THE HON. J. W. STRUTT ON THE THEORY OF RESONANCE. 
89 
The variable part of the pressure on the tube side of the piston 
The equation to the motion of the air in the mouth is therefore 
Q d # el'll q 
c dt dx dt ’ 
or, on integration, 
7E+H (m 
This is the condition to be satisfied when x=0. 
Substituting the values of b and we obtain 
cos 2 nut ^A^-f-B^ +(3 sin 2 t;P=0, 
which requires 
A^ + B = 0, (3 — 0. 
If there is a node at x= — l 
A cos Id —f B Jc sin 0 ; 
tan «=-£=-£. (15) 
This equation gives the fundamental note of the tube closed at x = — but it must be 
observed that l is not the length of the tube, because the origin a= 0 is not in the 
mouth. There is, however, nothing indeterminate in the equation, although the origin 
is to a certain extent arbitrary, for the values of c and I will change together so as to 
make the result for Jc approximately constant. This will appear more clearly when we 
come, in Part II., to calculate the actual value of c for different kinds of mouths. In 
tne formation of (14) the pressure of the air on the positive side at a distance from the 
origin small against a has been taken absolutely constant. Across such a loop surface 
no energy could be transmitted. In reality, of course, the pressure is variable on account 
of the spherical waves, and energy continually escapes from the tube and its vicinity. 
Although the pitch of the resonant note is not affected, it may be worth while to see 
what correction this involves. 
e must, as before, consider the space in which the transition from plane to sphe- 
rical waves is effected as small compared with A. The potential in free space may be 
taken 
4/=^ cos (fo'+rj-Zvnt), (16) 
expressing spherical waves diverging from the mouth of the pipe, which is the origin 
of r. The origin of x is still supposed to lie in the region of plane waves. 
