt.ens-cachan.fr

Pierre Ladev Èze

B Éatrice Faverjon

An adaptive numerical strategy for the medium-frequency analysis of Helmholtz's problem

Keywords: Medium-frequency, Helmholtz equation, acoustics, adaptive numerical strategy, variational theory of complex rays

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In recent years, the study of the vibrational behavior of mechanical systems has become a cornerstone of the design of industrial products and of the optimization of their performances. In the automotive industry, for instance, vehicle bodies have become thinner and thinner and, therefore, more prone to vibrations. Aerospace structures have become lighter and lighter, giving the study of vibrations a major role in design processes. The cost of experimental tests and the growing demand for prediction tools have made Computer Aided Engineering (CAE) tools essential. Today, the finite element method (FEM) [START_REF] Zienkiewicz | The Finite Element Method[END_REF] and the boundary element method (BEM) 2 are the CAE tools of reference for the analysis of vibration problems.

The FEM consists in using a large (but finite) number of small elements to discretize the problem. The dynamic field variables are described in terms of polynomial shape functions. However, since these shape functions are not exact solutions of the governing differential equations, a refined discretization is required in order to eliminate the associated pollution error [START_REF] Deraemaeker | Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions[END_REF] and achieve reasonable prediction accuracy at higher frequencies. This results in large numerical models whose resolution requires prohibitively large computational resources. As a result, the FEM is limited to low-frequency applications. [START_REF] Freymann | Advanced Numerical and Experimental Methods in the Field of Vehicle Structural-Acoustics[END_REF] A variety of techniques have been proposed in order to circumvent these problems, including modifications of the standard finite element method such as predefined reduced bases, [START_REF] Soize | Reduced models in the medium frequency range for the general dissipative structural dynamic systems[END_REF] the Galerkin least-squares FEM, [START_REF] Harari | Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains[END_REF] the quasi-stabilized finite element method, [START_REF] Babuska | A generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution[END_REF] the partition of unity method (PUM), [START_REF] Melenk | The partition of unity finite element method: Basic theory and applications[END_REF][START_REF] Melenk | Approximation with harmonic and generalized harmonic polynomials in the partition of unity method[END_REF] the generalized finite element method, [START_REF] Strouboulis | The generalized finite element method: An example of its implementation and illustration of its performance[END_REF] and residual-free bubbles. [START_REF] Hughes | Multiscale phenomena: Greens functions and the dirichlet-to-neumann formulation and subgrid scale models and bubbles and the origins of stabilized methods[END_REF][START_REF] Franca | Residual-free bubbles for the helmholtz equation[END_REF] Compared to standard finite elements, these techniques have been shown to reduce computational costs. However, the frequency ranges which they are capable of addressing are still lower than what would be desirable.

The BEM is based on a boundary integral formulation of the problem, so only the boundary of the domain being considered needs to be discretized. Within the boundary elements which constitute the model, some boundary variables are expressed in terms of simple polynomial shape functions. Compared to FE models, the application of the boundary conditions results in a small system of equations to be solved for the nodal values at the discretized boundary. Once these nodal values have been determined, the field variables inside the domain can be reconstructed in a postprocessing step using the boundary integration formulation. Since the boundary alone has to be discretized, the BEM is exempt from some of the problems mentioned concerning the FEM. However, even though some improvements have been made to the BEM in order to increase its efficiency, [START_REF] Greengard | A fast algorithm for particle simulations[END_REF][START_REF] Bonnet | A multi-level fast multipole BEM for 3-D elastodynamics in the frequency domain[END_REF] the construction of the frequency-dependent and fully populated matrices is very time consuming. Moreover, the integration of singular Green functions is the source of numerical difficulties. [START_REF] De Langre | Fonctions de transfert de plaques par équations intégrales. Test de validation et de performance[END_REF][START_REF] Harari | A cost comparison of boundary element and finite element methods for problems of time-harmonic acoustics[END_REF] Today, there are also dedicated computational strategies for the resolution of mediumfrequency problems, known as Trefftz methods, [START_REF] Trefftz | Ein gegenstuck zum ritzschen verfahren[END_REF] which differ from the FEM and the BEM in that in many cases the shape functions are exact solutions of the governing differential equations. These approaches include, for example, a special use of the PUM, [START_REF] Laghrouche | Short wave modelling using special finite elements[END_REF][START_REF] Strouboulis | The generalized finite element method for helmholtz equation: Theory and computation and and open problems[END_REF][START_REF] Strouboulis | Partition of unity method for helmholtz equation: q-convergence for plane-wave and wave-band local bases[END_REF] the ultra-weak variational method, [START_REF] Cessenat | Application of an ultra weak variational formulation of elliptic pdes to the two-dimensional helmholtz problem[END_REF] the least-squares method, [START_REF] Monk | A least-squares method for the helmholtz equation[END_REF] the discontinuous enrichment method (DEM), [START_REF] Farhat | The discontinuous enrichment method[END_REF] the element-free Galerkin method, [START_REF] Bouillard | Element-free galerkin solutions for helmholtz problems: Formulation and numerical assessment of the pollution effect[END_REF] the wave boundary element method, [START_REF] Perrey-Debain | Wave boundary elements: A theoretical overview presenting applications in scattering of short waves[END_REF] and the wave-based method. [START_REF] Desmet | An indirect trefftz method for the steady-state dynamic analysis of coupled vibro-acoustic systems[END_REF] All these numerical techniques use oscillating functions to solve the problem. The main differences concern the treatment of the transmission conditions between substructures and the boundary conditions. The variational theory of complex rays (VTCR), which is the subject of this paper, belongs to that category.

The VTCR was proposed in Refs. 27 and 28 for the resolution of vibration problems in the medium-frequency range. It is based on an original variational formulation of the problem which was developed in order to enable approximations within subdomains to be a priori independent of one another. In each subdomain, any type of shape function can be used, which brings this approach great flexibility and robustness. For vibration problems, the VTCR approximates the solution through a set of wave functions, called complex rays, which verify the governing differential equations exactly. Therefore, no residual error affects the governing partial differential equations inside the subdomains. However, the functions may violate the boundary conditions. Setting the residual boundary errors to zero using the previous variational formulation leads to a small matrix equation whose solution gives the angular distribution of the waves in space. In Ref. 29, the VTCR was used to predict the vibrational response of a 3D plate assembly. In Ref. 30, plates with heterogeneities were taken into account. In Ref. 31, that theory was extended to shell structures. The calculation of the vibrational response over a range of frequencies was presented in Ref. 32. The use of the VTCR for transient dynamic problems was addressed in Ref. 33. Finally, the extension to acoustic problems was presented in Ref. 34. Many examples have shown that this approach is capable of producing an accurate solution using only a small number of degrees of freedom (DOFs).

This paper shows how the efficiency of the VTCR for the study of interior acoustic problems governed by a Helmholtz-type partial differential equation can be improved. A first error indicator for controlling the calculation is derived based on the residuals of the boundary conditions calculated in a subcavity. Using this error indicator, an adaptive version of the VTCR is formulated. The adaptive nature of this new version is interesting in some cases where the directions of propagation of the waves are not distributed uniformly in space. Indeed, there are structures in which the boundaries guide the waves along specific directions, in which case there is no need for the waves to be densely distributed in all spatial directions. The VTCR is very well-suited for such anisotropic discretizations and the proposed error indicator enables one to determine whether a refined angular discretization of the waves is necessary or not. Based on these considerations, the adaptive VTCR works as follows: A first calculation is performed using a coarse angular wave discretization; next, a second calculation is carried out in order to evaluate the need for a refined mesh in each angular direction; a third and final calculation is then carried out with a mixed coarse and refined angular discretization. A new error calculation is performed, and the refinement process can start again.

The paper is organized as follows: Sec. 2 briefly describes the general vibration problem to be solved and the VTCR approach. Section 3 presents the error indicator for the VTCR. Section 4 describes the adaptive version of the VTCR. Section 5 shows numerical examples of applications of the method. Finally, Sec. 6 concludes the paper with general remarks and proposes some perspectives for future research.

The Variational Theory of Complex Rays for Helmholtz Problems

This section reviews the basic principles of the VTCR modeling methodology for the resolution of Helmholtz's equation (for more details, see Ref. 27). First, the general problem to be solved is defined. Then, the different steps of the VTCR computational approach are presented in detail.

The reference problem

For the sake of simplicity, let us consider a general two-dimensional steady-state interior dynamic problem in an acoustic cavity Ω divided into two subcavities Ω 1 and Ω 2 with an interface Γ (see Fig. 1).

The cavity Ω is filled with an incompressible fluid characterized by its sound velocity c 0 and its density ρ 0 . The steady-state dynamic behavior of Ω is studied at a fixed circular frequency ω. All the descriptive quantities can be expressed as complex numbers: An amplitude Q(x) is associated with the quantity Q(x)e iωt , where x represents the position and i = √ -1 the imaginary unit. Each subcavity Ω E (E ∈ {1, 2}) is subjected to excitations applied along

∂Ω E = ∂ p Ω E ∪ ∂ v Ω E ∪ ∂ Z Ω E ∪ Γ in
L v [ E ] = i ρ 0 ω ∂ E ∂n E = i ρ 0 ω n T E .∇( E ), (1)
where n is the outward normal to Ω E . The reference problem to be solved consists in finding the acoustic pressure (p

1 , p 2 ) ∈ H 1 (Ω 1 ) × H 1 (Ω 2 ) such that ∆p E + k 2 p E = 0 ∈ Ω E , E ∈ {1, 2}, (2)
p E = p dE over ∂ p Ω E L v [p E ] = v dE over ∂ v Ω E , E ∈ {1, 2} p E -Z E L v [p E ] = h dE over ∂ Z Ω E , ( 3
)
p 1 -p 2 = 0 over Γ L v [p 1 ] + L v [p 2 ] = 0 over Γ. ( 4
)
Equation ( 2) is the homogeneous Helmholtz equation, with k = ω/c 0 being the wave number. Equations ( 3) and ( 4) are related to the boundary conditions of the problem and the continuity conditions at the interface between the two subcavities Ω 1 and Ω 2 . Z E is an impedance coefficient.

Variational formulation of the problem

A weak form of the reference problem introduced in Sec. 2.1 can be derived using a variational formulation introduced in Ref. 27. This formulation uses power quantities and verifies the boundary conditions ( 3) and (4) in a weak sense. Thus, the resulting approximation includes exact solutions of the Helmholtz equation (2).

