Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 2, 297–305

On the Low Wave Number Behavior of Two-Dimensional Scattering Problems for an Open Arc

R. Kress

Dedicated to Professor Lothar von Wolfersdorf

Abstract. The low wave number asymptotics for the solution of the Dirichlet problem for the two-dimensional Helmholtz equation in the exterior of an open arc is analyzed via a singlelayer integral equation approach. It is shown that the solutions to the Dirichlet problem for the Helmholtz equation converge to a solution of the Dirichlet problem for the Laplace equation as the wave number tends to zero provided the boundary values converge. Keywords: Helmholtz equation, exterior boundary value problems, integral equation methods,

low wave number limits, cosine substitution

AMS subject classification: Primary 35 J 05, secondary 45 B 05

1. Introduction

The study of the low wave number behavior of solutions to exterior boundary value problems for the Helmholtz equation in two dimensions via integral equation methods has a long history. This is due to difficulties arising from the following two facts. Firstly, in two dimensions the fundamental solution of the Helmholtz equation does not converge to the fundamental solution of the Laplace equation if the wave number k tends to zero. Secondly, the fundamental solution of the Laplace equation is not bounded at infinity. For the Dirichlet problem in the exterior of a finite number of disjoint closed contours the limiting behavior was investigated by MacCamy [13], Werner [15] and the present author [10]. Here, the analysis can be based on reducing the boundary value problem to an integral equation of the second kind via a double-layer or a combined double- and single-layer approach.

In the present paper we want to examine the low wave number limit for the corresponding Dirichlet problem in the exterior of an open arc. In this case, as opposed to the case of closed contours, we need to work with an integral equation of the first kind via a single-layer approach. The existence and uniqueness of solutions to the Dirichlet problem in the exterior of an open arc is well established (c.f. [5] or the appendix in the Russian translation [2] of [1] and the references therein). More recently, the author [12] has suggested to base the analysis of the single-layer integral equation for an arc on the cosine transformation which has been introduced by Multhopp [14] and by Yan and Sloan [16] for the corresponding integral equation in the potential theoretic case of the Laplace equation. Here, we will adopt this approach to investigate the low wave number limit. Our analysis will be in a Sobolev space setting instead of a Hölder space setting as in [12]. Clearly, the results can be extended to the Dirichlet problem in the exterior of a finite number of disjoint open arcs.

R. Kress: Inst. für Num. und Ang. Math. der Univ., Lotzestr. 16-18, D - 37083 Göttingen ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

For corresponding investigations in two-dimensional elasticity we refer to Hsiao and Wendland [7, 8].

2. Scattering from an open arc

Assume that Γ ⊂ ℝ^{2} is an arc of class C^{2}, that is, Γ = {z(s) : s ∈ [−1, 1]} where z : [−1, 1] → ℝ^{2} is an injective and twice continuously differentiable function with z′(s) ≠ 0 for all s ∈ [−1, 1]. The mathematical treatment of the scattering of timeharmonic acoustic or electromagnetic waves from thin infinitely long cylindrical obstacles is modelled by the following boundary value problem:

Given a function f_{k} ∈ C^{1}(Γ), find a solution u_{k} ∈ C^{2}(ℝ^{2} \ Γ) ∩ C(ℝ^{2}) to the Helmholtz equation

Δu_{k} + k^{2}u_{k} = 0  in ℝ^{2} \ Γ  (2.1)

with wave number k > 0 satisfying the Dirichlet boundary condition

u_{k} = f_{k}  on Γ  (2.2)

and the Sommerfeld radiation condition

lim_{r→∞} \sqrt{r}(\frac{∂u_{k}}{∂r} − iku_{k}) = 0  (r = |x|)  (2.3)

uniformly in all directions \frac{x}{|x|}. Note that we do not explicitly assume any edge condition for the behavior of the solution at the two end points of Γ.

The limiting case k = 0 corresponds to the following static boundary value problem: Given a function f_{0} ∈ C^{1}(Γ), find a solution u_{0} ∈ C^{2}(ℝ^{2} \ Γ) ∩ C(ℝ^{2}) to the Laplace equation

Δu_{0} = 0  in ℝ^{2} \ Γ  (2.4)

satisfying the Dirichlet boundary condition

u_{0} = f_{0}  on Γ  (2.5)

and the boundedness condition

u_{0}(x) = O(1)  (|x| → ∞),  (2.6)

uniformly in all directions \frac{x}{|x|}. Again, no edge conditions are explicitly assumed.

