412 ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES [408
these occur appreciably, strict periodicity is lost. Further, if we abandon
the restriction to symmetry, a new term, r~*d?<f>ld6*, enters in (13) and the
solution involves a new factor cos (nd + e) in conjunction with the Bessel's function Jn in place of J"0.
The particular form of the differential equation exhibited in (13) is appropriate only when the section of the stream is circular. In general we have
g + g + p-n + .O, ..................... (16)
dx* dy~ ^ ' T
the same equation as governs the vibrations of a stretched membrane (Theory of Sound, § 194). For example, in the case of a square section of side b, we have
<i = cos . cos .e* <*»«+**>, ..................... (17)
T b b ^
vanishing when x—±^b and when y = ± %b. This represents the principal vibration, corresponding to the gravest tone of a membrane. The differential equation is satisfied provided
&2-/32 = 27rV&s, ........................... (18)
the equation which replaces (15). It is shown in Theory of Sound that provided the deviation from the circular form is not great the question is mainly one of the area of the section. Thus the difference between (15) and (18) is but moderate when we suppose irR2 equal to 62.
It may be worth remarking that when V the wave-velocity exceeds a, the group-velocity U falls short of a. Thus in (15), (18)
T/. ka TT d(0V) dk {3a
]/ «*- _ / / — _______ _^ _ i_ — rt _ — . _ •
0' dp ~ d/3~ k '
so that UV=a? .................................. (19)
Returning to the jet of circular section, we may establish the connexion between the variable pressure along the axis and the amount of the swellings observed to take place between the disks. From (9)
z=Wz + HJ3-1 sin fa . Ja and ] = H*/(W*/a*-l). sin J3z. JQ' (2-405) ............. (20)
The latter equation gives the radial velocity at the boundary. If denote the variable part of the radius of the jet,
[1 fd<f>\ , Hcosfa /fWz _\ T//nAn^
J w (£1 dz=~ -jw- v b ~ l) • Jo (2'40o)- • • -onger oscillates, but settles rapidly down into complete uniformity. This is of course the usual case of gas escaping from small pressures.at happens when the motion is strictly in one dimension. It is true that then a wave can be stationary in space only when the stream moves with' the velocity of sound; but here the motion is not limited to one dimension, as is shown by the swellings between the disks. Indeed the propagation of any wave at all is inconsistent with uniformity of pressure within the jet.l as of the particles of fluid with each other, is more directly shewn by an experiment on the continuance of a column of mercury, in the tube of a barometer, at a height considerably greater than that at which it usually stands, on account oi the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.