Let S E ad denote the space of the functions which verify Helmholtz's homogeneous equation (2) throughout cavity Ω E :

S E ad = {p E ∈ H 1 (Ω E ) | ∆p E + k 2 p E = 0, ∀ x ∈ Ω E }. ( 5
)
The partial differential equation problem defined by (2), ( 3) and (4) can be expressed as follows:

Find (p 1 , p 2 ) ∈ S 1 ad × S 2 ad such that Re    E=1,2 ∂pΩ E (p E -p dE ) • L v [δp E ]ds + E=1,2 ∂vΩ E (L v [p E ] -v dE ) • δp E ds + 1 2 E=1,2 ∂ Z Ω E (((1 -Z E L v )[p E ] -h dE ) • L v (δp E ) + ((L v -1/Z E )[p E ] + h dE /Z E ) • δp E )ds + 1 2 Γ ((p 1 -p 2 ) • L v [δp 1 -δp 2 ] + L v [p 1 + p 2 ] • (δp 1 + δp 2 ))ds    = 0, ∀ (δp 1 , δp 2 ) ∈ S 1 ad × S 2 ad , (6)
where and Re{ } designate respectively the complex conjugate and the real part of the complex quantity . The generalization of this variational formulation to an assembly of n subcavities is straightforward.

This formulation can be viewed as the expression of a power balance at each boundary of the subcavities Ω E . This is a mixed asymmetrical formulation which enables the approximations in the subcavities to be independent of one another. This property grants the strategy great flexibility and efficiency because no Lagrange multiplier or penalty term is necessary in order to ensure interelement continuity.

Since (6) is not a classical variational formulation, proof of the equivalence with reference problem (2), (3), (4), whose solution clearly satisfies (6), must be given. In order to do that, let us consider two solutions (p 1 , p 2 ) and (p 1 , p 2 ) of the variational formulation and let p = (p 1 , p2 ) denote the differences between the two. Because of ( 6), these two solutions must verify the following equation:

Re    E=1,2 1 2 ∂pΩ E (p E ) • L v [δp E ]ds + E=1,2 ∂vΩ E L v [p E ]δp E ds + E=1,2 1 2 ∂ Z Ω E (((1 -Z E L v )[p E ]) • L v (δp E ) + ((L v -1/Z E )[p E ]) • δp E ) ds + 1 2 Γ ((p 1 -p2 ) • L v [δp 1 -δp 2 ] + L v [p 1 + p2 ] • (δp 1 + δp 2 )) ds    = 0. (7)
Taking δp = p, (7) simplifies trivially to:

a(p, p) = Re    E=1,2 ∂Ω E pE L v [p E ]ds - E=1,2 1 2 ∂ Z Ω E Z E L v [p E ]L v [p E ] + 1 Z E pE pE ds    = 0. ( 8
)
The Strokes formula applied to each subcavity Ω E yields:

a(p, p) = Re    E=1,2 - i ρ 0 ω Ω E (∇p E • ∇p E + pE ∆p E ) dΩ - E=1,2 1 2 ∂ Z Ω E Z E L v [p E ]L v [p E ] + 1 Z E pE pE ds    = 0, (9)
which, since p ∈ S 1 ad × S 2 ad , becomes:

a(p, p) = Re    E=1,2 - i ρ 0 ω Ω E (∇p E • ∇p E -k 2 pE pE ) dΩ - E=1,2 1 2 ∂ Z Ω E Z E L v [p E ]L v [p E ] + 1 Z E pE pE ds    = 0. ( 10
)
Therefore, the real part of (10) yields:

- E=1,2 ∂ Z Ω E Z E L v [p E ]L v [p E ] + 1 Z E pE pE ds = 0. ( 11
)
The last equation shows that pE = 0 and

n T E •∇p E = 0 throughout ∂ Z Ω E .
Reintroducing that result into (7), one gets:

Re    E=1,2 ∂pΩ E (p E ) • L v [δp E ]ds + E=1,2 ∂vΩ E L v [p E ]δp E ds + 1 2 Γ ((p 1 -p2 ) • L v [δp 1 -δp 2 ] + L v [p 1 + p2 ] • (δp 1 + δp 2 )) ds    = 0. ( 12
)
Since δp E and its normal gradient are arbitrary, this leads to pE = 0 over

∂ P Ω E , to L v [p E ] = 0 over ∂ v Ω E
and to the transmission continuity (4) for pE . Therefore, pE verifies the initial Helmholtz reference problem with homogeneous conditions, i.e. pE is equal to zero, which proves the uniqueness of the solution of ( 6) and, consequently, the equivalence of ( 6) with ( 2), ( 3), (4).

Since the reverse of that implication is obvious, the equivalence between the reference problem ( 2), ( 3), ( 4) and the variational problem ( 6) is proven.

This result is similar to that given in Ref. 27 except that in the case of structures the uniqueness of the solution results from the hysteretic damping introduced into the constitutive law (which means that the structure dissipates energy). In acoustics, damping is localized primarily at the boundaries and taken into account by the Robin equation (3). This is clearly the case in buildings, where absorption takes place in insulation materials. Therefore, the normal impedance, especially its real part Re(Z E ), is essential because it is what ensures the uniqueness of the solution. Of course, any type of damping can be taken into account in the reference problem. In that case, the wave number k has a complex part, which leads to a new term associated with inner dissipation within the cavity, as in Ref. 27. The conclusion of the proof, however, remains the same.

The bilinear form a(p, δp) can be decomposed into a Hermitian part a H (p, δp) and an anti-Hermitian part a A (p, δp). One can easily prove that the Hermitian part takes the form

a H (p, δp) = W (1) Z (p 1 ) + W (2) Z (p 2 ) with W (E) Z(p E ) = Re{-∂ Z Ω E p E .v E .n E ds} (see Ref. 35).
Therefore, (6) becomes:

Find p ∈ S 1 ad × S 2 ad such that δ(W (1) Z (p 1 ) + W (2) Z (p 2 )) + p, δp = (l d , δp), ∀ δp ∈ S 1 ad × S 2 ad , (13)
where (l d , •) is the linear form associated with the acoustic boundarys conditions, •, • is the anti-Hermitian part of a(p, δp), and δ( ) is a small variation of the quantity . One can see that the Hermitian part of the formulation is associated directly with the damping introduced in ∂ Z Ω E . Consequently, this quantity must be calculated with care because it is what ensures the uniqueness of the solution.

The discretized problem

In order to solve (6), one must use approximations p a E for each subcavity Ω E . These approximations must be chosen in S a,E ad ⊂ S E ad . Therefore, the discretized version of ( 6) is:

Find p a = (p a 1 , p a 2 ) ∈ S a,1 ad × S a,2 ad such that δ(W (1) Z (p a 1 ) + W (2) Z (p a 2 )) + p a , δp a = (l d , δp a ), ∀ δp a ∈ S a,1 ad × S a,2 ad . ( 14
)
In the VTCR, the approximations p a are sought in the space of the propagative waves:

p a E (x) = C E A a E (k E )•e ik E •
x dC E , where A a E denotes the amplitude of a wave, k E is the wave vector in Ω E and C E is the circle which corresponds to all the possible wave propagation directions in the 2D cavity. One can see that in the VTCR the solution is considered to be a superposition of propagative waves. If necessary, other types of waves can be introduced (for example, see Ref. 31). Furthermore, the presence of the superscript a in A a E alone shows that the VTCR approximates (and, therefore, discretizes) only the amplitudes of the waves, but not their spatial shapes. Therefore, the VTCR belongs to the multiscale category of numerical approaches in which only the slowly oscillating quantities are discretized. Finally, one can observe that all the propagative waves are taken into account thanks to the integral distribution of their directions over the whole domain. This is important because the wave propagation directions are not known a priori.

A a E (k E ) represents the discretized amplitudes of the propagative waves over the circle C E . This discretization can take many forms: A a E (k E ) can be sought as a piecewise constant function

p a E (x) = C E A a E (k E )e ik E •x dC E = L E l=1 a l E C l E e ik E .
x dC E (15) or a piecewise Dirac function

p a E (x) = C E A a E (k E )e ik E •x dC E = L E l=1 a l E e ik l E •x . ( 16
)
Therefore, the approximate solution is sought as

p a E (x) = a T E .ϕ E (x), where ϕ E = [ϕ E,1 ϕ E,2 • • • ϕ E,L E ]
T is the vector of the shape functions and a E the vector of the associated amplitudes. In the case of Approximation (15)

ϕ E,l (x) = θ l+ 1 2 θ l-1 2
e ik E (θ).x dθ, and in the case of Approximation ( 16) ϕ E,l (x) = e ik E (θ l ).x . In both cases, S a,E ad = Vect{ϕ E,l (x)} l∈{1,...,L E } . Introducing (15) or ( 16) into ( 14), one gets a set of L 1 + L 2 algebraic equations whose solution is the vector of generalized amplitudes a = a1 a2 , of size L 1 + L 2 , which verifies:

K 11 K 12 K 21 K 22 • a 1 a 2 = b 1 b 2 , ( 17
)
where K and b denote respectively the matrix of the bilinear forms and the vector of the linear form of variational formulation (6).

The resolution of ( 17) followed by the back-substitution of the result into function p a E (x) leads to the numerical solution of the acoustic problem.

Several remarks can be made:

• Since the variational formulation is not symmetrical, the VTCR is not a discontinuous Galerkin method. • Since the complex rays {ϕ E,l (x)} l∈{1,...,L E } are defined over the entire acoustic subcavity Ω E , the matrices of the model are fully populated. However, in the case of more than two cavities, the off-diagonal submatrix [K] ij is zero if Cavity i is not connected to Cavity j. • Since the complex rays depend implicitly on the frequency, so do the matrices of the model. Therefore, they cannot be partitioned into frequency-independent submatrices. However, strategies have been developed in order to calculate the solution over a range of frequencies without calculating the response at each frequency (see Ref. 32). • The use of piecewise constant functions in the VTCR to approximate the amplitudes of the waves is more involved numerically than the use of piecewise Dirac functions. In the latter case, many integrations can be carried out analytically. However, in the case of wavebands, semi-analytical strategies can be used, as in Ref. 27. These strategies are based on the approximation of variable angular functions as piecewise angular functions which can be integrated analytically. • As mentioned previously, no Lagrange multiplier is required in order to ensure the continuity of the solution. This is taken into account automatically in the variational formulation (6). • Two different approaches can be used and will be illustrated in the following examples: the h-approach (which consists in keeping the number of waves constant and decreasing the size of the subcavities Ω E , hence the denomination h in reference to the FE technique in which the mesh is refined), and the p-approach (which consists in keeping the number of subcavities Ω E constant and increasing the number of waves L E in each subcavity, hence the denomination p in reference to the FE technique in which the polynomial degree of the shape functions is increased). With each approaches, the shape functions are

C l E e ik E .
x dC E (see 15) or e ik l E .x (see 16).