Theorem 2.1. The above exterior Dirichlet problems for an open arc both have at most one solution.

Proof. As in the case of the Dirichlet problem for the exterior of a closed contour (see [11: Theorem 6.11]), uniqueness for the Laplace equation relies on the maximumminimum principle for harmonic functions and the boundedness condition at infinity. For the Helmholtz equation, uniqueness follows from Rellich’s lemma and an application of Green’s theorem. The application of Green’s theorem can be justified by using an approximation idea due to Heinz analogously to the case of the exterior of a closed contour (see [3: Theorem 3.7])

For functions φ ∈ L^{p}(Γ) (1 < p < ∞) we introduce the operator M by

Mφ := φ − \frac{1}{|Γ|} ∫_{Γ} φ ds.

Then, for 0 < k < 1, we seek the solution of problem (2.1) - (2.3) in the form of a modified acoustic single-layer potential

u_{k}(x) = ∫_{Γ} Φ_{k}(x, y)\{(Mφ_{k})(y) − \frac{2π}{ln k} φ_{k}(y)\}ds(y)  (x ∈ ℝ^{2})  (2.7)

with a density φ_{k} ∈ L^{p}(Γ) and the fundamental solution to the Helmholtz equation in two dimensions given by

Φ_{k}(x, y) = \frac{i}{4} H_{0}^{(1)}(k|x − y|) (x ≠ y)

in terms of the Hankel function H_{0}^{(1)} of order zero and of the first kind. Since the Hankel function has the asymptotic behavior

\frac{i}{4} H_{0}^{(1)}(t) = \frac{1}{2π} ln \frac{2}{t} + \frac{i}{4} − \frac{1}{2π} − \frac{C}{2π} + O(t^{2} ln \frac{1}{t})  (t → 0)  (2.8)

where C = 0.5772 … denotes Euler’s constant, the acoustic single-layer potential (2.7) has a limit as k → 0 given by

u_{0}(x) = ∫_{Γ} {Φ_{0}(x, y)(Mφ_{0})(y) + φ_{0}(y)}ds(y)  (x ∈ ℝ^{2})  (2.9)

with the fundamental solution

Φ_{0}(x, y) = \frac{1}{2π} ln \frac{1}{|x − y|} (x ≠ y)

to the Laplace equation. Note that in the difference Φ_{k} − Φ_{0} the trouble making term ln k is eliminated due to ∫_{Γ} Mφ_{k} ds = 0.

The potentials (2.7) and (2.9) satisfy the Helmholtz and Laplace equations in ℝ^{2} \Γ, respectively. Furthermore, (2.7) fulfills the radiation condition and (2.9) is bounded at infinity because of ∫_{Γ} Mφ_{0} ds = 0. For densities in L^{p}(Γ) the single-layer potentials (2.7) and (2.9) are continuous throughout ℝ^{2} (see [6: p. 276]). Hence, (2.7) solves the Dirichlet problem (2.1) - (2.3) provided the density φ_{k} ∈ L^{p}(Γ) is a solution to the integral equation

∫_{Γ} Φ_{k}(x, y)\{(Mφ_{k})(y) − \frac{2π}{ln k} φ_{k}(y)\}ds(y) = f_{k}(x)  (x ∈ Γ).  (2.10)

Analogously, (2.9) solves the Dirichlet problem (2.4) - (2.6) provided φ_{0} ∈ L^{p}(Γ) is a solution to the integral equation

∫_{Γ} {Φ_{0}(x, y)(Mφ_{0})(y) + φ_{0}(y)} ds(y) = f_{0}(x)  (x ∈ Γ).  (2.11)

Substituting s = cos t into Γ = {z(s) : s ∈ [−1, 1]}, we define operators A and B which map functions on Γ into functions on the interval [0, π] by setting

\begin{matrix}(Aφ)(t) & = sin t |z′(cos t)| φ(z(cos t)) \\ (Bf)(t) & = f(z(cos t)) \end{matrix} \}

for 0 ≤ t ≤ π. The operator N given by

Nψ = ψ − [ ∫_{0}^{π} A1 dτ]^{−1}A1 ∫_{0}^{π} ψ dτ

is related to the operator M by NA = AM. By this substitution, the integral equation (2.10) is equivalently transformed into

∫_{0}^{π} H_{k}(t, τ)\{(Nψ_{k})(τ) − \frac{2π}{ln k} ψ_{k}(τ)\}dτ = g_{k}(t)  (t ∈ [0, π]),  (2.12)

where ψ_{k} = Aφ_{k} and g_{k} = Bf_{k} and where the kernel is given by

H_{k}(t, τ) = Φ_{k}(z(cos t), z(cos τ)) (t ≠ τ).

