And since the disturbance is small, P and P. are very nearly equal, therefore dy = 1 very nearly; (2) Let the fluid be of the kind denominated non-elastic, or liquid. The increment of the density of a liquid under a moderate pressure is found to be proportional to the pressure. Let, therefore, p + pressure p. Then K (p +2) d2y. da = mass of PQ = xp.d.w; and since μ the disturbance is very small, p is very small compared with up, therefore upd.y = upp very nearly, In liquids, the heat developed by compression is nearly insensible. cp up (3) Let AP be a rod vibrating longitudinally, the quantity by which the rod is shortened, or lengthened, when it is compressed longitudinally, or extended, by a pressure кр, the original length of the rod being c. Then dry.da length of PQ = = = - d&p, .. diy = μdzy. (1 p up. δα, In consequence of the variation of the thickness of a rod, when it is compressed longitudinally, or extended, the variation of its length (Pouillet Elemens de Physique, 463.) is 1,5 of what it would be if its thickness were invariable. It is the variation of the length of the rod, on the latter supposition, that is to be used in deducing the value of μ. μ. a2 64. The equation of the motion of the disturbance is of the form dya'day. The integral of this equation may be made to depend on that of dead, which we know to be x = (x + at), in the following manner: 65. Let the initial disturbance extend through a very small space 2a, that is from a to a; then at the beginning of the motion, or when t = 0, ad y + dy = px, ad ̧y dry = a; and the fluid at any point distant from A by a quantity greater than a, will be at rest, therefore d ̧y = 1, d¿y = 0; and therefore φω = a, x = a, as long as a does not lie between - a and a, Therefore (x + at) = a, except when a + at lies between and a; and (x − at) and a; hence dy = 0, except when one of the quantities a + at, x - at lies between a and a; and when at is less than or greater than a, x+at is greater than a; therefore if P, R be any two points in AP, the fluid at P will remain at rest till = a, except when a at lies between - α α α, AP - at = a, or till the end of the time - (AP – a), it will then a begin to move, and will return to a state of permanent rest when 1 AP – at= — a, or at the end of the time - (AP + a). In like α manner, the fluid at R will begin to move at the end of the (AR - a), and will return to a state of rest at the end of the time - (AR+a). Hence the fluid at R will begin = a to move a PR later than the fluid at P; therefore the velocity with which the disturbance is propagated 66. Sound is a repetition of such disturbances; and the velocity of sound in any medium, is the same as the velocity with which a small disturbance is propagated through it. When the disturbances are repeated at small and equal intervals the sound becomes a musical note. Hence, the velocity of sound in an elastic fluid, at the temperature T, is In dry air, √ {μ(1 + Ec) (1 + ET)}. √(u) = 916,45 feet, E = 0,00375, EC = 0,41365; therefore if v be the velocity of sound expressed in feet, v = 1090 √{1 + (0,00375) T}. The density of aqueous vapour = of the density of air, under the same pressure; therefore if Y be the tension of the vapour contained in the air at any time, II the atmospheric pressure, μ the ratio of the pressure to the density for dry air; the density of the moist air under the pressure II, will be equal to the density of dry air under the pressure II − Y + § Y ; and the ratio of its pressure to its density will be .. V = 1090 ( 1 + (1. (1 nearly; 8 П 3 Y √ {1+(0,00375) T}. 16 II When water saturated with air, at 8o, is pressed by a column of water 33,83 feet high, its density is increased by 0,000049589 of its original density, the force of gravity being 32,18 feet; ..м= (33,83) (32,18) ÷ (0,000049589), √(u) = 4686. The observed velocity of sound in water at 8°.1 is 4708 feet. SECTION VI. ON RESISTANCES. ART. 67. THE resistance of a fluid on a solid moving in it, is the resultant of the excess of the pressure of the fluid on the solid in motion, above the pressure of the fluid on the solid at rest. Let APB (fig. 32.) be a solid, moving in a fluid with the velocity V in the direction BA. Now if we communicate to the fluid and the solid a velocity V in the direction AB, the pressure of the fluid on APB will not be altered; and the solid will be at rest in a fluid moving in the direction AB with a velocity V. Hence the force with which a fluid in motion impels a solid immersed in it, is equal to the resistance of a stagnant fluid on a solid in motion, the velocity of the fluid, in one case, being equal to the velocity of the solid in the other. So also, when both the solid and fluid are in motion, the resistance on the solid, is equal to the force with which the solid at rest would be impelled by a stream moving with the relative velocity of the fluid and solid. Let an enveloping cylinder parallel to AB touch the solid in the curve PQR. The pressure on the surface RAPQ, will upon the whole be greater, and the pressure upon PBQR, less, than when the solid and fluid are relatively at rest. the following Articles we shall consider that part only of the resistance, which arises from the increased pressure on RAPQ. In It must be observed that the theory of resistances is very imperfect, and that it is useless to expect any close agreement between the results deduced from it, and those obtained by experiment. 68. To find the force with which a stream impels a plane, the plane being perpendicular to the direction of the stream. Let P (fig. 33.) be a point in the plane; EP, the direction of the stream, perpendicular to the plane; p' the pressure at P; p the pressure of the fluid at P before the plane was immersed, or, the pressure of the fluid at the point P in a plane moving with the same velocity, and in the same direction as the fluid; the density of the fluid; K the area of the plane. ρ Then p' Ρ will be the resistance on an unit of the plane 1 v2 = √, S − = p ; at P. Now (Art. 53.) 1 v2 = fsS ρ and after the plane is immersed, the velocity at P = 0, and the resistance on the plane = (p′ − p) K = 1⁄2 pv2K. pv2K is the weight of a column of fluid having the given plane for its base, and whose altitude is the space due to the velocity of the fluid. If the plane be made to move in a direction perpendicular to EP, it is manifest that the force with which the stream impels the plane, will not be altered. Hence the force with which a given fluid impels a given plane, depends only on that part of the relative velocity of the fluid and plane, which is perpendicular to the plane. Also, since the resistance, or impelling force of the fluid arises from the pressure of the fluid on the plane, it must act in a direction perpendicular to the plane. 69. A stream impinges obliquely on a plane; to find the force with which the stream impels the plane. Let P (fig. 33.) be a point in the plane; AP the direction of the stream; EP perpendicular to the plane; v the velocity of the stream; R the resistance, or the force with which the stream impels the plane. |