Reflectance of acoustic horns and solution of the inverse problem
(Color online) Horn diameter (D) as a function of axial distance (x) for various horn shapes. In the selected examples, the diameter is 1 cm at x = 0 and expands to make the cross-sectional area A = 10 cm2 for all horn shapes at x = 10 cm.
(Color online) Reflectance in the frequency-domain R(0,2π f). The reflectance functions were obtained from theoretical equations presented in the Appendixes. The arrow indicates the cutoff frequency.
(Color online) Reflectance in the time domain. The inverse Fourier transforms of frequency-domain reflectance were multiplied by the sampling rate to display values that are independent of sampling rate (solid lines) and are superimposed over the expressions for time-domain reflectance r(0,t) that are presented in the Appendixes (dashed lines). Good agreement between these two sets of curves completely obscures the dashed lines except for the parabolic horn. The inverse solution used only the time-domain reflectance calculated by inverse Fourier transform.
(Color online) Horn diameter inverse solution (solid lines) compared with true values (dashed lines), which were calculated from the expressions listed in Table I. The curves in this figure represent D(x).
(Color online) Horn inertance B(x) inverse solution (solid lines) compared with ɛ(x) (dashed lines), which was calculated from the expressions listed in Table I. The agreement between B(x) and ɛ(x) was unexpected.
(Color online) Horn diameter inverse solution (solid lines) compared with true values (dashed lines) for exponential horn with decreasing area function. The diameter functions obtained from D(x) = e− a x is also plotted as dashed lines, together with reproductions of increasing diameter function of Fig. 4 for comparison. The curves in this figure represent D(x).
(Color online) Evaluation of the limitations of the inverse solution. (a) A sampling rate of 1 MHz results in good diameter estimation and computational efficiency. (b) Decrease of the sampling rate to 100 kHz results in increased diameter deviation. (c) Using Type-II reflectance at x = 0 reduces the diameter deviations but makes the inverse solution less stable.
Horn diameter and its logarithmic gradient for various horn shapes. The value of α was selected (for parabolic, conical, and exponential horns) to make area A(10) = 10 cm while diameter D(0) = 1 cm. The units of α and ɛ are cm−1, while the unit of D is cm.
Radiation admittance and reflectance evaluated at x = 0.
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