Analogously, the parametrized version of (2.11) is given by

∫_{0}^{π} {H_{0}(t, τ)(Nψ_{0})(τ) + ψ_{0}(τ)} dτ = g_{0}(t)  (t ∈ [0, π]),  (2.13)

where

H_{0}(t, τ) := \frac{1}{2π} ln \frac{1}{|z(cos t) − z(cos τ)|} (t ≠ τ).

We will look for solutions of equations (2.12) and (2.13) in L^{2}[0, π]. For the corresponding density φ on Γ, related through ψ = Aφ, we then have

∫_{Γ} |φ(x)|^{p}ds(x) = ∫_{0}^{π} \frac{|ψ(t)|^{p}}{|z′(cos t)|^{p−1} sin^{p−1} t} dt

and, with the aid of Hölder’s inequality, can estimate

∫_{Γ} |φ(x)|^{p}ds(x) ≤ \frac{1}{inf_{s∈[−1,1]} |z′(s)|^{p−1}} [∫_{0}^{π} sin^{\frac{1−p}{1− \frac{p}{2}}} t dt]^{1− \frac{p}{2}} [∫_{0}^{π} |ψ(t)|^{2}dt]^{\frac{p}{2}} (2.14)

where the first integral on the right-hand side is finite provided 1 < p < \frac{4}{3}, that is, φ ∈ L^{p}(Γ) for 1 < p < \frac{4}{3} which ensures continuity of the single-layer potentials.

3. Low wave number limit

For the further discussion we rewrite equations (2.12) and (2.13) in operator notation as

K_{k}ψ = f_{k}

with

(K_{k}ψ)(t) = ∫_{0}^{π} H_{k}(t, τ)\{(Nψ)(τ) − \frac{2π}{ln k} ψ(τ)\}dτ (t ∈ [0, π])

for 0 < k < 1 and

(K_{0}ψ)(t) = ∫_{0}^{π} {H_{0}(t, τ)(Nψ)(τ) + ψ(τ)} dτ (t ∈ [0, π]).

In addition we introduce the integral operator

(Lψ)(t) = −\frac{1}{4π} ∫_{0}^{π} ln (\frac{4}{e^{2}} [cos t − cos τ]^{2})ψ(τ) dτ (t ∈ [0, π]).

In the sequel, by H_{e}^{1}[0, π] we denote the subspace of even functions in H^{1}[−π, π].

Lemma 3.1. The operator L : L^{2}[0, π] → H_{e}^{1}[0, π] is bounded and has a bounded inverse L^{−1} : H_{e}^{1}[0, π] → L^{2}[0, π]. The operator K_{0} − L : L^{2}[0, π] → H_{e}^{1}[0, π] is compact.

Proof. From the identity

ln (\frac{4}{e^{2}} [cos t − cos τ]^{2}) = ln (\frac{4}{e} sin^{2} \frac{t − τ}{2} ) + ln (\frac{4}{e} sin^{2} \frac{t + τ}{2} )

it follows that

∫_{0}^{π} ln (\frac{4}{e^{2}} [cos t − cos τ]^{2}) cos mτ dτ = ∫_{0}^{2π}ln (\frac{4}{e} sin^{2} \frac{t − τ}{2} ) cos mτ dτ.

Therefore, for f_{m}(t) = cos mt, from [9: p. 88] or [11: Lemma 8.21] we have that Lf_{m} = λ_{m}f_{m} where λ_{m} = \frac{1}{2 max(1,m)} (m ∈ ℕ_{0}). Hence L maps the function ψ ∈ L^{2}[0, π] with Fourier series ψ = ∑_{m=0}^{∞} a_{m}f_{m} into Lψ = ∑_{m=0}^{∞} λ_{m}a_{m}f_{m}. Since {f_{m}}_{ℕ_{0}} forms a complete orthogonal system for L^{2}[0, π] and for H_{e}^{1}[0, π], this implies boundedness of L : L^{2}[0, π] → H_{e}^{1}[0, π] and invertibility, with the bounded inverse L^{−1} given in terms of the Fourier series L^{−1}g = ∑_{m=0}^{∞} \frac{1}{λ_{m}} b_{m}f_{m} for g ∈ H_{e}^{1}[0, π] with g = ∑_{m=0}^{∞} b_{m}f_{m}. By Taylor’s formula

z(s) − z(σ) = (s − σ) ∫_{0}^{1} z′(σ + λ(s − σ)) dλ,

using the fact that z ∈ C^{2}[−1, 1] with z′(s) ≠ 0 for s ∈ [−1,], it can be verified that

