1916] ON THE DISCHARGE OF GASES UNDER HIGH PRESSURES 411
the pressure be constant, equal to that of the surrounding quiescent air, and this requires that the variable part of w vanish, since the pressure varies with the total velocity. Accordingly
/0{V(F2-a2)./3-R/aHO, ..................... (10)
which can be satisfied only when W > a, that is when the mean velocity of the jet exceeds that of sound. The wave-length (X) of the periodic features along the jet is given by X = 27T//3.
The most important solution corresponds to the first root of (10), viz. 2'405. In this case
2-406
The problem for the two-dimensional jet is even simpler. If b be the width of the jet, the principal wave-length is given by
The above is substantially the investigation of Prandtl, who finds a sufficient agreement between (11) and Emden's measurements*.
It may be observed that the problem can equally well be treated as one of the small vibrations of a stationary column of gas as developed in Theory of Sound, §§ 268, 340 (1878). If the velocity-potential, symmetrical about the axis of z, be also proportional to e* (kat+<3z) , where k is such that the wavelength of plane waves of the same period is 27r//c, the equation is § 340 (3)
^ + if£ + (J;=_m = 0, ..................... (13)
cfcr2 7- dr ^ ^ ' ^ > ^ '
and if k > ft
<£ = eW+^/0{V(/c2-/32).r} ..................... (14)
The condition of constant pressure when r — R gives as before for the principal vibration
*J(l<?-p}.R = 2-405 ......................... (15)
The velocity of propagation of the waves is ka/j3. If we equate this to W and suppose that a velocity W is superposed upon the vibrations, the motion becomes steady. When we substitute in (15) the value of k, viz, W@/a, we recover (11). It should perhaps be noticed that it is only after the vibrations have been made stationary that the effect of the surrounding air can be properly represented by the condition of uniformity of pressure. To assume it generally would be tantamount to neglecting the inertia of the outside air.
The above calculation of X takes account only of the principal vibration. Other vibrations are possible corresponding to higher roots of (10), and if
* When PF<a, j3 must be imaginary. The jet no longer 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.