Vibrations of Hi fie Barrels. 



333 



and the relation between p and s is 



*( 2 _ 5 i^44-4) a*)- 



2 \ j? V po z ' 



From (13) (14), using the above values for v m and /, 



po = 2-32 x 10 t3 f.s.s. £ = 0-00173 sec. 



The three cases are illustrated in diagrams 5, 6, 7, in which the 

 various curves show the pressure, velocity, and time elapsed since the 

 beginning of the motion during the passage of the shot through the 

 barrel. 



Diagrams 8, 9, 10 show the pressure in terms of time, and it is 

 these curves which have to be represented by a harmonic series. 



In order to avoid having a constant term at the beginning of the 

 series, the fundamental t is taken equal to '2%. 



Then by the ordinary rules for finding the coefficient of a Fourier 

 series, the succession of " battlements " which form the pressure curve 

 in case 1 (uniform acceleration), we find 



P = Po z { sin ' 2 ~ j + \ sm 3 j) + \ sin 5 27r J + &c. j> (16). 



In case 2, where the pressure curve is a succession of half-lengths of 

 a simple harmonic curve, the general coefficient of the nth term is 



2 ±n 



and the series is 



7T — 1 



*=^{jdn2^4 8 i tt2 (2*|) + & c.} (17). 



The series for case (3), where p = p Q ( 1 - 4r\ is 



01/ / 



= po ? | sin2 7T i + i sin 2 (2tt + 1 sin 3 ^ ^ ) + &c. j (18). 



The coefficients in series 17 and 18 soon become sensibly equal in the 

 corresponding higher terms of each. 



In the cases just considered, except the first, it is assumed that the 

 pressure at the muzzle is zero, which of course is not true, but the 

 existence of a terminal pressure can be readily represented by adding 

 a series of the form of (16) of suitable magnitude. The effect of this is 

 to increase the relative importance of the first and all the odd terms. 



We must now examine the forced vibrations which each term of the 

 series expressing the accelerating pressure would set up in the rifle, 



2 A 2 



