CONTEMPORARY ADVANCES IN PHYSICS 671 



be analyzed into a multitude of wave-trains traveling to and fro with 

 the speed it. 



The symbol V^ (to be read del or nahla- squared) stands for the 

 Laplacian operator which in rectangular coordinates is d^ldx^, or 

 (d-Jdx" + d'/dy^), or (d'^/dx^ + d^/dy^ + d'^/dz~), according as we are 

 dealing with one, two or three dimensions. In other coordinates than 

 rectangular, it naturally assumes other forms. Now in these problems 

 of two and three dimensions, the choice of coordinate-system and the 

 imposition of boundary-conditions are two decisions which cannot 

 be separated from one another. Were we to decree that the membrane 

 should be square or rectangular with its edges clamped, the suitable 

 coordinate-system would be the rectangular. The problem would then 

 be extremely simple (the reader can easily solve it for himself by 

 using the method adopted for the stretched string, and will arrive 

 at very similar results) but not so instructive to us as the problem of 

 the circular membrane with clamped edge. For this we must adopt 

 polar coordinates (with the origin at the centre of the membrane, 

 naturally). In these, the Laplacian operator assumes the form: 



dr^^ rdr^ r'dd^ ^^"^ 



We restate the fundamental differential equation (120) in this fashion; 

 we essay a tentative solution in the form of a product of a function 

 f(r) of r exclusively, a function F(9) of 6 exclusively, and a function g(/) 

 of / exclusively; and we discover as before that each of these functions 

 is subjected to a differential equation of its own. The procedure is 

 like that already used in the case of the stretched string clamped at 

 its ends. First we have 



fdf~rJdr^r^Fdd~ u'~gdt^ ' ^ ^ 



for, since the first member of this triplet does not depend on /, the 

 second not on r nor on 6, both must be independent of all three 

 variables and equal to a constant which, as before, I denote by — ni". 

 The differential equation for the factor dependent on / has the solution : 



g{t) =^ A cos niut -{- B sin muf. (124) 



Our experience with the stretched string suggests that m will be 

 restricted to certain Eigemverte, derived from the boundary-conditions; 

 and this is true; but before arriving at these, we must attend to the 

 differential equation governing the functions / and F. This assumes 

 the form: 



