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HIGH-FREQUENCY ASYMPTOTICS FOR THE HELMHOLTZ EQUATION IN A HALF-PLANE

Base on the integral representations of the solution being derived via Fokas' transform method, the high-frequency asymptotics for the solution of the Helmholtz equation, in a half-plane and subject to the Neumann condition is discussed. For the case of piecewise constant boundary data, full asymptotic expansions of the solution are obtained by using Watson's lemma and the method of steepest descents for definite integrals.


Introduction
There is huge mathematical and engineering interest in acoustic and electromagnetic wave scattering problems, driven by many applications such as modeling radar, sonar, acoustic noise barriers, atmospheric particle scattering, ultrasound and VLSI [4]. Many problems of scattering of time-harmonic acoustic or electromagnetic waves can be formulated as the Helmholtz equation, supplemented with appropriate boundary conditions. Much effort has been put into the development of efficient numerical schemes and approximate methods to deal with the Helmholtz equation involving high wave-numbers (i.e., high-frequencies) [1-3, 5, 9, 10, 13, 14, 16-18]. It is noted in [5,15], that a question yet to be fully resolved, is to obtain accurate approximations of the solutions with a reasonable computational cost in the high-frequency case. Therefore, it seems desirable, and of difficulty, to consider the highfrequency asymptotics of the solutions to the Helmholtz equation. This is one of the main motives of the present investigation. Applying the theory of asymptotic analysis [6,19,22], one may achieve a high degree of accuracy with only a few leading terms in the asymptotic expansions of the solution being involved.
A novel flexible approach was introduced by Fokas in the late 1990s, to solve the initial/boundary value problems for various two-dimensional linear and integrable nonlinear partial differential equations (PDEs). In particular, the method has been applied to the Laplace equation in a convex polygon and to other linear two-dimensional PDEs and nonlinear PDEs, including the Helmholtz and modified Helmholtz equations. Readers are referred to [8] for a systematic exposition of the method, its various applications and more references therein.
For linear PDEs, the new method of Fokas can be viewed as an extension of the classical transform method that produces the analogue of Green's integral representation (IR) in the transform (or Fourier) space. Indeed, whereas Green's IR given the solution as an integral involving the boundary values, the new IR given the solution as an integral involving transforms of the boundary values [21]. For some specific domains it is possible to remove either the Dirichlet or the Neumann data, thus providing an integral representation of the Neumann or the Dirichlet problem of the corresponding linear PDE. Various asymptotic methods for definite integrals might then be applied to derive asymptotic formulas of the solutions. As mentioned in [21], the solutions obtained by the Fokas method has two significant features: (i) The integrals can be deformed to involve exponentially decaying integrands. (ii) The expression is uniformly convergent at the boundary of the domain. These features give rise to both analytical and numerical advantages in comparison with classical methods [7]. It will be shown here and see [11,12,20] that the novel integral representations are suitable for the numerical evaluation and more convenient to extract the asymptotic behavior of the solution. This is a consequence of the fact that it is possible, using simple contour deformations in the complex k-plane, to obtain integrals involving integrands with a strong decay for large k.
The objective of the present paper is to consider the following Neumann boundary value problem of the Helmholtz equation in the upper half-plane Ω: where n is the outer normal vector, To ensure the unique solvability, we assume that the following radiation condition at infinity is satisfied (cf. [21]): As a first step, we given the integral representation for the solution of the Neumann boundary value problem (1.1)-(1.3) for general h derived by Fokas' transform method. This is actually a special case in [21,Chap. 4].
And then, we focus on the high-frequency asymptotics with respect to a specific Neumann data, namely, where [a, b] refers to a finite interval and E is an arbitrary constant. We need to give a note that the problem considered here is not a realistic scattering problem, but serves as an illustrative example of the asymptotic investigation via the Fokas method. We choose the illustrative example (1.1) and (1.3)-(1.4) to keep the derivation as non-technical as possible. The results can actually be generalized and may involve an oscillatory double integral. To deal with such kind of problems, one may need a higherdimensional version of the steepest descents method, and it is rather difficult. We will focus on this research elsewhere.
The rest of the paper is arranged as follows: In Sec.

The Integral Representation for the Helmholtz Equation
The method of Fokas has two basic ingredients: one is the integral representation and other is the global relation. The integral representations for linear PDEs obtained by the Fokas method can be derived in several ways including (a) spectral analysis of the appropriate Lax pair; (b) substituting particular IRs of the fundamental solution (obtained by classical transforms) into Green's IR. The interested reader is referred to [8,Chap. 11;21] for derived in detail.

