Acoustic Wave Filters

Acoustic Wave Filters Composed of a Series of Like Sections.---(1) Theory. Taking the impedance of any part of an acoustic circuit to be equal to the complex ratio of the applied pressure difference to the rate of change of volume displacement, it is shown that, neglecting dissipative forces, it is possible to construct a filter having limiting frequency values of no attenuation determined by the formul\ae{} $\frac{{Z}_{1}}{{Z}_{2}}=0$ and $\frac{{Z}_{1}}{{Z}_{2}}=\ensuremath{-}4$, where ${Z}_{1}$ is the impedance of the transmitting conduit circuit and ${Z}_{2}$ of each branch of each section. The impedance of any section depends on the inertance $M$ of dimensions mass per unit area squared, and the capacitance $C$ which has the dimensions of stiffness per unit area squared. If $M$ and $C$ are in parallel, $Z=\frac{\mathrm{iM}\ensuremath{\omega}}{(1\ensuremath{-}MC{\ensuremath{\omega}}^{2})}$, whereas if they are in series, $Z=i(M\ensuremath{\omega}\ensuremath{-}\frac{1}{C\ensuremath{\omega}})$. For instance, in the case of a closed chamber or resonator, $M$ and $C$ are in series and are equal to $\frac{\ensuremath{\rho}}{C}$ and $\frac{V}{\ensuremath{\rho}{a}^{2}}$ respectively where $\ensuremath{\rho}$ is the density of the medium, $c$ is the conductivity of the mouth, $a$ the velocity of sound and $V$ the volume. Formul\ae{} are derived for various assumed cases. On account of the uncertainty as to whether a tube may be considered sa having the equivalent inertness and capacitance connected in parallel or in series, the application of these formul\ae{} to actual cases is somewhat empirical. (2) Construction and test of filters of three types. Low-frequency-pass filters were made, for example, by two concentric cylinders joined by walls equally spaced and perpendicular to the axis. Each chamber thus formed had a row of apertures in the inner cylinder which served as the transmission tube. In one case the volume of each chamber was 6.5 ${\mathrm{cm}.}^{3}$, the radius of the inner tube 1.2 cm. the length between apertures, 1.6 cm. A chamber and one such length of the inner tube is called a section. Four such sections were found to transmit 90 per cent. of the sound from zero to approximately 3,200 d.v. where the attenuation became very high, resuiting in zero transmission up to about 4,600 d.v. where transmission again appeared, Other similar filters of different dimensions attenuated through wider or narrower ranges. The lower limit of attenuation was found to correspond within 8 per cent. with the formula: $f=(\frac{1}{\ensuremath{\pi}}){({M}_{1}{C}_{2}+4{M}_{2}{C}_{2})}^{\ensuremath{-}\frac{1}{2}}$. The upper limit was not predicted theoretically. High-frequency pass filters were made with a straight tube for transmission and short side tubes, for example, 0.5 cm. long and 0.28 cm. in diameter, opening through a hole with conductivity 0.08 into a tube 10 cm. long and 1 cm. in diameter. Six sections of such a filter would transmit about 90 per cent. of sounds above 800 but would refuse transmission to sounds of lower frequency. As would be expected, the cut off is not sharp. Filters with other dimensions were found to have an upper limit of attenuation varying from 450 to 2,300 d.v., agreeing with the formula $f=(\frac{1}{2\ensuremath{\pi}}){(\frac{1}{4{M}_{2}{C}_{1}}+\frac{1}{{M}_{1}{C}_{2}})}^{\frac{1}{2}}$, within about 13 per cent., on the average. The single-band filters made were a combination of the other two types, having side tubes leading to chambers of considerable volume. For instance, three sections each 5 cm. long and 0.5 cm. in diameter, with side tubes of the same size and 2.2 cm. long leading to a volume of 28 ${\mathrm{cm}.}^{2}$, transmitted between 270 and 370 d.v. The frequencies of the edges of the band of small attenuation are determined by the following formulae, ${2\ensuremath{\pi}f={[{M}_{2}{C}_{2}(1+\frac{{{M}_{2}}^{\ensuremath{'}}}{{M}_{2}})]}^{\ensuremath{-}\frac{1}{2}} \mathrm{and}}{2\ensuremath{\pi}f={[{M}_{2}{C}_{2}{(1+\frac{4{M}_{2}}{{M}_{1}})}^{\ensuremath{-}1}(1+\frac{{{M}_{2}}^{\ensuremath{'}}}{{M}_{2}}+\frac{4{{M}_{2}}^{\ensuremath{'}}}{{M}_{1}})]}^{\ensuremath{-}\frac{1}{2}}. \mathrm{Such}\mathrm{filters}}$ exhibit the same variations from theoretlcai performance as would be expected from a combination of the other two types. However, the agreement of each type with theory is sufficient to enable filters to be designed to fulfill set conditions. The attenuation secured with only four sections is very great, the transmission being certainly less than ${10}^{\ensuremath{-}7}$ in the attenuated region, while it may rise to 90 per cent. in unattenuated regions. Possible applications of these simple filters in laboratory work and in connection with specking devices, are briefly suggested.