Example of convergence

There are many examples in the literature showing the convergence of the VTCR (see, e.g. Refs. 27-34). Here we present another example of the convergence of the method which takes into account the impedance Z along boundary ∂ Z Ω. This example will be used in the following sections. The domain Ω is a square acoustic cavity of width L (ρ 0 = 1, 3 kg • m -3 and c 0 = 330 m • s -1 ) subjected along all four sides to a Robin condition with Z = ρ 0 c 0 and a prescribed excitation h d such that the exact solution p ex (x) = e -ike.x (with k e = k[cos θ ex , sin θ ex ] T ) corresponds to the propagative wave in direction θ ex = 12 • . Two circular frequencies were considered for the excitation so that kL = 80 (which corresponds to about 13 wavelengths in the cavity) and kL = 160 (which corresponds to about 25 wavelengths in the cavity), respectively. The approximations were based on piecewise constant functions. In applying the VTCR, as mentioned in the last remark of Sec. 2.3, two different approaches were used: the h-approach and the p-approach. For the VTCR h-approach, the geometry (here a simple square) was divided into squares of width h such that L/h = n, n being a positive integer. For the VTCR p-approach, the cavity was either considered as a whole or divided into two rectangular subcavities of equal size.

Once the exact solution is known, it is possible to calculate the actual error pp ex L 2 (Ω) . Figures 2 and3 show the convergence curves of the actual relative error pp ex L 2 (Ω) / p ex L 2 (Ω) for the FEM, the VTCR h-approach and the VTCR p-approach as a function of the number of DOFs. Several remarks can be made about these figures:

• The resolution of an acoustic problem with a Robin condition using the VTCR presents no difficulty. The approximate solution converges towards the exact solution. • The VTCR p-approach produces more accurate results with fewer DOFs. This is even more the case when the frequency increases. • The FEM requires a huge number of DOFs in order to alleviate the pollution effect.

For example, for kl = 160, keeping kL.(kh) 2 below 1 requires 1000 elements along each edge (see Ref. 3). The VTCR h-approach suffers less from the pollution effect because it uses oscillating wave functions. The comparison between the VTCR solution with 32 wavebands and the VTCR solution with 52 wavebands shows that the pollution decreases when the number of waves increases.

This simple numerical test agrees with the results already presented in Ref. 36: The convergence of the p-approach is exponential, whereas the convergence of the h-approach is slower. This is the reason why the VTCR leads to good results with only a few DOFs.

The Error Indicator

Construction of an error indicator for the VTCR

As mentioned in Sec. 2, it is very important to quantify the accuracy of the approximate solution. Using the actual error would be an interesting way to do that, but it is impracticable because the problem (2), (3), (4) usually has no analytical solution. Furthermore, the definition of an error indicator for each subcavity Ω E is a difficult task because some subcavities Ω E may not be connected to the boundary ∂Ω at all and, therefore, it is impossible to verify in these subcavities an equation such as (3), which is attached only to the boundaries and is the only reliable equation. The only way to assess the accuracy of the approximate solution in a subcavity Ω E is to verify the transmission condition (4) with all the other subcavities in its vicinity. However, the verification of such an equation is hampered by the fact that the solutions in these neighboring subcavities are not exact solutions.

The error indicator we propose in this paper is based on the total energy of each Ω E , denoted E T,Ω E , i.e. the sum of the time averages of the kinematic and potential energies, and is equal to the dimensionless quantity:

I Ω = max E I Ω E = max E E T,Ω E (p a E -p pv E )/mes(Ω E ) E E T,Ω E (p a E )/mes(Ω) , ( 18
)
where mes(Ω) and mes(Ω E ) designate the measures of Ω and Ω E , respectively, and p pv E refers to the solution of the problem in Ω E under prescribed pressure and velocity along the boundaries of Ω E which correspond to the pressure and the velocity in all the Ω E adjacent to Ω E . This is achieved by applying the variational formulation (6) to Ω E , with both the pressure and the velocity prescribed along the same boundary thanks to the additional term

Re 1 2 ∂pvΩ E ((p E -p dE ) • L v [δp E ] + L v [p E ] -v dE • δp E ) ds .
One can see that I Ω measures the total energy of the relative difference between the approximate solution and the exact solution of the problem with prescribed pressure and velocity, and quantifies the maximum global error in energy per subcavity. Therefore, it quantifies the consistency of the calculated solution in Ω E and, thus, in Ω. Indeed, a too large I Ω E indicates that the data at the boundaries of Ω E are incompatible and, therefore, were poorly calculated.

Furthermore, since I Ω E is defined in each subcavity Ω E , it can be used to determine the locations in which the discretization must be refined, leading to an adaptive version of the VTCR which will be presented in the next section.

Assessment of the VTCR error indicator for a highly directional example

First, let us go back to the previous numerical example (see Sec. Therefore, the exact wave amplitude is A ex (θ) = δ(θθ ex ). Figure 4 shows the exact solution of this first numerical example. This example is interesting because the problem is highly directional with respect to the directions of propagation of the waves. The problem was solved with the VTCR using a uniform decomposition of Ω into 9 = 3 × 3 square subcavities (see Fig. 4). The solution was obtained by extending (6) to the 9 subcavities.

In order to illustrate the effectiveness of error indicator I Ω = max E I Ω E , Fig. 5 shows how it compares with the actual relative errors in energy defined in each Ω E (obtained by substituting p ex for p pv E in ( 18)). One can see that error indicator 18) is a valid measure of the quality of the calculated solution. Indeed, I Ω E is very close to the actual relative error in energy for each subcavity and at each frequency considered. This numerical test shows that I Ω (18) can provide an indication of the accuracy of the solution in an example whose solution is highly directional.

I Ω = max E I Ω E (
On these curves, one can see that sometimes the errors stop decreasing beyond a certain large number of DOFs. This is a known behavior which is present in many strategies using oscillating functions as the approximate fields (see Refs. 20, 23, 26 and 34). The problem occurs when the number of waves being used becomes too large: The waves are so close to one another that the computer, even using double-precision arithmetics, cannot tell the difference numerically. Then, the conditioning number reaches the limit of the machine capacity and the process ceases to converge.

Assessment of the VTCR error indicator for an example with a continuous wave distribution

Now let us consider the numerical example of a square acoustic cavity of width L (ρ 0 = 1, 3 kg • m -3 and c 0 = 330 m • s -1 ) subjected along all four sides to a Robin condition with Z = ρ 0 c 0 and a prescribed excitation h d such that exact wave amplitude at the two circular frequencies determined by kL = 80 and kL = 160 is

A ex (θ) =    (1 + i) sin (4θ) for θ ∈ - π 4 ; π 2 0 e l s e . ( 19
)
Figure 6 shows the exact solution of this second numerical example for kL = 80. This example is interesting because it involves a continuous distribution of waves rather than some discrete directions of propagation. Again, in order to solve the problem using the VTCR, we used a uniform distribution and divided Ω into 9 = 3 × 3 square subcavities.

Figure 7 shows the error indicator (18) in each subcavity Ω E compared with the actual relative error in energy. One can see again that error indicator

I Ω = max E I Ω E (18) is a valid
measure of the quality of the calculated solution. This numerical test shows that I Ω (18) can provide an indication of the accuracy of the solution in an example whose solution presents a continuous angular wave distribution.

In this example, one can observe that the error is distributed relatively evenly among the subcavities. This is a consequence of the VTCR discretization, which is the same in all the Ω E . If one wants to use a coarser discretization in one of the Ω E (which the VTCR allows, see Ref. 34), the error would be localized in that subcavity. Of course, the error indicator I Ω E would reflect that situation.

An Adaptive Version of the VTCR

Presentation of the method

It is clear from Sec. 2 that the unknowns of the VTCR are the amplitudes A a E (θ) of the waves traveling in all directions θ ∈ [0; 2π[ of the 2D acoustic cavity. Of course, these amplitudes are not necessarily uniformly distributed: For specific internal cavity shapes whose boundaries guide the waves, the amplitudes can be sparsely distributed in some angular directions and densely distributed in others. Therefore, in the case of a piecewise constant angular approximation for the amplitudes (15), it would surely be advantageous to use different discretizations depending on the propagation direction.

The idea of the adaptive version of the VTCR stemmed from that remark. The objective is to enable one to calculate the VTCR using different angular discretizations: A refined angular discretization when the wave amplitudes are densely distributed and a coarse angular discretization when the amplitudes are sparsely distributed. Therefore, the process is completely analogous to that used in the adaptive FEM [START_REF] James | h-Adaptive finite element computation of time-harmonic exterior acoustics problems in two dimensions[END_REF] and consists of three steps (see Fig. 8):

(1) In the first step, a global analysis of the problem is carried out using a uniform, lowdensity angular wave distribution based on a coarse angular grid τ M (see Fig. 8(a)). (2) The objective of the second step is to calculate the proper angular discretization. The quality of the approximation of the first step is quantified using error indicator I Ω (18), which indicates whether a new angular mesh is necessary. If that is the case, a local analysis is performed in subcavity Ω E using a refined angular grid τ m and applying the (3) The third step is a new complete calculation using angular grid τ Mm (see Fig. 8(c)).

If the accuracy of the last calculation is insufficient, the procedure can be repeated until the desired accuracy is reached.

For the second step, it can be useful to set two limit levels m 0 and m 1 : if I Ω E < m 0 , the quality of the solution is considered to be good and no rediscretization of subcavity Ω E is necessary. If m 0 < I Ω E ≤ m 1 , the error is moderate, but still too high and a new refined angular discretization is necessary. If m 1 < I Ω E , the solution is seriously inaccurate and the boundary conditions of Ω E must be recalculated more precisely, which requires a new first step (see Fig. 8(a)). In practice, one often chooses m 0 = 10% and m 1 = 40%.

Calculation of the adapted angular discretization

As explained before, a large I Ω E indicates a poor solution in Ω E due to too coarse an angular discretization of the wave amplitudes. A new adapted angular discretization is required. In order to obtain that, one uses a refined angular grid τ m in Ω E . The pressures and velocities calculated in Step 1 are prescribed along the boundaries (except if an edge of Ω E corresponds to the boundary of Ω, in which case the boundary conditions of the initial problem are used). Then, an energy criterion based on the amplitudes of the waves is used to determine which angular wave directions in Ω E are important and require the refined discretization. If the propagation interval [0; 2π[ is discretized into L E angular sectors, the criterion is defined with a parameter η as follows:

• If the amplitude of the wave verifies

|a l E | 2 < ηa 2 max , then the angular interval [θ l-1 2 ; θ l+ 1 2
[ can be discretized using the coarse angular grid τ M . • If the amplitude of the wave verifies |a l E | 2 > ηa 2 max , then the angular interval [θ l-1 2

; θ l+ 1 2

[ can be discretized using the refined angular grid τ m . The final angular discretization is defined using the appropriate grid (τ M or τ m ), denoted τ Mm , for each angular sector of Ω E in [0; 2π[. In practice, one often chooses η = 0.1.

