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  • 11
    Electronic Resource
    Electronic Resource
    On the principles of limiting absorption and limit amplitude for a class of locally perturbed waveguides. Part 1: Time-independent theory (1988)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 10 (1988), S. 125-144 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: Let Ω be a local perturbation of the n-dimensional domain Ω0 = Ropf;n - 1 × (0, π). In a previous paper8 we have introduced the notion of an admissible standing wave. We shall prove that the principle of limiting absorption holds for the Dirichlet problem of the reduced wave equation in Ω at ω ≥ 0 if Ω does not allow admissible standing waves with frequency ω. From Reference 8, this condition is satisfied for every ω ≥ 0 if Ω ≠ Ω0, and v·x′ ≤ 0 on δΩ, where x′ = (x1,…, xn - 1, 0) and v is the normal unit vector on δΩ pointing into the complement of Ω. In contrast to this, the principle of limiting absorption is violated in the case of the unperturbed domain Ω0 at the frequencies ω = 1,2,… if n ≤ 3. The second part of our investigation, which will appear in a subsequent paper, is devoted to the principle of limit amplitude.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670100203
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  • 12
    Electronic Resource
    Electronic Resource
    Resonances in periodic media (1991)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 14 (1991), S. 227-263 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the propagation of linear acoustic waves (a) in an infinite string with a periodic material distribution, (b) in an infinite cylinder with a meterial distribution that is periodic in the longitudinal direction and does not depend on the transverse coordinates. We assume that the wave field is generated by a time-harmonic force distribution of frequency ω acting in a compact set. We show in both cases that resonances of order t1/2 occur for a discrete set of frequencies and that the solution is bounded as t→∞ for the remaining frequencies. In case (a) ω is a resonance frequency if and only if ω2 is a boundary point of one of the spectral bands of the corresponding spatial differential operator of Hill's type. A similar characterization of the resonance frequencies is given in case (b).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670140403
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  • 13
    Electronic Resource
    Electronic Resource
    Eine Integralgleichungsmethode für akustische Reflexionsprobleme an lokal gestörten Ebenen (1981)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 3 (1981), S. 128-144 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In this paper we consider the reflection of acoustic waves at an unbounded surface which coincides with a plane outside a sufficiently large sphere. We prove uniqueness and existence theorems for the corresponding boundary value problems for the reduced wave equation with Dirichlet and Neumann data by employing integral equation methods.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670030111
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  • 14
    Electronic Resource
    Electronic Resource
    Lösungen der poissongleichung und harmonische vektorfelder in unbeschränkten gebieten (1983)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 5 (1983), S. 233-255 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In the first part of this paper we consider generalised solutions of the Poisson equation Δ U = F in open subsets of Rn(n ≥ 3) with Dirichlet or Neumann boundary data. We prove existence and uniqueness theorems, not only for the corresponding interior and exterior problems, but also for domains with boundaries extending to infinity. In the second part we discuss generalised harmonic fields in open subsets of R3 with vanishing Dirichlet or Neumann boundary condition.
    Additional Material: 2 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670050117
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  • 15
    Electronic Resource
    Electronic Resource
    Ein resonanzphänomen in der theorie akustischer und elektromagnetischer wellenHerrn Professor Karl Nickel zum 60. Geburtstag am 9. 2. 1984 gewidmet (1984)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 6 (1984), S. 104-128 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We discuss the asymptotic behaviour of acoustic and electromagnetic waves, generated by given time-harmonic exterior forces with frequency ω, in the unbounded region between the parallel planes X3 = 0 and X3 = 1, and show that the principle of limiting amplitude is violated if ω = πn(n = 1, 2,…). For these values of the frequency, forces with compact support can be chosen such that the amplitudes of the waves increase with a logarithmic rate as t → ∞.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670060109
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  • 16
    Electronic Resource
    Electronic Resource
    The Helmholtz equation in disturbed half-spaces (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 312-323 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider boundary value problems for the Helmholtz equation in domains with boundaries coinciding with a plane outside a sufficiently large sphere. As opposed to the method of Gartmeier [6] (see also [12]) our analysis leads to an integral equation of the second kind with a compact operator. In the last chapter we study the set of far field patterns which are generated by entire incident fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090124
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  • 17
    Electronic Resource
    Electronic Resource
    On the scattering frequencies of time-dependent potentials (1986)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 8 (1986), S. 576-584 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: This paper discusses the scattering frequencies associated with the scalar wave equation and a time-periodic, real, potential. It is shown that the scattering frequencies form a discrete set in the complex plane and depend continuously on the potential. Existence of the scattering frequencies is proved for periodic potentials which are perturbations of a time independent, nonnegative, potential.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670080137
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  • 18
    Electronic Resource
    Electronic Resource
    Resonance phenomena for a class of partial differential equations of higher order in cylindrical waveguides (1989)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 11 (1989), S. 697-723 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We consider a domain Ω in ∝n of the form Ω = ∝l × Ω′ with bounded Ω′ ⊂ ∝n-l. In Ω we study the Dirichlet initial and boundary value problem for the equation ∂t2 u + [(- ∂12 -… - ∂l2)m + (- ∂l+12 -… - ∂n2)m]u = fe-iωt. We show that resonances can occur if 2m ≥ l. In particular, the amplitude of u may increase like tα (α rational, 0〈α〈1) or like in t as t∞∞. Furthermore, we prove that the limiting amplitude principle holds in the remaining cases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670110601
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  • 19
    Electronic Resource
    Electronic Resource
    On the instability of resonances in parallel-plane waveguides under local perturbations of the boundaryHerrn Professor Erhard Meister zum 60. Geburtstag am 12.2.1990 gewidmet (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 12 (1990), S. 1-34 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: We study the large-time asymptotics for solutions u(x, t) of the wave equation with Dirichlet boundary data, generated by a time-harmonic force distribution of frequency ω, in a class of domains with non-compact boundaries and show that the results obtained in [11] for a special class of local perturbations of Ω0 ≔ ∝2 × (0,1) can be extended to arbitrary smooth local perturbations Ω of Ω0. In particular, we prove that u is bounded as t → ∞ if Ω does not allow admissible standing waves of frequency ω in the sense of [8]. This implies in connection with [8]. Theorem 3.1 that the logarithmic resonances of the unperturbed domain Ω0 at the frequencies ω = πk (k = 1, 2,…) observed in [14] can be simultaneously removed by small perturbations of the boundary. As a main step of our analysis, the determination of admissible solutions of the boundary value problem ΔU + κ2U = - f in Ω, U = 0 on ∂Ω is reduced to a compact operator equation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670120102
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  • 20
    Electronic Resource
    Electronic Resource
    Zero resonances in local perturbations of parallel-plane waveguidesHerrn Professor Rolf Leis zum 60. Geburtstag am 22·7·1991 gewidment (1990)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 13 (1990), S. 111-136 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: In a joint paper with K. Morgenröther [9] we have studied the propagation of scalar waves in domains of the type Ω = Ωo-B̄ with Ωo:=∝2 × (0, 1), where B is a smooth bounded domain with B̄ ⊏ Ωo. In particular, we have shown that the solution of Neumann's initial and boundary value problem for the wave equation with time-independent right-hand side ƒ increases with a logarithmic rate as t → ∞ if the integral over ƒ does not vanish. The main purpose of the present paper is to extend this result to arbitrary smooth local perturbations of Ωo.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670130203
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