៛•¢¶ the more condensed portions of the wave gain continually on the less condensed, the tendency being apparently towards the production of a discontinuity, somewhat analogous to a " bore " in water-waves. Before this stage can be reached, however, dissipative forces (so far ignored), such as viscosity and thermal conduction, come into play. In practical acoustics the 'results are also modified by the diminution of amplitude due to spherical divergence § 2. Wave- Propagation in General. We have next to consider the processes of wave-propagation in two or three dimensions. The simplest case is that of air-waves. When terms of the second order in the velocities are neglected, the dynamical equations are dtl dp dv dp dw dp ^at-~di- ""Jt-'Ty' ""Tt^'ai• and the " equation of continuity " (see Hydromechanics) is (I) dp, (du dv dw
(2) If we write p=po(i +s), p = pD+ks, these may be written aM ^ds dv jflj dw j3£ dt ^ dx' dl~ ^ dy' dt~ ^ dz
where c is given by § i (13), and dt^ dx^dy'^ dz)'• • the latter equation expressing that the condensation i is diminishing at a rate equal to the " divergence " of the vector (w, v, w) (see Vector Analysis). Eliminating u, v, w, we obtain (3) (4) d?='^ •
(5) where v* stands for Laplace's operator d'ldx'+d'ldy'+d-ldz^ This, the general equation of sound-waves, appears to be due to L. Euler (1759). In the particular case where the disturbance is symmetrical with respect to a centre O, it takes the simpler form ^-..^
(6) dp dr where r denotes distance from O It is easily deduced from (i) that in the case of a medium initially at rest the velocity (a, v, w) is now wholly radial. The soludon of (6) is fjct-r) . F(cl+r) r "* r
(7) This represents two spherical waves travelling outwards and inwards, respectively, with the velocity c, but there is now a progressive change of amplitude. Thus in the case of the diverging wave represented by the first term, the condensation in any particular part of the wave continually diminishes as i/r as the wave spreads. The potential energy per unit volume [§ I (5)) varies as i^ and so diminishes in inverse proportion to the square of the distance from O. It may be shown that as in the case of plane waves the total energy of a diverging (or a converging) wave is half potential and half kinetic.
The solution of the general equation (5), first given by S. D . Poisson in 18 19, expresses the value of 5 at any given point P at time /, in terms of the mean values of s and s at the instant / = over a spherical surface of radius ct described with P as centre, viz. (8) where the integrations extend over the surface of the aforesaid sphere, dw is the solid angle subtended at P by an element of its surface, and f{ct), F(cO respectively denote the original values of s and i at the position of the element. Hence, if the disturbance be originally confined to a Hmited region, the agitation at any point P external to this region will begin after a time r, /c and will cease after a time r^fc, where ri, r^ are the least and greatest distances of P from the boundary of the region in question. The region occupied by the disturbance at any instant I is therefore delimited by the envelope of a family of spheres of radius cl described with the points of the original boundary as centres. One remarkable point about waves diverging in three dimensions remains to be noticed. It easily appears from (3) that the 'alue of the integral fsdt at any point P, taken over the whole time of transit of a wave, is independent of the position of P, and therefore equal to zero, as is seen by taking P at an infinite distance from the original seat of disturbance. This shows that a diverging wave necessarily contains both condensed and rarefied portions. If initially we have zero velocity everywhere, but a uniform condensation So throughout a spherical space of radius a, it is found that we have ultimately a diverging wave in the form of a spherical shell of thickness 2a, and that the value of j within this shell varies from isoa/r at the anterior face to —^Soa/r at the interior face, r denoting the mean radius of the shell. The process of wave-propagation in two dimensions offers some peculiarities which are exemplified in cylindrical waves of sound, in waves on a uniform tense plane membrane, and in annular waves The on a horizontal sheet of water of ^relatively) small depth, equauon of motioa is in all these cases of the form § p=c'Vi% . . •
![{\displaystyle s_{p}={\frac {t}{4\pi }}\int \int \mathrm {F} (ct)d\omega +{\frac {d}{dt}}\left[{\frac {t}{4\pi }}+\int \int f(ct)d\omega \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99feec3cd5a3331a9f11cd8e83e868d9e93a0444)
(9)
where Vi' = 3V3x»-f a=y9y'-In
the case of the membrane s denotes
the displacement normal to its plane;
in the application to watei-waves it
represents the elevation of the surface
above the undisturbed level.
The sol-
ution of (9), even in the case of symmetry
about the origin, is .analytically
much less simple than that of (6).
It
appears that the wave due to a transient
local disturbance, even of the simplest
type, is now not sharply defined in the
rear, as it is in the front, but has an
indefinitely prolonged "tail."
This is illustrated
by the annexed figures which
represent graphically the time-variations
in the condensation i at a particular
point, as a wave originating in a local
condensation passes over this point. The
curve A represents (in a typical case) the
effect of a plane wave, B that of a
cylindrical wave, and C that of a
spherical wave.
The changes of type
from A to B and from B to C are accounted
for by the increasing degree of mobility
of the medium.
The equations governing the displacements «, v, w oi a. uniform
isotropic elastic solid medium are
where
dhv
dA,
A- —1— -I- — dx'^dy^ dz (10) (II) From these we derive by differentiation ^=aVA (12) where and g=Wl. g = 6=V», . |y = 6'v'f. «. -?. f- a-i t»
