NOTE ON THE VIBRATION OF MEMBRANES
NOTE ON THE VIBRATION OF MEMBRANES
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关于膜振动的注意事项
DOI：
10.1093/qmath/os-11.1.63
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发表时间：
1940
影响因子：
0.7
通讯作者：
D. Christopherson
中科院分区：
文献类型：
--
作者：
D. Christopherson
—j+ gj JwH-A2^= 0(1) 出现在许多物理问题中，例如膜的振动问题，当边界条件（在受支撑的 e A^ h w= 0;(2) 和从下面加热的一层流体中的对流条件时，边界条件为 f^= 0,(2a)，其中 8v 是垂直于边界的单元。对于与以下相关的一个或其他边界条件，已获得 (1) 的精确解边界的几种形状，以及当边界是圆形时；但是瑞利在一篇有关热问题的论文中*指出，对于具有正六边形或等边三角形形式的边界，他不知道解决方案。六边形的（1）和（2a）的近似（“松弛”）解得出的结果表明，实际上确实存在精确的解析解；并且，因为一般情况下，模式 w 可以用下式表示：对于具有由直线定义的边的六边形，可以毫不费力地获得以下结果。
—j+ gj JwH-A2^= 0(1) occurs in a number of physical problems, such as that of the vibration of membranes, when the boundary condition (at a supported e A^ h w= 0;(2) and that of convection in a layer of fluid heated from below, the boundary condition then being f^= 0,(2a) where 8v is an element perpendicular to the boundary. An exact solution of (1) has been obtained for one or other boundary condition in relation to several shapes of boundary, and for both when the boundary is a circle; but Rayleigh, in a paper concerned with the thermal problem,* remarked that solutions were unknown to him for boundaries having the form of a regular hexagon or of an equilateral triangle.Results obtained in an approximate ('relaxation') solution of (1) and (2 a) for the hexagon suggested that an exact analytic solution does in fact exist; and, since on general grounds the mode w may be expected to be expressible in trigonometric functions, the following results were obtained without difficulty. For a hexagon having sides defined by the straight lines