90 
THE HON. J. W. STEUTT ON THE THEOEY OE EESONANCE. 
*4 TT- 
~ = rate of total flow across the surface of the sphere whose radius is r 
= — 4wA'[cos 2w^{cos {kr+g)+kr sin (fcr+g)} 4-sin 2w^{sin {Jcr+g)—Jcr cos {Jcr 4- </)}]• 
If the compression in the neighbourhood of the mouth is neglected, this must be the 
same as 
Q ^^ 0 =QA cos 2 unt. 
Accordingly 
AQ — — 4w A' { cos {Jcr ~rg)- f- Jcr si n {Jcr 4- g ) } , 
0 = sin {Jcr 4- g) — Jcr (cos Jcr 4 -g). 
These equations express the connexion between the plane and spherical waves. 
From the second, tan ( Jcr-{-g)=Jcr , which shows that g is a small quantity of the order 
{Jcr) 2 . From the first 
A'= — 
so that 
, AQ 
T >r 4ir r 
cos 2mt— 
AQ k 
47 T 
sin 2i mt, 
the terms of higher order being omitted. 
Now within the space under consideration the air moves according to the same laws as 
electricity, and so 
Q f/4 
c dx = 0 
— 4 / *=o + 4 / « 
A[/ 
dx — 0 
= A cos 2? mt, 
cos 2vnt-\-(3 sin 2 mt. 
Therefore on substitution and equation of the coefficients of sin 2 vut, cos 2 irnt, we 
obtain 
P=- 
AQ /c 
4tt 
When the mouth is not much contracted c is of the order of the radius of the mouth, 
and when there is contraction it is smaller still. In all cases therefore the term 
is very small compared to and we may put 
AQ 
c 
B, (3: 
AQk 
47 r 5 
( 17 ) 
* Throughout Helmholtz’s c paper the mouth of the pipe is supposed to lie in an infinite plane, so that the 
diverging waves are hemispherical. The calculation of the value of c is thereby simplified. Except for this 
reason it seems better to consider the diverging waves completely spherical as a nearer approximation to the 
actual circumstances of organ-pipes, although the sphere could never be quite complete. 
