Bauhaus-Universität Weimar

Titel:
On sound and atmospheric vibrations, with the mathematical elements of music
Person:
Airy, George Biddell
PURL:
https://digitalesammlungen.uni-weimar.de/viewer/image/lit38525/95/
DIVERGENT WAVES OF TWO DIMENSIONS. 79 
The sum = —+ - 
æn 
dr2 
1 dB m 
r * dr 9 
and the equation of Article 37 becomes 
_1 cPR_dm 1 dR 
a2 * dt2 dr2 r ' dr 
The solution of this equation has been exhibited 
under the form of the following definite integral ; where 
6 is a new or fictitious angle, introduced solely for the 
purpose of being the subject of integration ; where the 
integration is to be taken from 9 = 0 to 9 = ir\ and 
where 8 denotes the definite integral so taken : 
B = Se . {at + r cos 6) 
+ Se (■at + r cos 6). log (r sin20). 
The solution may be verified without much trouble, 
remarking that differentiations with regard to t and r 
may be performed under the sign of integration with 
regard to 6. 
It is difficult to extract an intelligible result from this 
6 
expression. If for r cos 6 we put r — 2r sin2 - or 
2 
Q 
- r + 2r cos2-, we can expand in terms of u or v and 
A 
of integrable functions of 6 ; but the series proceeds by 
increasing powers of r, and is unfitted for great dis¬ 
tances.
        

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