220 DR. LOUIS YESSOT KING ON THE PROPAGATION OF SOUND IN THE FREE 
Taking the lower sign, corresponding to a wave propagated along the positive 
direction of the ai-axis, we have 
U = F (a) = C+ [2 a/(y— l)] a * 
(v-i) 
(25) 
C being a constant of integration which takes into account a possible progressive 
motion of the medium as a whole. If the velocity of the medium as a whole is U 
when undisturbed by sound-waves, we have u — U when a = 1 , which gives 
C = U — 2a/(y— l), so that the relation between the particle velocity and the density 
in the medium is given by 
2 a 
u = U- 
y 
-l 
i-G 
Po 
1(7-1)' 
(26) 
It is not difficult to verify that if f is any arbitrary function corresponding to 
u — f(t) when x = 0, 
u =/ 
a 
= U- 
2 a 
7 
-1 
[l_ a -Uv-D] 
(27) 
is a solution of Earnshaw’s equation which may be written in the form 
au/dt = a 2 a ~ (v+1) da./dx, .(28) 
From equation (25) we derive 
ci(ca/dx) = —a :j( ' y+1) du/dx, . (29) 
du/dt = -C(a' i ' (v+1) du/dx .(30) 
Differentiating (27) we have 
/'[...].(31) 
so that (28) becomes 
cm 
di 
i-i( y+ D«*«-'(|)( |f 
CU 
« — = 
ox 
. rJ l (7+1) (y + 1) Xa 
i(7-i) da- 
dx 
■a 
*‘ ,+ll +i(r + i)-“ v l f 
a ox_ 
giving 
era 
— aa - Ky+ ’ — = 
du 
dec 
l_4( y+ l) a «»->>*§“ 
a ox. 
(32) 
Since 3a/3 1 = d 2 y/dxdt = du/dx, we notice that (3l) and (32) are identical, from 
which it follows that (30) is satisfied, i.e., that (27) is a complete solution of 
Earnshaw’s equation. From equation (26) we may write the solution in the form 
u=f 
t — 
x/a 
{l+l(y-l)( W -U)M 
7 + 1 
7-1 
(33) 
