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View Preview. Learn more Check out. Related Information. Close Figure Viewer. Ill, in dealing with wave propagation in acoustic media. P is not a constant; it is a function of the impedance, and hence of the frequency. Acoustic Filters Following the development of electrical filters, G. Stew- art devised in Phys. The acoustic filters are merely iterated resonators, and presumably for every electrical filter known there is an acoustic analogue, though all are not equally practicable.
In Figs. In each case the filter is supposed to have an infinite number of successive equal elements; the oscillations are exactly out of phase from one element to the next, in the mode considered. The simple formulae for limiting frequency, assuming inertia and stiffness entirely concentrated at a point, and taking no account of resistance, are shown adjacent to the diagrams.
John- son, "Circuits for Telephonic Communication," Chaps. XI and XVI. Zobel, ibid. Note that in Stewart's formulae sub- scripts i relate to properties of the "transmission conduit" while subscripts 2 relate to branch elements which are in shunt with the transmission conduit. The high-pass filter is shown schematically in Fig. The more accurate formula preferred by Stewart is This is directly obtainable from the formula of Fig.
The damping characteristic of one section of a good electrical filter can be made small, e.
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Stewart has very properly made the main trans- mission conduit a relatively wide tube; but the side conductiv- ities 2 are likely to introduce complications particularly as regards dissipation. The side conductivities in the low-pass filters are due to a number of small holes; and we should expect considerable dissipation therein. In the high-pass filter the side conductivities will cause additional losses due to radiation.
Al- lowance for these complications and for the fact that the inertia and stiffness elements are much more definitely "lumped" in the electrical case may explain the relative shortcomings of the acoustic filter. A disadvantage due to multiple resonance 2 in the resonators which accounts for the failure of the low-pass filter to suppress very high frequencies is apparently inherent, due to the con- tinuous structure of the acoustic medium within the resonator.
This disadvantage of course is lacking in discretely loaded structures, such as the string of beads. See K. Johnson, "Transmission Circuits," p.
Finite String; Normal Coordinates; Normal Functions We conclude with a discussion of the motion of a finite stretched string of uniform density. There is a certain interest in the problem as such, and in addition, we can use it to intro- duce a further consideration of more general matters.
According to the straightforward method cf. This was the method followed by Lagrange, to whom we owe the solution of the beaded string problem.
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The reader interested in the rapidity of convergence for the fundamental mode will find an interesting table in Ray- leigh, Vol. I, It appears, for example, that for the case of equal discrete loading at as many as six points, the frequency is within about I per cent of that of the continuously loaded string of the same total mass, length and tension. We shall not carry the matter further.
One of the interesting questions which we wish to consider is the relation of such dynamical developments as that of to the more purely mathematical notions of the properties of Fourier's Series. To obtain , we have consciously used [apart from the fact that sines and cosines are standard solu- tions of 91! This is that any finite sum of solutions of a linear differential equation is also a solution. If this is granted, and if the summation is extended to include a very large number of separate solutions, then in solving the problem of a uniform, continuous string we have virtually a dynamical proof of the validity of the expansion of the arbitrary initial configuration of the string into a sine or cosine series.
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From the practical standpoint it is reassuring to note that "the physical induction has been most fully corroborated by independent mathematical proof. The reader will enjoy discovering for himself, if he has not already done so, the parallelism all along the way between the analytical theory of Fourier's series and the purely physical ideas involved in the application of these series to concrete problems.
To follow the ideas we have in mind, in making these brief suggestions, the chapter on Fourier's Theorem in Lamb will be found very much to the point. Again consider the matter of normal coordinates.
In dealing with the continuously loaded string we have arrived, as it hap- pens, at a solution in terms of normal coordinates; we began, with the string loaded with discrete particles, using coordinates that were not normal. In 10 p. The difference between the factors sin and Jo kjr in the two cases is not material at the moment. It seems then, that for the string and the membrane in both of which the mass is uniformly distributed the use of normal coordinates and the Principle of Superposi- tion in attacking the problem is a natural procedure; whereas in the case of non-uniform distributions of mass, it would be NORMAL FUNCTIONS 81 impracticable.
In general, while we may not always be success- ful in applying the method of normal coordinates to the vibra- tions of a system with uniformly distributed constants, it is to such systems that we look for the principal applications of the method. Finally, the success of the method of normal coordinates, and the resulting gain of considering each natural oscillation as independent of the others, depends on the availability of a set of related normal functions in terms of which the shape of the vibrating body can be stated at any moment of its history.
Mathematically a set of functions FI X , F 2 x.
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A set of Bessel's functions can likewise be normalized. We state these results briefly, and without proof, in order to give the reader an idea of the general situation in regard to the use of normal functions in solving the problem of normal modes of oscillation in certain systems. Fortunately, the ap- propriate normal functions are known for the uniform string, and the uniform membrane. In the case of vibrations of a rectangular plate ', the normal func- tions are unknown; but in most problems with which we have to deal, the properties of a certain set of known normal func- tions are the key to the solution.
Following the chapter on Fourier's Theorem in Lamb, the more general treatment in Whittaker and Watson's "Modern Analysis" will provide the necessary background in the theory of normal functions. Solve the problem of 21 [i. The moving member induces in the circuit an electromotive force 2 c. Taking the constants of the vi- brating member as a y b y c in the familiar notation , find the driving point impedance of the electrical circuit when the vibrating member is in resonance.
Find the natural frequency and the damping coefficient, taking the velocity of sound in air at ordinary temperatures as 3. Compute the natural frequency, and the damping due to radiation under these new conditions; also give an estimate of the damping due to friction in the neck. In 25 a certain problem is solved for free oscillations.
Adjusting the fundamental of this string to agree with that of problem 15 , compare the motion of the three corre- sponding points spaced along the 2 strings. Use the first four terms which do not vanish of the Fourier's series corresponding to For the filter sketched in Fig. Devise a low-pass torsional filter suitable for coupling between two rotating shafts, and determine its upper limiting frequency. Neglecting damping in the inner orifice, find the natural frequencies of the coupled system, and the overall velocity amplification of the system at these frequen- cies.
How do these coefficients compare with the velocity amplification of the first resonator alone, at its resonant frequency?
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