Selection of the coarse and refined angular discretizations

A key point of the adaptive version of the VTCR is the choice of the coarse and refined angular discretizations, i.e. the number of waves to be used in directions where the evolution is smooth or irregular, respectively. An important point which must be understood is that it is not necessary to use the same number of DOFs in acoustic cavities of different sizes. Indeed, more waves are required in large cavities than in small cavities for similar accuracy results. Thus, an interesting heuristic criterion to define the number of waves to be used in Ω E is

L E = [T • k • diam (Ω E )], ( 20
)
where T is a positive real number, diam(Ω E ) the diameter of Ω E , and [ ] the integer part of . All the examples presented in Refs. 28-34 show that such a criterion guarantees similar accuracy with the VTCR in all the Ω E for a large range of frequencies.

Therefore, for the coarse and refined discretizations, the angular size of the grids is defined as

τ M E : ∆θ M E = 2π/[T M • k • diam(Ω E )] τ m E : ∆θ m E = 2π/[T m • k • diam(Ω E )]. (21)
Having considered numerous examples, the authors feel that T M ∈ [0; 0.8] and T m ∈ [1; 2] are good choices. The small T M used in the coarse angular discretization enables one to use only a few waves in directions which are not significant in terms of energy while the large T m used in the refined angular discretization enables the use of many waves in directions which are significant in terms of energy for the solution.

An important consideration is the need for very stable coefficients a l E . Indeed, problems can occur when the number of waves is too large, leading to ill-conditioning of the matrices and an unstable solution a l E even though solution p E is good. In order to improve robustness, one must use a regularization technique (see Ref. 35). The purpose of such a technique is to find a regularized solution of problem (17):

K 11 + λ p H 1 K 12 K 21 K 22 + λ p H 2 . a 1,λp a 2,λp = b 1 b 2 , ( 22
)
where H E is a regularization operator which represents the norm of the second θ-derivative of the amplitudes a E (θ), and λ p a weighting coefficient. This technique will be developed in a forthcoming paper.

Assessment of the Performance of the Adaptive Version of the VTCR

In this section, the performance of the adaptive VTCR is evaluated based on two benchmark acoustic problems considered at two different frequencies each. These problems are formulated in such a way that the exact solutions are known, which enables one to calculate the exact convergence curves.

Numerical results of an example based on a propagative wave

Let us go back to the first numerical example of Sec. Formulae (21) determine the number of waves to be used in the coarse and refined angular discretizations based on the definitions of τ m E and τ M E . Figure 9 shows the convergence curves as functions of parameter T (see (20)) for subcavity Ω 5 (see Fig. 4), which is the most critical subcavity because it is attached to no reliable boundary condition. This curve shows that the heuristic law (20) works because the number of waves to be used is independent of the frequency being considered. Indeed, the parameter T to be used to achieve a given error level is almost the same. The behavior of the curves corresponding to the other subcavities Ω E is similar.

In order to evaluate the convergence of the strategy, we tested four values of parameter T M : T M 1 = 0.2, T M 2 = 0.4, T M 3 = 0.6 and T M 4 = 0.8 (see Fig. 9). The convergence of the VTCR was tested by making T m → ∞. The energy criterion was used as explained in Sec. 4.2 to determine the dominant propagation directions in terms of energy for the solution. The results are compared to the nonadaptive VTCR with a uniform angular wave discretization in Fig. 10.

These curves show that the convergence rate of the adaptive VTCR is much better. For this example in which the solution is highly directional, this technique is very efficient. Furthermore, one can see that the efficiency was highest with the coarsest angular discretization. This is due to the shape of the exact solution being sought. Indeed, the wave propagating in direction θ ex = 12 • is the only wave needed. The adaptive procedure detects that specificity and focuses the refined angular discretization on that angle. The coarse discretization concerns all the other directions, which are not needed. Therefore, this coarse discretization should be as coarse as possible, or even eliminated. T M1 = 0.2, T M2 = 0.4, T M3 = 0.6, T M4 = 0.8 (see Fig. 9).

Numerical results of an example based on a continuous distribution of propagative waves

Now let us go back to the second numerical example of Sec. 3.3: A square acoustic cavity of width L(ρ 0 = 1, 3 kg • m -3 and c 0 = 330 m • s -1 ) subjected along all four sides to a Robin condition with Z = ρ 0 c 0 and a prescribed excitation h d such that the exact wave amplitude is A ex (θ) = (1 + i) sin (4θ) if θ ∈]π/4; π/2[, and A ex (θ) = 0 otherwise. Two circular frequencies such that kL = 80 and kL = 160 were considered. Figure 6 shows the exact solution of this second numerical example with kL = 80. Again, we solved this problem using the VTCR with a uniform decomposition of Ω into 9 = 3 × 3 square subcavities.

As mentioned previously, this example is interesting because in the exact solution it involves a continuous distribution of propagative waves rather than some specific discrete propagation directions. Furthermore, the angular shape of the exact solution is sufficiently irregular to show how the adaptive method behaves in the case of rapidly oscillating representations.

We tested four values of parameter T M : T M 1 = 0.2, T M 2 = 0.4, T M 3 = 0.6 and T M 4 = 0.8. A very large T m was used in order to evaluate the performance of the adaptive VTCR (see Fig. 11).

Figure 12 shows the adaptive solution obtained with kL = 160 at convergence. The angular distribution of the waves shows that the amplitudes of the waves are greatly influenced by the origin of the spatial coordinates, in this case the center of Cavity Ω 1 (see the vector x 0 in Fig. 6). Indeed, the phase of the solution changes drastically far away from the origin. In Cavity Ω 1 , however, the numerical solution obtained matches the exact solution perfectly.

Figure 13 shows the convergence curves as functions of the numbers of DOFs used compared with the nonadaptive VTCR with a uniform angular wave discretization. One can observe that the coarse grid greatly influences the quality of the solution. The more refined this grid, the better the solution. If parameter τ M is too large, the strategy is no longer efficient. In that case, the error due to the coarse grid is greater than that due to the refined grid. In this situation, it would be more beneficial to increase T M than to increase T m . Of course, one can easily imagine that the procedure described previously is interesting only when the waves follow specific directions of propagation. Otherwise, a regular angular distribution of the waves is preferable. This is what the proposed strategy tends to do in such a case.

Conclusion

This paper discusses a new development of the VTCR for the study of interior steady-state dynamic problems governed by a Helmholtz-type partial differential equation. The objective is to improve the efficiency of the numerical handling of propagative waves in the case of highly directional solutions. In order to do that, we introduced an error indicator based on the consistency of the boundary conditions calculated in a subcavity. Then, we developed an adaptive wave approach which consists of three steps. In the first step, a calculation is performed with a coarse regular angular wave discretization, leading to a first approximation of the solution. In the second step, the angular errors are estimated using the error indicator and the angular zones where the wave discretization needs to be densified are determined. The third step consists in a new calculation with only a few waves in the directions in which few waves are needed, but many waves where the solution is highly directional. In order to do that, an energy-based angular error indicator was introduced to localize the angular directions where further refinement is needed. The methodology is presented in such a way that it can be applied to a general Helmholtz problem.

This strategy was applied successfully to two numerical examples of different complexity (a highly directional example and a more regular example). These examples prove the effectiveness of the error indicator and the capability of the proposed adaptive strategy to handle the different solutions: Using the error indicator we proposed, the angular directions which require further refinement could be determined; thanks to this refinement being limited to the necessary directions alone, it was possible to calculate the vibrational behavior of the closed acoustic cavity efficiently. Both examples show the advantages of angular adaptivity in terms of convergence. Of course, the strategy also lends itself to the analysis of problems in which the angular discretization is completely regular.

The examples given showed that this strategy works at least for problems with between 13 and 25 wavelengths in the characteristic dimension of the problem. In particular, they showed that for a problem with a highly directional wave distribution, the adaptive strategy is very efficient and the coarse angular wave discretization can be chosen to be arbitrarily coarse. Predictably, however, for examples with continuous wave distributions, the adaptive strategy is less interesting and the coarse angular wave discretization must be refined in order to improve the accuracy. Some subjects for future investigations are the extension of this adaptivity strategy to plate applications (for which edge waves exist) and the extension of the VTCR to 3D problems.

Fig. 1 .

 1 Fig. 1. A general 2D bounded Helmholtz problem.

Fig. 2 .

 2 Fig. 2. Example of a square (see Sec. 2.4). Comparison between the VTCR and the FEM. kL = 80.

Fig. 3 .

 3 Fig. 3. Example of a square (see Sec. 2.4). Comparison between the VTCR and the FEM. kL = 160.

  2.4): A square acoustic cavity of width L(ρ 0 = 1, 3 kg • m -3 and c 0 = 330 m • s -1 ) subjected along all four sides to a Robin condition with Z = ρ 0 c 0 and a prescribed excitation h d such that the exact solution p ex (x) = e -ike•x (with k e = k[cos θ ex , sin θ ex ] T ) corresponds to the propagative wave in direction θ ex = 12 • at two circular frequencies determined by kL = 80 and kL = 160, respectively.

Fig. 4 .

 4 Fig. 4. The exact solution of the first example (see Sec. 3.2). (a) The wave amplitude Aex(θ) and (b) the real part of the pressure.

Fig. 5 .

 5 Fig. 5. Comparison between error indicator(18) and the actual relative error in energy for the first example with Aex(θ) = δ(θθex) (see Sec. 3.2 and the solution and the discretization of Fig.4). Two circular frequencies such that kL = 80 (nine top curves) and kL = 160 (nine lower curves) are considered.

Fig. 6 .

 6 Fig. 6. The exact solution of the second example (see Sec. 3.3). (a) The wave amplitude |Aex(0)| and (b) the real part of the pressure.

Fig. 7 .

 7 Fig. 7. Comparison between error indicator(18) and the actual relative error for the second example with Aex(θ) = (1 + i) sin (4θ) if θ ∈]π/4; π/2[ and Aex(θ) = 0 otherwise (see Sec. 3.3 and the solution and the discretization of Fig.6). Two circular frequencies such that kL = 80 (nine top curves) and kL = 160 (nine lower curves) are considered.

Fig. 8 .

 8 Fig. 8. The three steps of the adaptive VTCR strategy. (a) Step 1, (b) Step 2 and (c) Step 3.

3 . 2 :

 32 A square acoustic cavity of width L (ρ 0 = 1, 3 kg •m -3 and c 0 = 330 m •s -1 ) subjected along all four sides to a Robin condition with Z = ρ 0 c 0 and a prescribed excitation h d such that the exact solution p ex (x) = e -ike•x (with k e = k[cos θ ex , sin θ ex ] T ) corresponds to the propagative wave in direction θ ex = 12 • at two circular frequencies such that kL = 80 and kL = 160, respectively. The problem was solved using a 3 × 3 spatial discretization of Ω. The exact solution is shown in Fig.4. Again, this example is interesting because it shows the behavior of the adaptive version of the VTCR for an example in which the solution being sought is highly directional in terms of the wave propagation.

