The principal feature of this communication is the discovery of an integral of a certain class of differential equations. This class includes, as a particular case, the differential equation of motion when a disturbance is transmitted through a uniform elastic medium confined in a horizontal tube. If the equation
dy
/
dt
= F(
dy
/
dx
) be differentiated with regard to
t
, it produces the equation
d
2
y
/
dt
2
= {F' (
dy
/
dx
)}
2
·
d
2
y
/
dx
2
; which, by means of the general function F', can be made to coincide with any proposed differential equation in which the ratio between
d
2
y
/
dt
2
and
d
2
y
/
dx
2
is dependent on
dy
/
dx
only. The integral obtained in this manner is that which arises from the elimination of (
a
) between the two following equations,—
y
=
ax
+ F (
a
) ·
t
+
φ
(
a
), 0 =
x
+ F' (
a
) ·
t
+
φ
' (
a
). This integral, though not found by the direct integration of the differential equation, and though evidently not the general symbolical integral of it, is proved to be the general integral for wave-motion, from its affording the means of satisfying all the necessary equations of initial disturbance and wave-motion.
Unidirectional wave motion in a nonlocally and nonlinearly elastic medium: the KdV, BBM, and CH equations
