HOEIZONTAL LAYER OF FLUID
443
the plane into similar parts or cells, such as squares, equilateral triangles, or regular hexagons. But in the present problem it is perhaps more appropriate to consider divisions of the plane with respect to which w is symmetrical, so that dwjdn is zero on the straight lines forming the divisions of the cells. The more natural analogy is then with the two-dimensional vibration of air, where w represents velocity-potential and the divisions may be regarded as fixed walls.
The simplest case is, of course, that in which the cells are squares. If the sides of the squares be STT, we may take with axes parallel to the sides and origin at centre
w = cosx + cosy, ...........................(55)
being thus composed by superposition of two parts for each of which /c2= 1. This makes dw/dfc = — smai, vanishing when so = ± IT. Similarly, dw/dy vanishes when y = + IT, so that the sides of the square behave as fixed walls. To find the places where w changes sign, we write it in the form
•y
_
w = 2 cos
cos
.(56)
2 'ww 2
giving x + y = ±7r, x — y=±7r, lines which constitute the inscribed square (fig. 1). Within this square w has one sign (say +) and in the four right-angled triangles left over the — sign. When the whole plane is considered, there is no want of symmetry between the + and the — regions.
The principle is the same when the elementary cells are equilateral triangles or hexagons; but I am not aware that an analytical solution has been obtained for these cases. An experimental determination of 7c2 might be made by observing the time of vibration under gravity of water contained in a trough with vertical sides and of corresponding section, which depends upon the same differential equation and boundary conditions*. The particular vibration in question is not the slowest possible, but that where there is a simultaneous rise at the centre and fall at the walls all round, with but one curve of zero elevation, between.
In the case of the hexagon, we may regarjl it as deviating comparatively little from the circular form and employ the approximate methods then applicable. By an argument analogous to that formerly developedf for the boundary condition w = 0, we may convince ourselves that the value of Jc* for the hexagon cannot differ much from that appropriate to a circle of the same area. Thus if a be the radius of this circle, k is given by J0' (ka) = 0,
* See Phil. Mag. Vol. i. p. 257 (1876); Scientific Papers, Vol. i. pp. 265, 271. t Theory of Sound, § 209 ; compare also § 317. See Appendix.
Fig. 1.he nodal system (w = 0), where the .membrane may be regarded as held fast; and we may suppose the nodal system to divideem to one or two particular problems. The special limitation which characterizes them is the neglect of variations of density,