Fig. 9 .

 9 Fig. 9. The relative error in L 2 norm as a function of parameter T (20) in subcavity Ω 5 (see Fig. 4) for the first adaptive example (see Sec. 5.1).

Fig. 10 .

 10 Fig.10. The actual relative error of the adaptive VTCR in Ω as a function of the number of DOFs, and comparison with the nonadaptive VTCR (uniform discretization) for the first adaptive example (see Sec. 5.1). T M1 = 0.2, T M2 = 0.4, T M3 = 0.6, T M4 = 0.8 (see Fig.9).

Fig. 11 .

 11 Fig. 11. The relative error in L 2 norm as a function of parameter T (20) in subcavity Ω 5 (see Fig. 6) for the second adaptive example (see Sec. 5.2).

Fig. 12 .

 12 Fig. 12. The numerical solution for the second adaptive example (see Sec. 5.2) with kL = 160: (left) the angular wave distributions obtained in the various subcavities; (right) the angular distributions used in the various subcavities.

Fig. 13 .

 13 Fig.13. The actual relative error of the adaptive VTCR as a function of the number of DOFs, and comparison with the nonadaptive VTCR (uniform discretization) for the second adaptive example (see Sec. 5.2). T M1 = 0.2, T M2 = 0.4, T M3 = 0.6, T M4 = 0.8 (see Fig.11).

  the form of a prescribed pressure p dE over ∂ p Ω E , a prescribed normal velocity v dE over ∂ v Ω E , a mixed excitation over ∂ Z Ω E and the conditions imposed by the other subcavities. The mixed condition applied to part ∂ Z Ω E of ∂Ω E , called a Robin condition, introduces absorption through its real part. Let us define operator L v [ E ] as


Bettina Migge

Margot Van Den Berg

Creole learner varieties in the past and in the present: Implications for Creole development

Keywords: Surinamese Creoles, creolization, negation, tense & aspect, L1 & L2 acquisition

Second language (L2) acquisition is widely assumed to have played a role in the emergence of creole genesis. However, the impact of L2 acquisition may not have been restricted to its genesis. In Surinam, newcomers outnumbered locally-born speakers of the Creole throughout the 18 th century. To date we know little about the effects that this disproportion of non-native vs. native speakers may have had in the initial and subsequent stages of development of these Creoles. In this paper we combine historical and contemporary data in order to investigate the impact of L2 acquisition and use on developing creoles. We examine several linguistic features in contemporary native (L1) as well as non-native (L2) creole speech in order to reveal the differences in the underlying L1 and L2 systems. These are then compared with their equivalents in the available historical sources. The findings suggest that L2 acquisition affected the development of some linguistic subsystems while others appear little influenced.

Introduction

Discussions about the role of second language (L2) learning strategies have a long history in the literature on the genesis and development of creoles. Since the beginning of the 21 st century there is renewed interest in the study of the shared processes involved in both L2 learning and creole genesis [START_REF] Winford | An introduction to contact linguistics[END_REF][START_REF] Lefebvre | L2 Acquisition and Creole genesis: Dialogues[END_REF][START_REF] Siegel | The Emergence of Pidgin and Creole Languages[END_REF].

L2 acquisitionists working with Clive Perdue and Wolfgang Klein in particular provide a valuable framework for comparing proto-typical L2 practices and developing creoles. According to these scholars, outcomes of second language learning, so-called learner varieties, are systems, characterized by a particular lexical repertoire and by a particular interaction of organizational principles that principally differ from native varieties in their underlying organizational principles [START_REF] Klein | The basic variety (or: Couldn't natural languages be much simpler?)[END_REF]Dimroth & Starren 2003;[START_REF] Hendriks | The Structure of Learner Varieties[END_REF].

In pidgin and creole (P/C) studies, the nature of learner varieties and their role in shaping creole grammars have hardly been explicitly studied because there is a longstanding bias towards discovering the grammatical system of creoles, and because locally-born children are traditionally assumed to be the main agents of creole genesis. Following [START_REF] Bickerton | Roots of Language[END_REF], it is widely assumed that the structural elaboration differentiating a complex creole from its simple pidgin ancestor results from nativization, namely the acquisition of native speakers by a language. While foreign-born adults (non-native speakers) contribute innovative features to the developing language, locally-born children play a crucial role, via first language acquisition, in stabilizing and elaborating the linguistic system (see for example DeGraff 1999 among others). This classic view of creole development has in recent years been further developed by researchers working on expanded Pacific Pidgins and Hawai'ian (cf. Roberts 2000;[START_REF] Siegel | The Emergence of Pidgin and Creole Languages[END_REF]). For instance, [START_REF] Siegel | The Emergence of Pidgin and Creole Languages[END_REF] posits a three-generational rather than a two-generational model, and argues that substrate calquing and second language use, rather than first language acquisition, played an important role in the development of creoles. He posits the following model: that the first generation who was dominant in their ancestral language introduced new morphosyntactic features to the pidgin through substrate calquing. The second generation, who were bilingual in an ancestral language and in the emerging contact language and who also made frequent use of the pidgin, assigned new functions to such features mostly based on models found in their ancestral languages. Finally, the third generation who were mostly monolingual in the contact variety further systematized and firmly established their use.

In the case of the Surinamese Creoles, the picture is more complex and a similar clear-cut three generational development model finds little support (cf. [START_REF] Arends | Syntactic developments in Sranan: Creolization as a gradual process[END_REF]Essegbey & Bruyn to appear;[START_REF] Migge | The origin and development of possibility in the creoles of Suriname[END_REF][START_REF] Kramer | Gradualism in the transfer of tone spread rules in Saramaccan[END_REF][START_REF] Van Den Berg | A grammar of Early Sranan[END_REF]).2 However, Siegel's emphasis on the role of substrate influence via transfer in the development of creoles is clearly of interest for the Surinamese context. Demographic data suggest that Sranan was being learned and practiced mostly as an L2 by newly arriving enslaved Africans, free Europeans and later indentured labourers for a considerable amount of time after its initial emergence. Even in late 18 th century Surinam, over a century after initial colonization, only 30% of the black population was locally-born (Arends 1995: 269). The Maroon Creoles nativized more quickly, but they, particularly Ndyuka and to a lesser degree the other Eastern Maroon varieties, have also been and continue to be practiced as L2 varieties (cf.

Léglise 2007 for the modern context). To date it is still unclear what effect this numerical dominance of non-native over native speakers has had on the development of these creoles.

In this paper we will take up this issue, drawing on historical as well as contemporary data. In contrast to other creoles, a remarkable corpus of 18 th century language materials in and on Sranan is available that provides us with a unique window on 18 th century Sranan language practices. 3 The contemporary data come from recordings and discussion with native and non-native speakers of the Creoles of Surinam in French Guiana. In order to explore the similarities and differences in the underlying L1 and L2 systems, we first examine three linguistic features, namely negation, and imperfective and past time marking, in the contemporary varieties and then compare them with their Early Sranan equivalents. The aim of this study is to improve our understanding of the role and relative impact of strategies of second language learning and use on the development of creole grammar.

The paper is organized as follows. Part One provides some background on the data and outlines the research methodology in more detail. In Parts Two to Four we compare the distribution of the three linguistic features in our contemporary and historical corpora. Part Five summarizes the findings and discusses the implications for theories of creole development.

Data & Methodology

1. The contemporary language data

The French overseas department of French Guiana (Guyane Française) is highly multilingual. Current estimates put the number to about 30 languages belonging to several families of languages (cf. [START_REF] Léglise | Language naming practices, ideologies and linguistic practices: Toward a comprehensive description of language varieties[END_REF]. Besides French, the official language, they include Amerindian languages from the Cariban (Kali'na, Wayana), Tupi-Guarani (Emerillon, Wayampi), and Arawak (Lokono, Palikur) family of languages, various European languages such as Brazilian Portuguese, (Surinamese) Dutch and Spanish, English-lexified creoles (Aluku, Ndyuka, Pamaka, Saamaka, Sranan Tongo, Guyanese Creole), French-lexified creoles (the Creoles of French Guiana, Martinique and Guadeloupe & Haitian Creole), and Asian languages like Hmong and varieties of Chinese. In the last 30 years, French Guiana has received considerable numbers of migrants from the Guiana region (Guyana, Surinam, Brazil) and from outside of the region (Haiti, China), and processes of urbanization have given rise to a fair amount of internal migration. Speakers of Surinamese Creoles are at the centre of such migration movements. They currently constitute about 40% of the overall population and make up an even higher proportion in the western part of the department. The three related Eastern Maroon Creoles (EMC) Aluku, Ndyuka and Pamaka are spoken natively by more than 30% of the population [START_REF] Léglise | Des langues, des domains, des régions, pratiques, variations, attitudes linguistiques en Guyane[END_REF]. Native speakers of Sranan Tongo are restricted to the few Arawak villages and a few Kali'na villages.

The Creoles of Surinam are also widely used as a lingua franca in western French Guiana where they are locally referred to by the generalizing terms Takitaki or Businenge Tongo which may refer to all the Creoles of Surinam or just to a subset such as the three related EMCs (cf. [START_REF] Léglise | Language naming practices, ideologies and linguistic practices: Toward a comprehensive description of language varieties[END_REF]. A school survey showed that more than 65% of 6 th grade children in the west declare speaking Ndyuka or Takitaki as an L2 [START_REF] Léglise | Des langues, des domains, des régions, pratiques, variations, attitudes linguistiques en Guyane[END_REF].

The data for this study come from a wide range of interactions involving persons of European, Guianese, Haitian, Chinese, Brasilian and Amerindian (Kali'na) origin. Some of the interactions had an interview character while others were service encounters or informal chats (cf. Migge & Leglise in prep.). People's competence in the language, learning trajectories and usage patterns also differ. One person was essentially at the one-word stage while most others had progressed to an intermediate or high level of competence. Some people learned 'Takitaki' in their youth while others only learned it later in adult life. Finally, some used 'Takitaki' only for a limited number of work-related interactions while others used it regularly for a wide range of interactions, including peers, partners, in-laws, and some formal interactions.

Historical sources

The texts that were consulted for the present study were retrieved from the Sranan Section of the Surinam Creole Archive. 4 They include a) religious texts such as bible translations and hymns (Schumann 1781); b) judicial documents such as transcripts of interrogations and witness reports (Court Records); c) official documents such as a peace treaty; d) travel reports and e) documents that were created for the purpose of language instruction such as dictionaries and language manuals by a Moravian missionary (C. L. Schumann) as well as secular persons (J.D. Herlein, P. van Dyk, J.

