High frequency asymptotic solutions of the reduced wave equation on infinite regions with non-convex boundaries

• Main Author: Bloom Clifford O.
• Format: Article
• Language: English
• Published: Hindawi Limited 1996-01-01
• Series: Mathematical Problems in Engineering
• Subjects: High frequency radiation; scattering; global approximate solution; uniform asymptotic approximation; caustics; geometrical optics; inhomogeneous medium; anisotropic medium; reduced wave equation
• Online Access: http://www.hindawi.net/access/get.aspx?journal=mpe&volume=2&pii=S1024123X96000385

<p>The asymptotic behavior as <math alttext="$lambda o infty $"> <mrow> <mi>&lambda;</mi> <mo>&rarr;</mo> <mi>&infin;</mi> </mrow> </math> of the function <math alttext="$Uleft( {x,lambda } ight)$"> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> that satisfies the reduced wave equation <math alttext="$L_lambda left[ U ight] = abla cdot left( {Eleft( x ight) abla U} ight) + lambda ^2 N^2 left( x ight)U = 0$"> <mrow> <msub> <mi>L</mi> <mi>&lambda;</mi> </msub> <mrow> <mo>[</mo> <mi>U</mi> <mo>]</mo> </mrow> <mo>=</mo> <mo>&nabla;</mo> <mo>&sdot;</mo> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>&nabla;</mo> <mi>U</mi> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>&lambda;</mi> <mn>2</mn> </msup> <msup> <mi>N</mi> <mn>2</mn> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mi>U</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> on an infinite 3-dimensional region, a Dirichlet condition on <math alttext="$partial V$"> <mrow> <mo>&part;</mo> <mi>V</mi> </mrow> </math> , and an outgoing radiation condition is investigated. A function <math alttext="$U_N left( {x,lambda } ight)$"> <mrow> <msub> <mi>U</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> is constructed that is a global approximate solution as <math alttext="$lambda o infty $"> <mrow> <mi>&lambda;</mi> <mo>&rarr;</mo> <mi>&infin;</mi> </mrow> </math> of the problem satisfied by <math alttext="$Uleft( {x,lambda } ight)$ "> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> . An estimate for <math alttext="$W_N left( {x,lambda } ight) = Uleft( {x,lambda } ight) - U_N left( {x,lambda } ight)$ "> <mrow> <msub> <mi>W</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> <mo>&minus;</mo> <msub> <mi>U</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> on <math alttext="$V$ "> <mi>V</mi> </math> is obtained, which implies that <math alttext="$U_N left( {x,lambda } ight)$"> <mrow> <msub> <mi>U</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> is a uniform asymptotic approximation of <math alttext="$Uleft( {x,lambda } ight)$ "> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> as <math alttext="$lambda o infty $ "> <mrow> <mi>&lambda;</mi> <mo>&rarr;</mo> <mi>&infin;</mi> </mrow> </math>, with an error that tends to zero as rapidly as <math alttext="$lambda ^{ - N} left( {N = 1,2,3,...} ight)$ "> <mrow> <msup> <mi>&lambda;</mi> <mrow> <mo>&minus;</mo> <mi>N</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi>N</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>...</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math>. This is done by applying a priori estimates of the function <math alttext="$W_N left( {x,lambda } ight)$ "> <mrow> <msub> <mi>W</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> in terms of its boundary values, and the <math alttext="${ext{L}}_{ext{2}} $"> <semantics> <mrow> <msub> <mtext>L</mtext> <mtext>2</mtext> </msub> </mrow> <annotation encoding='MathType-MTEF'> </annotation> </semantics></math> norm of <math alttext="$rL_lambda left[ {W_N left( {x,lambda } ight)} ight]$ "> <mrow> <mi>r</mi> <msub> <mi>L</mi> <mi>&lambda;</mi> </msub> <mrow> <mo>[</mo> <mrow> <msub> <mi>W</mi> <mi>N</mi> </msub> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> </mrow> </math> on <math alttext="$V$ "> <mi>V</mi> </math>. It is assumed that <math alttext="$Eleft( x ight)$ "> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math>, <math alttext="$Nleft( x ight)$ "> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math>, <math alttext="$partial V$ "> <mrow> <mo>&part;</mo> <mi>V</mi> </mrow> </math> and the boundary data are smooth, that <math alttext="$Eleft( x ight) - I$ "> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>&minus;</mo> <mi>I</mi> </mrow> </math> and <math alttext="$Nleft( x ight) - 1$ "> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>&minus;</mo> <mn>1</mn> </mrow> </math> tend to zero algebraically fast as <math alttext="$r o infty $ "> <mrow> <mi>r</mi> <mo>&rarr;</mo> <mi>&infin;</mi> </mrow> </math>, and finally that <math alttext="$Eleft( x ight)$ "> <mrow> <mi>E</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math> and <math alttext="$Nleft( x ight)$ "> <mrow> <mi>N</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </math> are slowly varying; <math alttext="$partial V$ "> <mrow> <mo>&part;</mo> <mi>V</mi> </mrow> </math> may be finite or infinite. </p><p>The solution <math alttext="$Uleft( {x,lambda } ight)$ "> <mrow> <mi>U</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>&lambda;</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> can be interpreted as a scalar potential of a high frequency acoustic or electromagnetic field radiating from the boundary of an impenetrable object of general shape. The energy of the field propagates through an inhomogeneous, anisotropic medium; the rays along which it propagates may form caustics. The approximate solution (potential) derived in this paper is defined on and in a neighborhood of any such caustic, and can be used to connect local &#8220;geometrical optics&#8221; type approximate solutions that hold on caustic free subsets of <math alttext="$V$ "> <mi>V</mi> </math>.</p><p>The result of this paper generalizes previous work of Bloom and Kazarinoff [C. O. BLOOM and N. D. KAZARINOFF, <emph>Short Wave Radiation Problems in Inhomogeneous Media: Asymptotic Solutions</emph>, SPRINGER VERLAG, NEW YORK, NY, 1976].</p>