On the short-wave asymptotic theory of the wave equation (∇2 + k2)ø = 0

1957 ◽  
Vol 53 (1) ◽  
pp. 115-133 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Green’S Function ◽  
Green's Function ◽  
Short Wave ◽  
Regularity Conditions ◽  
Convex Curve ◽  
Normal Velocity ◽  
Sound Waves ◽  
Small Kernel ◽  
Image Position

ABSTRACTThe theory of time-periodic wave problems falls into two parts. On the one hand there is the rigorous formulation in terms of differential wave equations, on the other there are approximate theories like geometrical optics. It should be possible, in principle, to deduce the latter from the former by a logical process; but this has been done only for a few simple configurations, e.g. the circle. A possible approach to the solution of the general problem is suggested here, and is applied to a typical two-dimensional acoustical example. An arbitrary closed convex curve (satisfying certain regularity conditions) is emitting short sound waves towards infinity, the normal velocity V(s) exp (− iωt) is prescribed on the curve as a function of the arc-length s, and the potential is to be found, first on the curve and then at any point in the sound field. (Only the first part of the problem is treated in detail.) The potential ø(s) exp (− iωt) on the curve satisfies all the integral equationswhere G(s, s′) is any Green's function of the problem, and V(a) is prescribed. All the equations corresponding to different Green's functions have the same solution. An asymptotic and convergent short-wave solution can be found by iteration if G can be chosen explicitly so that the integral equation has a small kernel for high frequencies. At any point of the curve draw the local circle of curvature; then the explicit known solution for a source on this circle is (with slight modifications) a possible Green's function, and the equation formed with it has a small kernel and can be solved rigorously by iteration. If V(a) is independent of the frequency, the leading term in the resulting asymptotic expansion iswhere c is the velocity of sound and 2πk−1 is the (short) wavelength corresponding to the frequency ω/2π. If V(a) varies rapidly, as in diffraction theory, the iterative solution still gives a convergent asymptotic expansion, but the first approximation is then practically useless in the shadow region. Diffraction problems are not treated in the present paper.The present work appears to be the first practical and rigorous solution of a short-wave problem in optics or acoustics when a solution in closed form is not available. It is suggested that the technique (suitably combined with formal expansions) may be applicable to a wider class of radiation and diffraction problems.

scholarly journals Green's function of the clamped punctured disk

1977 ◽  
Vol 20 (2) ◽  
pp. 175-181 ◽  
Author(s):  
Mitsuru Nakai ◽  
Leo Sario
Keyword(s):  
Green’S Function ◽  
Circular Plate ◽  
Green's Function ◽  
American Mathematical Society ◽  
Point Load ◽  
Thin Elastic Plate ◽  
Constant Sign ◽  
Boundary Circle ◽  
Simply Supported ◽  
Image Position

If a thin elastic circular plate B: ∣z∣ < 1 is clamped (simply supported, respectively) along its edge ∣z∣ = 1, its deflection at z ∈ B under a point load at ζ ∈ B, measured positively in the direction of the gravitational pull, is the biharmonic Green's function β(z, ζ) of the clamped plate (γ(z, ζ) of the simply supported plate, respectively). We ask: how do β(z, ζ) and γ(z, ζ) compare with the corresponding deflections β0(z, ζ) and γ0(z, ζ) of the punctured circular plate B0: 0 < ∣ z ∣ < 1 that is “clamped” or “simply supported”, respectively, also at the origin? We shall show that γ(z, ζ) is not affected by the puncturing, that is, γ(·, ζ) = γ0(·, ζ), whereas β(·, ζ) is:on B0 × B0. Moreover, while β((·, ζ) is of constant sign, β0(·, ζ) is not. This gives a simple counterexmple to the conjecture of Hadamard [6] that the deflection of a clampled thin elastic plate be always of constant sign:The biharmonic Gree's function of a clampled concentric circular annulus is not of constant sign if the radius of the inner boundary circle is sufficiently small.Earlier counterexamples to Hadamard's conjecture were given by Duffin [2], Garabedian [4], Loewner [7], and Szegö [9]. Interest in the problem was recently revived by the invited address of Duffin [3] before the Annual Meeting of the American Mathematical Society in 1974. The drawback of the counterexample we will present is that, whereas the classical examples are all simply connected, ours is not. In the simplicity of the proof, however, there is no comparison.

Parabolic Representations of Scattering Green’s Functions

1999 ◽  
Author(s):  
Paul E. Barbone
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Green’S Function ◽  
Inhomogeneous Medium ◽  
Green's Function ◽  
Free Space ◽  
Green's Functions ◽  
Green’S Functions

Abstract We derive a one-way wave equation representation of the “free space” Green’s function for an inhomogeneous medium. Our representation results from an asymptotic expansion in inverse powers of the wavenumber. Our representation takes account of losses due to scattering in all directions, even though only one-way operators are used.

Estimates on the Green's function of second-order elliptic operators in ℝN

1998 ◽  
Vol 128 (5) ◽  
pp. 1033-1051
Author(s):  
Adrian T. Hill
Keyword(s):  
Green’S Function ◽  
Heat Kernel ◽  
Hypergeometric Function ◽  
Green's Function ◽  
Confluent Hypergeometric Function ◽  
Elliptic Operators ◽  
Polar Coordinates ◽  
Pointwise Estimates ◽  
Image Position ◽  
Second Order Elliptic Operators

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equationfor λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

THE PRESSURE PULSE PRODUCED BY A LARGE EXPLOSION IN THE ATMOSPHERE

1961 ◽  
Vol 39 (7) ◽  
pp. 993-1009 ◽  
Author(s):  
V. H. Weston
Keyword(s):  
Fourier Transform ◽  
Green’S Function ◽  
Green's Function ◽  
Pressure Pulse ◽  
Closed Surface ◽  
Excess Pressure ◽  
Normal Velocity ◽  
Harmonic Frequency ◽  
Large Explosion ◽  
Ring Source

The pressure pulse produced by a large explosion in the atmosphere is investigated. The explosion is represented in terms of the excess pressure and normal velocity on a closed surface, outside of which the hydrodynamical equations are linearized. The pulse is represented in terms of a Fourier transform of the associated harmonic frequency problem, for which a ring-source Green's function is obtained in terms of an expansion of the discrete modes. It is shown that the excess pressure may be represented in terms of an integral (containing the Green's function) over the surface surrounding the source. The gravity wave portion of the pressure pulse at the ground is computed for various ranges from the source, which is located at various altitudes, and for three models of the atmosphere. In calculating the head of the pulse a new asymptotic technique is introduced which gives very good results for intermediate and long ranges.

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