Nepveu and G. C. Weygandt). Because of this variety of text types, variation within and among the texts may correspond to different dimensions, ranging from diachronic to social, stylistic as well as geographical. Furthermore, variation within and among the texts may be linked to the different speech events represented in these texts, ranging from recorded, recalled to imagined and invented. While recorded texts are the most reliable (van den Berg & Arends 2004), texts belonging to other text types need to be assessed carefully in terms of representativeness and validity. Detailed assessments can be found in the works of [START_REF] Smith | The genesis of the Creole Languages of Suriname[END_REF], [START_REF] Arends | Syntactic developments in Sranan: Creolization as a gradual process[END_REF][START_REF] Arends | Introduction to Part I[END_REF], [START_REF] Bruyn | Grammaticalization in Creoles: The Development of Determiners and Relative Clauses in Sranan[END_REF] and [START_REF] Van Den Berg | A grammar of Early Sranan[END_REF] among others. The sources are presented in table 1. (w = word list; dl = dialogue; pl = play; dc = dictionary; we = Sranan words and expressions in text in another language; pt = peace treaty)

3. Methodology

We have selected three morphosyntactic features for comparison between contemporary native and non-native creole speech and Early Sranan: Negation, Imperfective aspect and Past time reference. We focus on these aspects of the verb phrase because they have figured prominently in the literature on L2 learning.

First, we examine these features in contemporary native and non-native creole speech in order to reveal the similarities and differences in the underlying L1 and L2 systems. Subsequently, their Early Sranan equivalents are studied. We posit that similarities between L1 practices and Early Sranan (but not L2 practices) suggest that a feature emerged rapidly and that its emergence was mostly likely not characterized by a protracted period of variation preceding stabilization. By contrast, we take similarities between L2 practices and Early Sranan (but not L1 practices) to be indicative of a slow process of development of the investigated feature; its emergence involved one or more stages, and/or several different models were in competition prior to the feature's stabilization. Similarities between L1, L2 and Early Sranan suggest that the investigated feature may have emerged via converging developmental paths in L1 and L2 acquisition. Differences between L1, L2 and Early Sranan call for other explanations, such as contact-induced changes modelled on languages not encountered in the other setting. For example, Dutch and German may have exerted some influence on (the representation of) Early Sranan, but only (Surinamese) Dutch influences L1 and L2 practices in French Guyana, indirectly through Sranan and directly because some L1 speakers did all or part of their education in Surinam.

Negation

Negation markers are among the few closed class items that are typically found in the Basic Variety (BV), an early but relatively stable state in the process of spontaneous (adult) second language acquisition [START_REF] Klein | The basic variety (or: Couldn't natural languages be much simpler?)[END_REF]. They tend to precede the part of the utterance over which they have scope and occur at the topic-focus boundary (Klein & Perdue 1997: 318). Because the placement of negation (and other scope particles) depends to a certain extent on the topic-focus structure of the utterance, and on the position of the elements over which they have scope, its position "indicates which part of the utterance is to be affected by the particle, and its scope can be described as "adjacent and to the right"." (Dimroth & Watorek 2000: 309).

The expression of negation evolves via the use of holophrastic (or anaphoric) negation in the pre-BV (stage 1), to a focus operator and later topic-focus linker in the BV (stage 2) to forms that correspond more closely to the syntax and morphology of negation in the target language (stage 3).

But already in the pre-BV, negation is integrated in the utterance structure, as it is either a comment on a topic X or it has in its focus an element X. Once lexical categories and thematic arguments are acknowledged by the learner, the item that expresses negation, the negation operator, begins to act as a focus sensitive operator, being placed in pre-focus (typically pre-verbal) position. When focus is no longer the main drive behind the placement of negation in pre-verbal position, a syntactic motivation is assumed for the placement of negation in this position (see Dimroth et al. 2003).

1. Negation in contemporary L1 varieties

In the L1 varieties of the EMCs and Sranan clausal negation is expressed by a negation operator or negative particle, ná or á(n) in the EMCs and no in Sranan, that directly precedes the verb and its auxiliaries (1). However, our EM L1 data involves variation between EMC and Sranan forms. This is most prominent among young urbanized Maroons who are at pains to project an urban identity (cf.

Migge 2007).

( The negative constituent adds emphasis (cf. Huttar & Huttar 1994: 253). Compare for instance the sentences in ( 4): (4a) with constituent negation has an emphatic reading while (4b) without it does not.

In (4a), everybody without exclusion is referred to while in (4b) it is implied that only some, usually the majority, are included.

(4) a.

Negation in contemporary L2 varieties

The expression of negation in the L2 varieties of the Surinamese Creoles is similar to that found in the L1 varieties. In the L2 varieties (7), as in the practices of urbanized younger EMs, the EMC and the Sranan clausal negators are used interchangeably.

( The comparison shows that L2 varieties differ from L1 varieties in the relative frequency of use of negative constituents and quantifiers and the rules governing their co-occurrence with clausal negation. The placement of negative constituents and quantifiers in L2 varieties mainly serves the function of focus marking which is reminiscent of the behavior of negation markers in the BV as reported by [START_REF] Dimroth | The scope of additive particles in basic learner languages[END_REF] and Dimroth et al. (2003).

Negation in historical sources

Basic negative clauses do not differ much across the various sources of Early Sranan; the negation operator no is placed in between the subject and the verb, preceding markers of tense mood and aspect, and auxiliaries (11a). 5 In copula-less clauses, no precedes the nominal predicate (11b).

( These findings suggest that no wan was not a lexicalized negative quantifier that modified nouns in Early Sranan, even though pronominal no wan 'no one; nobody' (< Eng. no one) is encountered from the mid-18 th century onwards. They further suggest a more syntactic rather than a more focus-sensitive motivation underlying the use of the clausal negation. The clausal negator seems to overrule the use of the constituent negator, and the negator no no longer appears adjacent to the item it has in focus.

Negation in L1, L2 and the historical sources

Our discussion of the expression of negation revealed some remarkable similarities between L1 EMC and Early Sranan varieties. In both sets of varieties a negated constituent in non-clause-initial position combines with preverbal clausal negation. In clause-initial position, it only combines with clausal negation when it is under focus. This contrasts with modern Sranan and contemporary L2 varieties where clause-initial negated constituents may always combine with clausal negation. We hypothesize that this difference between L1 EMC/Early Sranan and modern Sranan/L2 varieties is due to the latter having being heavily influenced by other languages for a prolonged period of time, and therefore involving a greater degree of variation. Finally, modern L2 varieties differ from L1 EMC and L1

Sranan in that they do not make frequent use of constituent negation; preverbal negation is the preferred strategy. This is also true for Early Sranan. This may be because of convergence: In the L1 linguistic system, preverbal negation is the unmarked strategy to express negation, while constituent negation is the marked strategy, bringing an item into focus in an explicit manner. In L2 learning the preverbal position is the pre-focus position, where the negator occurs when it begins to act as a focussensitive operator [START_REF] Dimroth | The scope of additive particles in basic learner languages[END_REF]Dimroth et al. 2003). Because the syntactic and focussensitive positions overlap, and constituent negation is relatively marked, the preverbal position is the preferred position for the negator. Thus, we conclude that the domain of negation must have stabilized early onwards.

Tense and aspect

Tense and aspect are studied extensively in L1 and L2 acquisition research, uncovering a strong relationship between inherent lexical aspect of verbs and the acquisition of tense-aspect morphology [START_REF] Andersen | The primacy of aspect in first and second language acquisition: The pidgin-creole connection[END_REF][START_REF] Sugaya | The acquisition of progressive and resultative meanings of the imperfective aspect marker by L2 learners of Japanese: Transfer, Universals, or Multiple Factors? Studies in Second Language Acquisition n o 29[END_REF][START_REF] Bardovi-Harlig | Tense and Aspect in Second Language Acquisition: Form, Meaning, and Use[END_REF]. The Aspect hypothesis summarizes this relationship, predicting that at the early stages of acquisition, learners predominantly use past tense and perfective aspect forms with punctual and telic verbs and progressive aspect forms with activity verbs [START_REF] Sugaya | The acquisition of progressive and resultative meanings of the imperfective aspect marker by L2 learners of Japanese: Transfer, Universals, or Multiple Factors? Studies in Second Language Acquisition n o 29[END_REF][START_REF] Andersen | The primacy of aspect in first and second language acquisition: The pidgin-creole connection[END_REF]. Li & Shirai (2000: 50) summarize the predicted order of development of tense-aspect morphology across different types of verbs as follows: 'He is having an extramarital affair with her.' Aspect is marked in example (23a), while it is not marked in (23b). Example (23a) is a paraphrase of the sentence 'he/she has a relationship with him/her'; (23b) conveys the same meaning, but qualifies the relationship as an extramarital one. Towards the end of the 18 th century de is used increasingly to mark imperfective aspect. For example, more instances of de are found in Weygandt's language primer, which is an extended revision of Van Dyk's (c1765) language primer, than in the original Van Dyk. A detailed comparison of these sources (van den Berg in prep.) reveals that Weygandt corrected some uses of de in Van Dyk. He also inserted de where unmarked verbs occur in Van Dyk, compare (24a) with (24b). In conclusion, we summarize that de emerged as an imperfective aspect marker from the mid-18 th century onwards, but it appears to be an optional, rather than a categorical item. Further research is needed to determine what precisely governs the use of de in Early Sranan.

Expression of past time reference

The Surinamese Creoles have a relative tense system in which the tense locus may be either the moment of utterance (speech time) or some other reference point.

Contemporary L1

In the contemporary Surinamese Creoles past time reference is either expressed by the particles ben (Sranan) and be (EMC) which precede the verbal head (25a) or by the unmarked verb (25b) optionally modified by temporal adverbials. Constructions using the unmarked verb occur once reference time has been established (cf. Huttar & Huttar 1994: 491-492).

(25) a. Di wi anga bakaa be e wooko a be tan so. Verbs that have more of a state-type interpretation such as sabi 'to know' and tan 'to stay' generally require be to locate the state in the past because their default interpretation is present time reference.

when
When ben/be precede an activity-denoting verb in a past context, ben/be locate the activity as having taken place prior to the point of reference in the past, i.e. 'a past before past' or background information (cf. Winford 2000: 401 for Sranan). If ben/be are combined with o, sa or wani, they express a hypothetical or counterfactuality meaning (cf. Winford 2000: 409).

Past time reference in L2 varieties

The situation seems to be relatively similar in L2 varieties where predicates are marked for past time by the marker be or ben. However, once the time frame has been set, such marking is no longer necessary, particularly with activity-denoting verbs.

(

Early Sranan

Relative past is expressed by variants of ben in the early Sranan sources. Prototypically, ben locates a certain situation as occurring prior to the time reference point established in the discourse, that is the moment of speech or some reference point in the past. In the dialogue represented in (29) the reference point is the period of departure from France. Thus, ben indicates that the talking took place prior to the leaving.

