C. K. Wead—Intensity of Sound. 183 
t a 
(15) u=ato =; 
(16) uae Hat bet sin {4/p? —}a+e’F et} : 
if we wish to make ¢ more than a few periods replace z’ in the 
exponent by $(z’+2’’), where 2” is the amplitude at the time ¢. 
Obviously 
(17) 2’=a'e ttt He +2) ht 
_ AsIdo not find any discussion of such an equation as (16) 
It 1s well to show what is involved in it. Differentiating twice 
and dropping the accent from 2’ we get 
: (18) @+aa+bzu+p?u=o. 
Represent the mean value of w during a quarter period by (@); 
of u® by (u®); and the mean value of ()i, that is, the mean 
value of the product of the velocity at any time and the mean 
velocity, by ((a)u); and neglect the second term under the 
Big 8 2? 
(19) (=i Pe, (20) ((%)x) st (™)=+ i Xp 5: 
Whence | 
(21) atau (an +p*u=0. 
The mean value of the third term = 
AD og 
(22) aT el 3 
velocity, and as (a?) the mean square of the Marsa oe 
Course we get the usual equation of harmonic motion in @ 
resisting medium. . 
* Tn strictness equations (19) and (20) are true only when a=d=0; but the 
error here is very small. | 