2)
and the curves L, oriented as in Fig. 1, is defined as

Asymptotic Approximations
Our goal in this section is to study the asymptotic behavior of the solution to the Neumann problem of the Helmholtz equation, as the frequency approaches to infinity. This large-β asymptotic analysis will be based on the integral representation (2.4), and will be carried out by using the method of steepest descents. First we note that to separate the integrand in (2.4) into pieces, the curve L is dented to keep away from k = −i; cf. Fig. 2. We denote this dented curve by L 1 . Therefore, q(z) in (2.4) can be expressed as the sum of two integrals of the form The phase function φ(k) has a pair of saddle points k = ±e −iθ with θ ∈ (0, π), determined by φ (k) = 0, lying symmetrically on the unit circle. The steepest descent path passing through the saddle point k + is determined by requiring that and that Re φ(k) decreases as k goes away from the saddle; cf. [6,19,22] for the definition and for basic background of the method of steepest descents. For θ ∈ (0, π/2), the path is the curve Γ connecting the origin and the infinite, as illustrated in Fig. 3. Furthermore, we need to decide how Γ turns to the origin and infinite. Writing in polar coordinates k = |k|e iϕ 1 , one has Γ : (|k| + 1/|k|) cos(ϕ 1 + θ) = 2, while Re φ = −r(|k| − 1/|k|) sin(ϕ 1 + θ) → −∞ as k turns to 0 or ∞ along Γ. Hence ϕ 1 = arg k → −π/2 − θ ∈ (−π, −π/2) as k turns to 0 on Γ, and ϕ 1 → π/2 − θ ∈ (0, π/2) as k turns to infinity on Γ. Hence we can deform the integration path L 1 in (3.1) into the oriented steepest descent path Γ by applying Cauchy's integral theorem. Hence we have Fig. 3. The steepest descent path Γ into which the integration path L 1 deformed: 0 < θ < π/2 (left), π/2 < θ < π (right).

Case (i) 0 < θ < π/2. In this case, we start from (3.3) and employ a standard argument of Watson's lemma. Firstly, we introduce a new integration variable
It is readily seen that τ ∈ [0, ∞) on the steepest descent path, and for each τ ∈ (0, ∞) there is a pair of points on Γ, say, k 1 and k 2 , satisfying (3.5). To be precise, we may specify 2 ) as τ → 0 + . Thus for τ ∈ (0, ∞), k 1 lies on the half steepest descent path connecting k + and ∞, while k 2 on the half joining k + and the origin. Now we have where Φ(τ ) = 1 k 2 1 + 1 which can be expanded into convergent series for small τ in the form where δ is a positive constant, and the coefficients c l = c l (r, θ) can be evaluated in view of (3.6), of which the first few are Waston's lemma implies that all contribution to the asymptotic behavior of the integrals in (3.7) comes from the saddle k = k + . Substituting (3.9) into (3.7) yields the full asymptotic expansion where Γ(t) is the usual Gamma function, Γ(l Case (ii) θ = π/2. From (3.1) and (3.2), in this case, we have cf. Fig. 3 for the integration path L 1 , dented at k = −i. The situation could be complicated since we have a pole of the integrand and a saddle of the phase function, both locate at k = −i. Denoting by Γ ε the half circle centered at k = −i, where ε is the radius. Then we observe that This fact can be seen by making a change of variable k → −1/k, which maps L 1 \Γ ε onto itself, with orientation being reversed, and keeps the integrand unchanged, thus implies J = −J. Hence we need only pick up the contribution from the circular part around k = −i, and we have

16)
as β → +∞, where r j and θ j are defined in (3.15). Two exceptional cases are
Picking up the main contribution to the asymptotic behavior, for fixed z = x + iy ∈ Ω, we have the following piecewise approximation:  One interesting phenomenon is that the exponential terms switch on and off as z crosses the vertical half lines Re z = a and Re z = b.

Conclusions
A new transform method for solving linear and certain nonlinear PDEs was introduced in the late 1990's by Fokas [8]. For linear PDEs it can be viewed as an extension of the classical transform method that produces the analogue of Green's Integral Representation (IR) in the transform (or Fourier) space [21]. In the present paper, based on the integral representations of the solution to the Helmholtz equation, in a half-plane and subject to Neumann condition obtained by Spence [21], full high-frequency asymptotic expansions for a specific Neumann condition (piecewise constant boundary data) are derived by using Watson's lemma and the method of steepest descents for definite integrals. The asymptotic expansion (3.16) in Theorem 1 is uniform in compact subsets of I, II and III, but not in neighborhoods of the boundaries. The piecewise results have jumps along two vertical lines. This can be explained mathematically, the reason is that there is a coalescing of saddle point with a simple pole in the integrand.