386 NOTES [Vol. VIII, No. 4
Eq. 3(c) must now be replaced by — H, , and, since the left side of Eq. 3(c) is known at
t = t0 , H,(t0) is known throughout the field. A finite difference approximation then
yields H(t0 -f At) throughout the field, so that the process may be repeated. Thus a
step-by-step solution may be obtained for all values of t.
A NOTE ON THE PRINCIPAL FREQUENCY OF A TRIANGULAR MEMBRANE* 1
By G. POLYA (Stanford University)
The principal frequency of a membrane of triangular shape is exactly known in
two simple cases: for the 45°, 45°, 90° and the 60°, 60°, 60° triangles.2 As will be shown
in this note, an exact solution of comparable simplicity exists also for the 30°, 60°, 90°
triangle-a result which, to the author's knowledge, has not been observed before.
The three lines the equations of which in rectangular coordinates x, y are
y = 0, z = a31/2/2, y = x3"1/2 (1)
delimit a triangle, one half of an equilateral triangle with side a. Define
u = sin 7r(y — 3~1/2x)/a ■ sin r{y + 3~1/2a;)/a • sin 2ir3'inx/a
(2)
• sin n(y — 3I/2a;)/3a • sin ir(y + 31/2x)/3a • sin 2iry/da
By elementary transformations we find that
-32m = cos 2ir(5y + 31/2x)/3a - cos 2x(4y + 3I/22x)/3a + cos 2ti {y + 31/23a;)/3a
(3)
— cos 2ir(by — 31/2z)/3a + cos 2ir(ty — 31/22x)/3a — cos 2ir(y — 3I/23x)/3a.
We see from (3) that u satisfies the equation
uxx uvv 9 '1127r2a \ = 0,
from (2) or (3) that u vanishes along the lines (1), and from (2) that u does not vanish
in the interior of the triangle between the lines (1). Therefore, u represents the principal
mode.
The well known solution for the two other triangles mentioned at the beginning
can be presented in a strictly analogous form. Any of these three triangles, repeated by
successive reflections, covers the whole plane without overlapping and there are no
further triangles of this kind. Therefore, there are no other triangles for which the
solution can be presented in a comparably simple form as a function of x, y without
singularity in the whole plane. The last remarks indicate also the heuristic reasoning
which led to the solution (2).
*Received April 17, 1950.
'Sponsored by the Office of Naval Research.
2Rayleigh, The theory of sound, Dover Publications, New York, 1945, §199.