H_{0,1}(t, τ) := \frac{1}{2π} ln \frac{1}{|z(cos t) − z(cos τ)|} + \frac{1}{4π} ln (\frac{4}{e^{2}} [cos t − cos τ]^{2}) (t ≠ τ)

can be extended in a continuously differentiable manner on [0, π] × [0, π]. Hence, the operator K_{0,1} : L^{2}[0, π] → H_{e}^{1}[0, π] defined by

(K_{0,1}ψ)(t) = ∫_{0}^{π} H_{0,1}(t, τ)(Nψ)(τ) dτ (t ∈ [0, π])

is compact. The operator K_{0,2} : L^{2}[0, π] → H_{e}^{1}[0, π] defined by

(K_{0,2}ψ)(t) = [LA1 + 1] ∫_{0}^{π} ψ(τ) dτ (t ∈ [0, π])

is bounded and has finite-dimensional range and therefore also is compact. Now compactness of K_{0} − L : L^{2}[0, π] → H_{e}^{1}[0, π] follows since K_{0} − L = K_{0,1} + K_{0,2}

Theorem 3.2. There exist constants c > 0 and 0 < k_{0} < 1 such that for all 0 ≤ k ≤ k_{0} the inverse operators K_{k}^{−1} : H_{e}^{1}[0, π] → L^{2}[0, π] exist and satisfy

∥K_{k}^{−1} − K_{0}^{−1}∥ ≤ \frac{c}{ln |k|} .

Proof. We first show that K_{0} : L^{2}[0, π] → H_{e}^{1}[0, π] has a bounded inverse. For a given g ∈ H_{e}^{1}[0, π], solving K_{0}ψ = g for ψ ∈ L^{2}[0, π] is equivalent to solving the equation of the second kind

ψ + L^{−1}(K_{0} − L)ψ = L^{−1}g

where the operator L^{−1}(K_{0}−L) : L^{2}[0, π] → L^{2}[0, π] is compact by the previous Lemma 3.1. Hence, by the Riesz-Fredholm theory, it suffices to show injectivity of K_{0} in order to establish bijectivity of K_{0} and boundedness of the inverse.

Let ψ_{0} ∈ L^{2}[0, π] be a solution to the homogeneous equation K_{0}ψ_{0} = 0. Then the corresponding potential u_{0} with density φ_{0} ∈ L^{p}(Γ) (0 < p < \frac{4}{3}) defined by (2.9) solves the homogeneous boundary value problem (2.4) - (2.6). Therefore by Theorem 2.1 we have u_{0} = 0 in ℝ^{2}. From (2.9) we have the asymptotic behavior

u_{0}(x) = ∫_{Γ} φ_{0}ds + o(1) (|x| → ∞)

uniformly for all directions. Therefore u_{0} = 0 in ℝ^{2} implies that ∫_{Γ} φ_{0}ds = 0. Now, in view of grad u_{0} = 0 in ℝ^{2} \ Γ, the potential theoretic jump relations yield

φ_{0}(x) = lim_{h→0} [ν(x) · grad u_{0}(x − hν(x)) − ν(x) · grad u_{0}(x + hν(x))] = 0

for almost all x ∈ Γ. Here, ν denotes the unit normal vector on Γ (chosen with one of the two possible orientations). That the jump relations for the gradient of the single-layer potential are valid for L^{p} densities can be seen by relating them to the Sokhotski–Plemelj formulae for the Cauchy integral (see [11: p. 116]). However, the latter are valid for L^{p} densities (see [4: p. 50]). Finally, φ_{0} = 0 almost everywhere on Γ implies ψ_{0} = 0 almost everywhere in [0, π] and the injectivity of K_{0} is proven.

From (2.8) and

\frac{i}{4} \frac{d}{dt} H_{0}^{(1)}(t) = −\frac{1}{2πt} + O(t ln \frac{1}{t})

for t → 0 it can be deduced that ∥K_{k} − K_{0}∥ = O(1/ ln |k|) (k → 0) for the difference K_{k} − K_{0} : L^{2}[0, π] → H_{e}^{1}[0, π]. Now the statement of the Theorem follows from the invertibility of K_{0} and the boundedness of the inverse K_{0}^{−1} by writing K_{k} = K_{0}[I + K_{0}^{−1}(K_{k} − K_{0})] and using the Neumann series

Corollary 3.3. For 0 ≤ k ≤ k_{0} the Dirichlet problem (2.1) − (2.3) is uniquely solvable.

