Surface reflection: On the convergence of a series solution to a modified Helmholtz integral equation and the validity of the Kirchhoff approximation
Surface reflection: On the convergence of a series solution to a modified Helmholtz integral equation and the validity of the Kirchhoff approximation
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表面反射：关于修正亥姆霍兹积分方程级数解的收敛性以及基尔霍夫近似的有效性
DOI：
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发表时间：
1986
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影响因子：
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通讯作者：
S. McDaniel
中科院分区：
文献类型：
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作者：
D. McCammon;S. McDaniel
在压力释放边界的散射问题中，Helmholtz积分方程可以表示为以未知表面速度为自变量的第二类Fredholm方程。Meecham [J. Ration.机械肛门5，323-333（1956）]以获得该方程的级数解，其中该级数的第一项实际上是解的无阴影基尔霍夫近似。米查姆试图通过确定何时可以忽略级数的其余项来制定基尔霍夫近似的有效区域，然而，他用于选择“小”的论点受到了质疑。正弦边界的积分方程的精确解是用来检查系列，逐项，收敛性;它发现，（1）交替性质的系列不会带来收敛时，系列绝对发散，和（2）没有任何传播的边订单…
In the problem of scattering from a pressure release boundary, the Helmholtz integral equation may be expressed as a Fredholm equation of the second kind with the unknown surface velocity as the independent variable. The method of successive approximations was employed by Meecham [J. Ration. Mech. Anal. 5, 323–333 (1956)] to obtain a series solution to this equation, where the first term of this series is, in fact, the unshadowed Kirchhoff approximation to the solution. Meecham attempted to formulate the region of validity of the Kirchhoff approximation by determining when the remaining terms of the series could be neglected, however, his arguments used to select ‘‘smallness’’ have been questioned. An exact solution to the integral equation for a sinusoidal boundary is employed to examine the series, term by term, for convergence; and it is found that (1) the alternating nature of the series does not bring convergence when the series diverges absolutely, and (2) the absence of any propagating side orders ...