This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1919 edition. Excerpt: ... the jet, say at x-0 and x = I. On substituting the exponential factor into the differential equation (416), we find that while the boundary conditions are satisfied if I = JnX, where n is any integer. Thus the different values of q are given by The vibration of lowest frequency is given by n = 1, and ...
Read More
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1919 edition. Excerpt: ... the jet, say at x-0 and x = I. On substituting the exponential factor into the differential equation (416), we find that while the boundary conditions are satisfied if I = JnX, where n is any integer. Thus the different values of q are given by The vibration of lowest frequency is given by n = 1, and q2 becomes negative when I first reaches the value This fixes the first dynamical point of bifurcation; as I increases more and more points of bifurcation occur, the complete set being given by (n = 1.2, 3...)...(420). Thus as I increases, one vibration after another loses its stability. The initial unstable motion of any vibration is one in which the matter of the jet tends to collect into nuclei or bunches at the iiodes of the wave. Or, alternatively, we may consider that a series of furrows tends to form in the jet, and that these get continually deeper. After passing the first point of bifurcation one furrow tends to form, namely a furrow between the jet and the main body; after passing the next point of bifurcation two furrows begin to form, and so on. 161. Clearly the formation of 1, 2, 3... furrows in succession in this problem is very closely analogous to the formation of 1, 2, 3... furrows in succession which occurs when the incompressible mass passes points of bifurcation corresponding to harmonics of orders 3, 4, 5 Although the formation of furrows in these two problems is closely analogous, it would be a mistake to suppose that the two solutions we have obtained merge gradually into one another as the compressibility of the primary mass gradually changes. We may notice that the breaking up of an incompressible mass takes place independently of its size, whereas the breaking up of the jet of matter formed from the atmosphere in...
Read Less
