. The Bell System technical journal . nedwith the same assumptions as above. If we have one conductingtube of area ^i, joined to a second of area S2, we can write ^5, = ^,82 or Fi = F2, (13) where ^1 is the linear velocity in the first tube and ^2 in the secondtube. We have also that the pressures in the adjoining tubes areequal. Hence p2 = pi and F2 = Fi. (14) This equation is of the same order of approximation as the second ap-proximation given above for a junction, since we measure the lengthfrom one change of area to the next change. I 264 BELL SYSTEM TECHNICAL JOURNAL Equation 14 has been
. The Bell System technical journal . nedwith the same assumptions as above. If we have one conductingtube of area ^i, joined to a second of area S2, we can write ^5, = ^,82 or Fi = F2, (13) where ^1 is the linear velocity in the first tube and ^2 in the secondtube. We have also that the pressures in the adjoining tubes areequal. Hence p2 = pi and F2 = Fi. (14) This equation is of the same order of approximation as the second ap-proximation given above for a junction, since we measure the lengthfrom one change of area to the next change. I 264 BELL SYSTEM TECHNICAL JOURNAL Equation 14 has been found to hold well as long as the change inarea is small while equation 12 holds well as long as the length of ajunction is less than half of a wave-length. III. Recurrent Filters With the aid of equations (10), (12), and (14), we can obtain thepropagation characteristics of any structure employing straight tubes,sidebranches, and changes in area of conducting tubes. Among the simplest of these are recurrent filters. Fig. 2 shows an. Fig. 2—A typical acoustic filter example of this type of structure, a main conducting tube, with equallyspaced sidebranches. In order to make the structure symmetrical,we let the distance L between one end and the first sidebranch equalone half the distance between two sidebranches. We can then writewith regard to the first tube Vi — V\ cosh aiL — i^Si sinh axL,p2 = pi cosh a\L — Vi -^sinh aiL, (15) where ai and Z^j refer to the conducting tube. For the junction, wehave by (12) Pz — p2- (16) REGULAR COMBINATION OF ACOUSTIC ELEMENTS 265 Combining with (15), we have Fs = Vi ( cosh aiL + ^^V^sinh aiL ZsSi - pxS, p3 = pi cosh aiL sinh aiL S^ cosh aiL Li ZgSi —^;—^sinh a\L. (17) The pressures and volume velocities pi and Vi at one half the distancebetween the first and second sidebranches are again (18) Vi = Vz cosh aiL — ^^-^sinh a^L, z . ^ pi = p3 cosh aiL — Fs-^sinh ajL. Combining with (17), we obtain / Z S Vi— Vi[ cosh 2«iL + y
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