Elements of natural philosophy (Volume 2-3) . roe orfour, &c, component curves, the resultant curve would ™™onentsbe determined by the same rule. § 58. Taking it, then, as a fact, that the disturbance _ Resultant action of every molecule produced by the coexistence of two of two equalor more causes will be the algebraic sum of the dis-waves on a particle; turbances which they would produce separately, let usconsider the nature of the displacement produced bythe superposition of the action of two waves of thesame length on the same molecule, the waves beingsupposed to come from any directions w
Elements of natural philosophy (Volume 2-3) . roe orfour, &c, component curves, the resultant curve would ™™onentsbe determined by the same rule. § 58. Taking it, then, as a fact, that the disturbance _ Resultant action of every molecule produced by the coexistence of two of two equalor more causes will be the algebraic sum of the dis-waves on a particle; turbances which they would produce separately, let usconsider the nature of the displacement produced bythe superposition of the action of two waves of thesame length on the same molecule, the waves beingsupposed to come from any directions whatever. We shall have for the displacement of the moleculeby the first wave, Eq. (19), d = °L . sin fan. Vt X+AA . /90^ placement bythe first wave; and by the second. a . ro V. t—x jffl d = - . sin 2 7T. _ +A\ ; x L X J (21) Same by the6econd; in which af and a,determine the intensi-ties of the sound inthe two waves at theunits distance ; and Aand A, the places ofthe maximum displace-ment at the expirationof the time t. Fig. Dlustration; 61 NATURAL PHILOSOPHY. Bum of Operations Taking the sum, and developing the circular function by the usual formula for the sine of the sum of twoarcs, we find, after reduction, VcnA + a-co&A) ft Tgjfl (aBln^+a-rin^) r gjgldisplacements; +« - x -ML ^ _V a, L \ J and making, Supposition; a . cos .A = ar COS ^4/ + # cos A, . . (a) a . sin A — a! sin J/ + a sin J_, ... (J) Notation; ^q a]}0ve becomes, after writing d for the total displace- ment, Totaldisplacement ; d = -• [cos sin (2 rr V-t-x\ + cos (2 it V-~)\ \ replacing the quantity within the brackets by its equal,viz.: the sine of the sum of the two arcs, we have Same; d = - - Sm f 2 * . 7,M + ... (22) Squaring Equations (a) and (b) and taking the sum, wefind, Transformations; a* = a2 + a* + 2 aa cos (^L - A) . . (23)and dividing Eqation (5), by Equation (a), we obtain deductions; tan J_ = .... (24). 0 . cos A + a . cos J. From Equation (22) we see that the l
Size: 2442px × 1024px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, booksubjectmechanics, booksubjectopticsandphoto