( Although the point of reference is not made explicit in the examples in (30), based on the discourse context it can be argued that it is not the time of speaking, but some time in the past. Thus, Nepveu's correction may not necessarily be an improvement; he may have mistaken the reference point as speech time under the influence of his native tongue.

Similar to a secondary (modal) use of ben in contemporary L1 Sranan [START_REF] Winford | Tense and Aspect in Sranan and the creole prototype[END_REF][START_REF] Wilner | Non-temporal uses of ben in Sranan Tongo[END_REF]), Early Sranan ben can convey a sense of hypotheticality or suggestion, to 'soften' the effect of a piece of advice, an invitation or a request, as in the example below. It is taken from the opening lines of a dialogue between two ladies, one visiting the other.

(31) Mi ben wan kom na misi na disi zabatim.

(VD c1765: 32) 1S PAST want come LOC lady LOC this evening 'I would like to visit the lady this evening.'

Conclusion

10 Benakese is a contracted serial verb construction, sen(i) aksi, which roughly translates as 'to send to ask'. The first verb sen(d)i is misprinted as ben. For a more detailed discussion, see van den [START_REF] Van Den Berg | A grammar of Early Sranan[END_REF] and [START_REF] Arends | Introduction to Part I[END_REF]. The word wen is unknown.

There are several interesting similarities and differences between the use of the imperfective aspect marker in the three data sets. Unlike the predictions presented in between grammatical morphemes and adverbial TMA marking is typical of emerging linguistic systems developed by L2 speakers. [START_REF] Slobin | From Ontogenesis to Phylogenesis: What Can Child Language Tell Us about Language Evolution[END_REF] credits their obligatory use to the agency of children acquiring the system as an L1. This has strong parallels with our case suggesting that the obligatory use of the Imperfective marker in L1 Eastern Maroon varieties was due to nativization. Further research is needed to clarify this.

The use of the relative past tense marker in the contemporary L1, L2 and Early Sranan varieties is very similar, although there is some indication of overuse in contemporary L2 varieties and Early Sranan. This is not atypical of L2 practices (cf. also [START_REF] Siegel | The Emergence of Pidgin and Creole Languages[END_REF].

Conclusions and implications

The findings presented in this paper show some remarkable similarities and differences between contemporary L1 and L2 varieties and Early Sranan. The three investigated areas of grammar are very similar in the three types of data. Each subsystem involves the same variants and their distribution is also broadly similar in the three types of data. This suggests that there is diachronic and synchronic continuity between these three varieties: The language(s) has/have not been subject to drastic change.

Continuity between them is particularly clear in the area of negation where we find close resemblance in the distribution of negative operators and negative constituents between (some) L1 (varieties), L2

varieties and varieties represented in the early texts. This suggests to us that some areas of grammar did not pass through several developmental stages, appear to have been little affected by L2 practices and consequently nativized quickly.

However, other subsystems of grammar such as imperfective marking and the use of the past time marker were subject to variation in the early sources. E alternated with zero and in the case of the past marker there was some tendency of using ben as an obligatory rather than as a relative past marker. These patterns of variation resemble those found in L2 varieties in general and those in contemporary L2 varieties of the Surinamese Creoles in particular, but differ from contemporary L1 patterns. This suggests at least two things in relation to the impact of L2 practices on creole development. First, L2 practices affected some areas of grammar in the early varieties, possibly those areas whose makeup differed in the native languages of the early users and creators of the languages, but were subsequently levelled probably as a result of increased nativization. Second, some areas of grammar stabilized relatively slowly, supporting the view that creole grammars emerged gradually via several stages rather than abruptly [START_REF] Arends | Syntactic developments in Sranan: Creolization as a gradual process[END_REF].

Abreviations

  eating more meat than bread.'

Table 1 The texts in the Sranan section of SUCA that were used in this study

 1

	text	year	document	page SR tokens	token total
			type
	Court Records	1707-1767	dl; we	-	500	-
	Herlein	1718	w; dl	3	200	400
	Nepveu	1762	pt	12	1.900	1.900
	Van Dyk	c1765	w; dl; pl	108	14.000	28.000
	Nepveu	1770	w; dl	8	700	1.800
	Schumann	1783	dl; dc	205	20.000	40.000
	Stedman	1790	we	-	300	-
	Weygandt	1798	w; dl; pl	144	15.000	30.000
	total			480	52.600	102.100

  Combinations of clausal negation and negative quantifiers such as notti 'nothing'(< Engl. nothing)

	2S NEG one place NEG LOC one place NEG master of 1S
	'You are not my master.' 'nowhere'
	and nimmere 'never' (< Dutch nimmer 'never') are encountered in most of the sources, see for
	example (12).
	(12) a. mi no habi notti	va	takki.	(Sch 1783: 125)
	1S NEG have nothing to	say
	'I have nothing to say.'
	b. Mino jerri zo wan zant nimmere	(VD c1765: 79)
	1S-NEG hear	so a	thing never
	'I have never heard such a thing.'
	(14) a. mi no de	go wanpeh,	OD. No wan peh mi de	go
	11) a. effi mi no ben takki gi 1S NEG IMPF go one/a-place / NEG one/a place 1S IMPF go ju, ju no ben sa sabi. (Sch 1783: 15)
	if 1S NEG PAST talk GIVE 2S 2S 'I am going nowhere.' (Sch 1783: 134) NEG PAST FUT know
	'If I didn't tell you, you would not have known.' b. dem no sa doe joe wan santie (CR 1760)
	b. jou no meester vor mi. 3P NEG FUT do 2S a	(CR 1707) thing
	'They will not harm you.'
	c. no wan peh,	no na wan peh	(Sch 1783 : 133)

However, when the negative quantifier occurs in clause-initial position, the clausal negation is dropped, as in (

13

). (

13

) nebretem mi ben du datti, OD. mi no ben du datti nebretem (Sch 1783: 121) never 1S PAST do that / 1SG NEG PAST do that never 'Never did I do that; I never did that.'

The negative construction no wan 'NEG one/a' differs from notti 'nothing' and nimmre/nebretem 'never' in that it cannot co-occur with clausal negation in basic negative sentences. Instead we find constructions such as (14a-b). 6 Furthermore, we find examples such as (14c), where the preposition na splits up the negative construction no wan, while the meaning of the entire construction appears unaffected by this change.

Table 2 .

 2 The development of tense-aspect morphologyThe main difference between L1 and L2 practices is that e appears to be optional in L2 varieties. Most L2 speakers use e in more than 60% of the cases and few use only or predominantly Ø(Migge & Léglise in prep.). At this stage, it appears that absence of e is more common if the verb is preceded by the Sranan negative marker no.While the examples in (22) show that de is an established imperfective aspect marker by the end of the 18 th century, other examples reveal that its use is optional rather than categorical throughout the 18 th century. All sources contain instances of unmarked verbs, while the discourse context suggests that de would have been appropriate if it had been obligatory. For example, we find in Nepveu (1770):

		State	Activity		Accomplishment	Achievement
	(Perfective) Past	4		3			2	1
	Progressive	?		1			2	3
	Imperfective	1		2			3	4
	Note: Numbers represent order of acquisition, from the earliest (1) to the latest (4).
	Below the expression of tense and aspect in L1, L2 varieties and Early Sranan is studied from this
	developmental perspective.
	(15) a. CON+ACT	On pe! Da a	mi anga i	e	nyan a
			Q place then FOC 1S with 2S	IMPF	eat DET
			pina	fu		saanan.
			suffering POSS Surinam
			'What! You and I suffer/get the bad part of Surinam.'
	b. PRO+ACT		A	tii	e	kali i.
			DET	elder IMPF call 2S
			'The elder is calling you.'
	c. HAB+ACT	Mi ná	e	go	a	ini	sama taki.
			1S	NEG	IMPF	go	LOC	in	person talk
			'I don't typically interfere with people's discussions.'
	d. INC+ACC		Den e	poli	den	pikin.
			3P IMPF spoil DET	child
			'They are spoiling the children.'
	e. CON+STATE Tii A.	e	de	fi en so mooi(n).

4.1.1. Imperfective aspect in the contemporary L1 varieties

Most Surinamese Creole L1 varieties express imperfective aspect with the element e

(Winford & Migge 2007: 85-91)

. 7 E can occur with a wide range of verbs such as state verbs (STATE), activity verbs (ACT), accomplishment-denoting verbs (ACC) and achievement-denoting verbs (ACH) expressing a progressive (PRO), habitual (HAB), continuous (CON) as well as inchoative (INC) occurrence of the state or event denoted by the main verb.

'As long as it does not have puss, it is better to cover it.'

In terms of the Aktionsart of the verb, absence of e appears with a wide range of verb types -absence is frequent with activity verbs, but they are also most frequent in the data.

IMPF have for die 'It's holidays that we are taking [right now, but] we're going to die.'

4. 1. 3. Imperfective in early sources

In Early Sranan, imperfective aspect meanings are expressed by (spelling variants of) de, auxiliaries such as tan 'to stay' or begin 'to begin; to start', or they are not expressed at all (zero, Ø). For limitations of space, we restrict our discussion to de. As in L1 varieties, it occurs with a wide variety of verbs in the sources, expressing a progressive, habitual, continuous as well as inchoative occurrence of the state or event denoted by the main verb. Some examples of de with state verbs, activity verbs, accomplishment verbs, achievement verbs as well as semelfactives are presented in

(22). 8

  Because I didn't want to go to the army. So I knew an association in St.Laurent, thus I came to work there. They made me go to the army in Cayenne.'There are a number of instances in the L2 data where the past marker appears to be overused in that the timeframe was established, but speakers still continued to use be instead of the unmarked verb