Proof. This follows from Theorems 2.1 and 3.2 by using the single-layer approach (2.7)

Now we are in a position to state the main and final result of this paper.

Theorem 3.4. Assume that ∥f_{k} − f_{0}∥_{C^{1}(Γ)} → 0 as k → 0. Then, for k → 0, the solution u_{k} of the Dirichlet problem (2.1) − (2.3) for the Helmholtz equation converges uniformly on compact subsets of ℝ^{2} to the solution u_{0} of the Dirichlet problem (2.4) − (2.6) for the Laplace equation.

Proof. We represent u_{k} and u_{0} in the form of the potentials (2.7) and (2.9) with densities φ_{k} and φ_{0}, respectively. For the corresponding solutions ψ_{k} = Aφ_{k} of the integral equations (2.12) and (2.13), that is, for ψ_{k} = K_{k}^{−1}g_{k}, using Theorem 3.2 and the triangle inequality, we can conclude that

∥ψ_{k} − ψ_{0}∥_{L^{2}[0,π]} → 0  (k → 0).  (3.1)

Here, we have used that

∥g_{k} − g_{0}∥_{H_{e}^{1}[0,π]} = ∥B(f_{k} − f_{0})∥_{H_{e}^{1}[0,π]} ≤ c_{1}∥f_{k} − f_{0}∥_{C^{1}(Γ)}

for some positive constant c_{1}. In view of (2.14), the convergence (3.1) implies that

∥φ_{k} − φ_{0}∥_{L^{p}(Γ)} → 0  (k → 0)  (3.2)

for 1 < p < \frac{4}{3}.

We split the difference u_{k} − u_{0} = v_{k} + w_{k} where, for x ∈ ℝ,
v_{k}(x) = ∫_{Γ} {Φ_{0}(x, y)(M[φ_{k} − φ_{0}])(y) + [φ_{k}(y) − φ_{0}(y)]}ds(y)

and

w_{k}(x) = ∫_{Γ} \{[Φ_{k}(x, y) − Φ_{0}(x, y)](Mφ_{k})(y) − [\frac{2π}{ln k} Φ_{k}(x, y) + 1]φ_{k}(y)\}ds(y).

For v_{k}, from the boundedness of K_{0} : L^{2}[0, π] → H_{e}^{1}[0, π] and (3.1), we have that

\begin{matrix}∥Bv_{k}∥_{∞} & = ∥K_{0}(ψ_{k} − ψ_{0})∥_{∞} \\ & ≤ c_{2}∥K_{0}(ψ_{k} − ψ_{0})∥_{H_{e}^{1}[0,π]} \\ & ≤ c_{3}∥ψ_{k} − ψ_{0}∥_{L^{2}[0,π]} \\ & → 0 \end{matrix} (k → 0)

for some constants c_{2} > 0 and c_{3} > 0, that is ∥v_{k}∥_{∞,Γ} → 0 as k → 0. For sufficiently large R > 0, by the Hölder inequality and (3.2), we can estimate

|v_{k}(x)| ≤ c_{4} ∥φ_{k} − φ_{0}∥_{L^{p}(Γ)} → 0 (k → 0)

for |x| = R and some constant c_{4} > 0 depending on R. Now, by the maximumminimum principle for harmonic functions from the two preceding estimates we derive that v_{k} → 0 (k → 0) uniformly on the disk {x ∈ ℝ^{2} : |x| ≤ R}.

Finally, for the function w_{k}, using (2.8) and the Hölder inequality, we can estimate

|w_{k}(x)| ≤ \frac{c_{5}}{ln |k|} ∥φ_{k}∥_{L^{p}(Γ)} → 0 (k → 0)

for |x| ≤ R and some constant c_{5} > 0 depending on R. This completes the proof

In the case where f_{0} = 1 we have that u_{0} = 1 in all of ℝ^{2}. On the other hand, the radiation condition implies that u_{k}(x) → 0 as |x| → ∞ for all k > 0. Therefore, in Theorem 3.4, we cannot have uniform convergence in ℝ^{2}.

From the proof of Theorem 3.4 it is obvious that we also have uniform convergence of the derivatives of arbitraty order on compact subsets of ℝ^{2} \ Γ.

On the Low Wave Number Behavior 305

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Received 11.12.1997