Nijboer-Zernike approach in acoustics: ANZ

Nijboer-Zernike approach in acoustics : ANZ

In recent years, the Nijboer-Zernike approach has been applied to solve forward and inverse problems in acoustical radiation from a flexible circular piston surrounded by a rigid infinite planar set (baffle) and from a flexible spherical cap on a rigid sphere. The flexibility is embodied by a non-uniform velocity profile v that  is assumed to be radially symmetric in the case of piston radiation and axially symmetric in the case of radiation from the cap. As in optical diffraction theory, the complex amplitude of the sound pressure in the baffled-piston case is given by a Rayleigh integral comprising the field point r in front of the baffle and the wave number k of the harmonic excitation applied to the piston with velocity profile v. The frequencies that are used in acoustics (20 Hz-20kHz in audio and up to 200 kHz in ultrasound) are so much lower than those used in optics that dispersion effects can be neglected. Furthermore, although the phenomenon of focusing does manifest itself in acoustics (especially at higher frequencies), this does not occur usually to an extent that one can speak of a focal volume as in optics. Despite all these differences with optics, a wealth of analytic results, with the basic result of the classical Nijboer-Zernike theory as cornerstone, have been obtained for piston radiation and  also for radiation from a spherical cap. In both cases, the non-uniform velocity profile is considered to be developed as a series involving the Zernike polynomials (appropriately modified in the case of spherical-cap radiation). The contribution of each of these Zernike terms to the pressure and various other acoustical quantities turns out to have a tractable form, and so the corresponding quantity for the velocity profile can be obtained in semi-analytic form by linear superposition. Due to the efficiency of the Zernike terms in representing velocity profiles through its expansion coefficients, this offers the opportunity to estimate an unknown velocity profile v on the level of expansion coefficients from measured data in the field. Here one can use a matching approach in which the unknown expansion coefficients are found by requiring an optimal match between the measured data and the theoretical expression comprising the coefficients.

Far-field pressure obtained from near-field measurements for baffled-piston radiation

The quality of a loudspeaker is often assessed by an inspection of the complex amplitude of the sound pressure in the far field due to a harmonic excitation. A direct measurement of the far field comes with special requirements on the measurement room so as to avoid contributions to the measured pressure from reflecting walls (anechoic room). The use of such an expensive anechoic room could be avoided when one would be able to predict the far field of the loudspeaker from measurements in the near field. It turns out that this is possible indeed by the facts that
1. the Zernike expansion coefficients of the velocity profile can be estimated from near-field, on-axis pressure data,
2. the pressure in the far field can be computed from the expansion coefficients

These two steps use a recently obtained analytic result, see [ANZ1] for both the on-axis pressure and the far-field pressure per Zernike term. In Fig. ANZ-1 the procedure of predicting the far-field pressure is illustrated. Hence, 10 on-axis pressure values were measured, and the expansion coefficients (4 in number) of the velocity profile were chosen such that a best match between the theoretical expression (via fact 1) and the data occurred. With these coefficients, the theoretical expression for the far field (via fact 2) was computed and evaluated at on-axis points.

Figure ANZ-1. 10 measured on-axis near-field pressure data points connected by the black solid line 'p meas', and the estimated on-axis pressure 'p rec', blue dotted line, as a function of the normalized distance r/a. The loudspeaker is a Vifa MG10 SD09-08 with membrane radius a=3.2cm and measured in an IEC-baffle at 13.72 kHz.

Comparing radiation from a loudspeaker and from a flexible spherical cap on a rigid sphere

It has been suggested by Morse and Ingard that the sound radiation of a loudspeaker in a box is comparable with that of a spherical cap on a rigid sphere when the volumes of box and sphere and the areas of the vibrating membrane and the cap are matched. In Fig. ANZ-2 three polar plots, with the angle θ between the acoustical axis and the line segment connecting center of the radiating surface and field point as variable, of the modulus of the pressure of 4 different frequencies in the far field are shown. The top figure shows the polar plot of a real loudspeaker, the middle figure shows the polar plot for baffled-piston radiation with a rigid piston (v is constant), and the bottom figure shows the polar plot for spherical-cap radiation with the cap moving uniformly parallel to the axis. The polar plots for the loudspeaker and the spherical cap resemble quite well.

Figure ANZ-2. Polar plots of the pressure at frequency 1kHZ (solid, black curves), 4kHZ (dotted, red curves), 8kHZ (dashed-dotted, blue curves), and 16 kHZ (dashed, green curves), normalized such that the pressure is 1 at θ = 0 for
a) loudspeaker (same as in Fig. ANZ-1) in a rectangular cabinet measured at 1 m distance,
b) rigid piston in an infinite baffle, piston radius a=3.2 cm,
c) rigid spherical cap with z-component  of the velocity profile equal to 1 m/s, cap aperture θ0 = π/8 , and r = 1.

This agreement continues to hold for several other acoustical quantities. In Fig. ANZ-3 this is demonstrated for the so-called directivity index as a function of the wave number k (R is the radius of the sphere containing the cap).

Figure ANZ-3. The directivity index DI (explained in the text) of a rigid spherical cap with various aperture angles of the cap: 5π/32 rigid (solid, black curve), π/8 (dotted, red curve), and π/10 (dashed-dotted, blue curve). The long-dashed, green curve starting for kR=0 at 3 (dB) is the directivity for a rigid piston in an infinite baffle.

This directivity index is a qualitative measure of how directive a particular sound radiator is and compares the far-field, on-axis power of the radiator to the total radiated power. The Figs. ANZ-2,3 for the spherical cap were produced by using a Zernike-based semi-analytic result, see [ANZ6], for the complex amplitude of the pressure at any field point on and around the sphere.

Acoustical spatial impulse response for baffled-piston radiation

In the case of baffled-piston radiation, the complex amplitude p(r;k) of the pressure at the field point r due to a harmonic excitation is given by a Rayleigh integral comprising the wave number k of the excitation. Since acoustical media are non-dispersive, as opposed to optical media, this integral expression is valid for the whole frequency range of interest without a need to correct for refraction effects. From well-established practices in physical signal processing, one can obtain a field-point dependent impulse response by simply performing a Fourier transformation with respect to the wave number k. In Fig. ANZ-4 plots for a constant and for a Gaussian profile v (σ) , 0≤σ a, on the circular piston are given as a function of the normalized time and the normalized distance of the field point from the axis at a constant normalized distance from the piston plane (normalization: radius a of the piston).

Figure ANZ-4. Spatial impulse response with constant value z/a = ½ of the axial variable as a function  of the normalized radial variable w/a and the normalized time variable ct/a for the cases that (a) v (σ) = 1, 0 ≤ a, (b) v (σ)=exp{-4 (σ/a)2}, 0 ≤σ a.

These plots were produced by expanding the velocity profile v into Zernike terms and using a recently derived analytic result, see [ANZ9], for the contribution of each of the Zernike terms to the spatial impulse response. The method can also be used in the reverse direction in which an unknown velocity profile is estimated on the level of its Zernike expansion coefficients from impulse response data measured at field points in the half-space in front of the baffle.

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