	26) A: Saide i the thing, he said he wanted to get a drink, he didn't have any change.' be kon a Guyane ? (FM)
	why b. No na ini wan bigi boto. A abi 2S PAST come LOC French Guiana ala sani	u e	wroko
	'Why did you come to French Guiana?' no LOC in a big boat 3S have all things 1P IMPF work
	B: bika sigisi wrokoman mi án	be kapten, a wani meki suudati, da mi be kapten be de wan	sabi
	because 1S NEG six workers	PAST capitain DET want make soldier then 1S PAST know capitain PAST COP a
	wan association na Soolan Jampanner ya, mi be	de	neen mi Ø kon barman. A no sowmaar a fi wooko na a
	one association LOC St. Laurent then 1S Japanese here 1S PAST COP barman 3S NEG nothing come for work LOC DET DET
	association, neen den Ø tya mi kon boto dati a no somaar […] (PI)	a	suudati a	Cayenne.
	Association then 3P boat that 3S NEG nothing carry 1S come LOC	soldier LOC Cayenne
	'No, inside a big ship. It had everything. We were six crew members and a captain, the captain was a Japanese, I was the barman, it wasn't any kind of ship.' c. A no so, mi be o gi yu, yu e kali mi 3S NEG so 1S PAST FUT give 2S 2S IMPF call 1S 'form. giiliman, yu no sabi. (StL)
	greedy.person 2S	NEG	know
	(27) A: ohoo ma i seefi, i kweki a 'It's not like that. I would have given you, [but then] you called me a greedy Haiti anda? (Kr)
	Oh person, you don't know.' but 2S self 2S raise LOC Haiti over-there
	'Oh, but you yourself, you grew up in Haiti?'
	B: ai baya dati na langa tin toli. Ai mi ben kweki na Haiti
	Yes friend that PRE long		time story yes 1S PAST raise LOC Haiti
	wel en		baka dati mi be	kon	na Sranan.
	Well and back that 1S PAST come LOC Sranan
	'Yes my friend, that's from a long time ago. Yes I grew up in Haiti and then I
	came to Surinam.'
	The L2 corpus also includes some examples where be is used as a 'past before past' or
	background marker (28a-b) and others where it marks hypothetical or counterfactuality in conditional
	clauses (28c), as in L1 varieties.
	(28) a. mi gi	a	man feifi euro a	tra	dei, i	sabi	a
	1S give DET man five Euro LOC other day you know DET
	man fu tyari a	sani, a taki a be	wani wan dringi a
	man for carry DET thing 3S say 3S PAST want one drink 3S
	ná be		abi pikin moni. (StL)
	NEG PAST have small money
	'I gave the guy five Euro the other day, you know the guy who was to bring

  The earliest attestation of ben is Herlein's (1718) text, see (30a) below. Nepveu (1770) corrects this sentence, leaving out ben among other alternations (30b). Indeed, sendi 'to send' is a non-stative verb; when unmarked it usually expresses 'past time reference', if the point of reference is speech time.Example (30c) is a dialogue from Weygandt's manual that is similar to the Herlein dialogue.

	(30) a. No mie ben benakese ta	entre ples	a reddi wen (HL 1718: 122 ) 10
		NEG 1S	PAST send-ask SAY other please again ?
		'No, I have already send to ask someone else if it would please her.'
		b. no mi sendi hakisi, na tara	plessi	a reddi (N 1770: 274)
		NEG 1S send ask	LOC other please/place already
		'No, I've already asked somewhere else' or:
		'No, I've already asked someone else if it would please her.'
		c. [A] Joe dé	go kiesie fiesietie	na joe tidee sabathem?
		2S	IMPF go get		company LOC 2S	today evening
		'Are you receiving company at your place this evening?'
		[B] Miesie Annaatje ben sen aksie mie ofoe mie sa	dé	na
		Miss Anna	PAST send ask	1S	if	1S	FUT COP LOC
		hoso, ofoe. Wan tra soema	kom	(Wey 1798: 121)
		house if	one other person come
		'Miss Anna send (someone) to ask me if I'll be at home or if another person
		is coming.'
	)	[A] Hoe zan den ben takki da.tem joe kommote na	frans
		Q	thing 3P	PAST say when 2S	come.out LOC	France
		konderi.
		country
		'What did they say when you were leaving France?'
		[B] Den ben takki van noeffe zomma disi ben go trouw
		3P	PAST say of	many person REL PAST go marry
		'They spoke of many people who were getting married.' (VD c1765: 36)

  Table1, our data do not exhibit a strong relationship between inherent lexical aspect of verbs and the occurrence of tense-aspect morphology. Our findings show that in both contemporary L2 varieties and Early Sranan the imperfective marker can occur with all possible types of verbs, expressing all possible kinds of aspectual meanings. The main difference is that unlike the contemporary L1 varieties it is optional in the contemporary L2 varieties and Early Sranan. It is not fully understood at present what precisely governs its distribution in these varieties. Two explanations suggest themselves. First, it may be possible that the distribution of the imperfective marker is determined by factors such as the presence of the negation operator and by discourse or pragmatic factors. This would explain why the imperfective marker is not obligatory: Its occurrence is not syntactically motivated but rather driven by the speaker's desire to underscore the status of the state/event in the discourse and/or by whether or not temporal information is easily recoverable from the context (cfBenazzo 2009: 15). Second, the variability in the presence of TMA markers may also be indicative of the fact that the use and functions of the imperfective marker were/are not yet fully conventionalised in early Sranan and contemporary L2 varieties. Essentially, they would be representing earlier developmental stages and the L1 varieties would represent later stages. For instance,[START_REF] Slobin | From Ontogenesis to Phylogenesis: What Can Child Language Tell Us about Language Evolution[END_REF] based on a comparison of the development of the future marker in Tok Pisin and TMA markers in Nicaraguan sign language that developed from several varieties of Home Signs and[START_REF] Benzzo | L'emergence de moyens grammaticaux pour exprimer les relations temporelles en L2[END_REF] based on the development of interlanguage systems of learners of Dutch and French conclude, that variation

  -Court Records 1667-1767. Nationaal Archief, The Hague. Inventaris van de archieven van de Raad van Politie (1669-1680) en de Raad van Politie en Justitie (1680-1683) en het Oud-Archief van het Hof van Politie en Criminele Justitie in Suriname (access code 1.05.10.02, inventory numbers 781-948); Overgekomen brieven en papieren uit het archief van de Sociëteit van Suriname, 1683-1715 (access code 1.05.04.01, inventory numbers 212-240), 1751-1767 (access code 1.05.04.06, inventory numbers 286-335). -Schumann, Christian Ludwig (1781) Die Geschichte unsers Herrn und Heilandes Jesus Christi, aus den vier Evangelisten zusammengezogen. Utrecht; MS 617. -Schumann, Christian Ludwig (1783) Neger-Englishes Wörter-Buch. Moravian archives, Utrecht/Paramaribo; MS 648. -Herlein, J. D. (1718) Beschrijvinge van de volksplantinge Zuriname: vertonende de opkomst dier zelver Colonie, etc. mitsgaders een vertoog van de Boschgrond, etc. Verrijkt met een landkaart (daar de legginge der Plantagien worden aangewezen) en kopere platen. Leeuwarden: Injema (2e druk). UB, Amsterdam. UBM: 1803 G 11. -Nepveu, Louis (1762). Sranan version of the Saramaka Peace Treaty. Nationaal Archief, The Hague. Inventaris van de archieven van de Raad van Politie (1669-1680) en de Raad van Politie en Justitie (1680-1683) en het Oud-Archief van het Hof van Politie en Criminele Justitie in Suriname (access code 1.05.10.02, inventory number 66, ff. 177 v o -183 v o . -Van Dyk, P. (c1765) Nieuwe en nooit bevoorens geziene onderwyzinge in het Bastert Engels, of Neeger Engels, Zoo als hetzelve in de Hollandsze colonien gebruikt word. Amsterdam: De Erven de Weduwe Jacobus van Egmont. Universiteitsbibliotheek Leiden: 1148 H 32.-Nepveu, Jean or Jan (1770) Annotatien op de Surinaamsche Beschrijvinge van A o 1718. MS.

	PRO Early Sranan Tongo Sources Progressive aspect
	Q	general question morpheme
	REL	relative marker
	S	singular
	SAY	the serial verb 'say'
	STATE	state-denoting verb
	1	first person pronoun
	2	second person pronoun
	3	third person pronoun
	ACC	accomplishment-denoting verb
	ACH	achievement-denoting verb
	ACT	activity-denoting verb
	BV	Basic Variety
	CON	continuative aspect
	COP	copula
	D	discourse marker
	DET Municipal Archives, Amsterdam. definite determiner
	FOC	focus marker
	FUT	future marker
	GIVE	the serial verb 'give'
	HAB	habitual aspect
	IDEO	ideophone
	IMPF	imperfective marker
	INC	inchoative aspect
	LOC	general locational preposition
	NEG	negation marker
	P	plural
	PAST	past time marker
	POSS	possessive marker
	PRE	presentative marker

-Weygandt, G. C. (1798) Gemeenzaame leerwyze, om het Basterd of Neger-Engelsch op een gemakkelyke wyze te leeren verstaan en spreeken. Paramaribo: W.W. Beeldsnijder. UB, Leiden; 799 E 18.

We would like to thank two anonymous reviewers and the editor of this volume, Sandra Benazzo, for their insightful comments and criticism on earlier versions of this paper. All remaining errors are the responsibility of the authors. bettinamigge@ucd.ie; m.v.d.berg@let.ru.nl

Note that[START_REF] Smith | English speaking in Surinam[END_REF] and[START_REF] Veenstra | Modeling Creole Genesis: Headedness in morphology[END_REF] propose a more abrupt scenario that is more in line with a three generational development model.

Historical texts for the Maroon Creoles are much less substantial, but due to close similarities between these languages, the Early Sranan corpus can also provide important insights into the development of the Maroon Creoles.

The Suriname Creole Archive (SUCA) is a joint project of the Radboud University Nijmegen, the University of Amsterdam and the Max Planck Institute Nijmegen supported by NWO (http://suca.ruhosting.nl, under construction).

The negation operator is occasionally represented as na in Schumann's (1783) dictionary and Van Dyk's (c.1765) language manual. Although these occurrences may just be unintentional spelling errors by the authors, or the type setter in the case of van Dyk, they remind us of the EMC negation operator ná. Since both Schumann and van Dyk include many features that can be associated with the vernacular of the population of African descent, in particular of the enslaved people on the plantations (van den Berg 2007), it may very well be the case that no and na were negation operator variants in some of the plantation varieties of Early Sranan.

The only instances in which clausal negation co-occurs with no wan in the sources is when the negated constituent is emphasized by means of placing intonational stress on wan. In Schumann's (1783) dictionary intonational stress is orthographically marked. In other sources, it can be inferred from the context that wan is stressed.

[START_REF] Seuren | Tense and aspect in Sranan[END_REF] was perhaps the first to trace the imperfective aspect marker (d)e to the copula/existential verb de which ultimately derives from the English locative adverb there[START_REF] Smith | The genesis of the Creole Languages of Suriname[END_REF]. It is being debated whether this process took place in Suriname[START_REF] Arends | Syntactic developments in Sranan: Creolization as a gradual process[END_REF][START_REF] Migge | The Origin of the Copulas (d/n)a and de in the Eastern Maroon Creole[END_REF] or elsewhere[START_REF] Smith | Voodoo chile. Differential substrate effects in Saramaccan and Haitian[END_REF], as it could have been imported by enslaved Africans prior to their arrival in Suriname. In Saamaka imperfective aspect is expressed by ta which derives from English stand (tan in the older sources).

A semelfactive is a type of verb that is used in reference to an event that happens only once.

While the reduplication may give the impression of non-semelfactivity, meki koffokoffo can be used as a semelfactive ('to give a cough' as well as a non-semelfactive ('to be coughing').