Denver, Colorado NOISE-CON 013 013 August 6-8 Transient, planar, nonlinear acoustical holography for reconstructing acoustic pressure and particle velocity fields a Yaying Niu * Yong-Joe Kim Noise and Vibration Group Acoustics and Signal Processing Laboratory KBR, Inc. Department of Mechanical Engineering 601 Jefferson Street Texas A&M University Houston College Station TX 7700-7900 TX 77843-313 yaying.niu@kbr.com joekim@tamu.edu ABSTRACT A steady-state, nonlinear nearfield acoustical holography procedure, based on the Westervelt Wave Equation (WWE), was developed by the authors of this article to accurately reconstruct nonlinear acoustic pressure fields. Here, a transient, nonlinear acoustic holography algorithm is introduced that can be used to reconstruct three-dimensional, nonlinear acoustic pressure as well as particle velocity fields from two-dimensional acoustic pressure data measured on a measurement plane. This procedure is based on the Kuznetsov Wave Equation (KWE) that is directly solved by applying temporal and spatial Fourier Transforms to the KWE. When compared to the WWE-based procedure, the proposed procedure can be used to reconstruct acoustic particle velocity fields in addition to acoustic pressure fields. It can be also applied to multi-frequency source cases where each frequency component can contain both linear and nonlinear components. The KWE-based procedure is validated by conducting four numerical simulations with: 1) an infinite-size panel vibrating at a single frequency, ) a pulsating sphere with a bifrequency excitation, 3) a finite-size, vibrating panel generating bended wave rays, and 4) an ultrasound transducer with a transient excitation. The numerical results show that holographically-projected acoustic fields match well with directly-calculated ones. 1. INTRODUCTION The linear Nearfield Acoustical Holography (NAH) procedures in Refs. 1-8 can be used to reconstruct three-dimensional (3-D), linear acoustic pressure and particle velocity fields based on two-dimensional (-D) acoustic pressure or particle velocity measurements. When an acoustic source generates high-level acoustic fields where nonlinear effects are not negligible, the linear NAH algorithms 1-8 cannot be applied to reconstruct the nonlinear acoustic fields accurately. In order to nonlinearly reconstruct acoustic pressure fields, a steady-state, planar, nonlinear NAH (NNAH) algorithm 9, using perturbation and renormalization methods 10,11 to solve the Westervelt Wave Equation 1,13 (WWE), was developed by the authors of this article. Although this WWE-based algorithm can be used to reconstruct nonlinear acoustic pressure fields directly at any reconstruction planes in a computationally efficient manner, it has the following limitations: (1) This NNAH algorithm can be used to only reconstruct a fundamental a A full version of this article is submitted to Journal of the Acoustical Society of America and under review. * This paper is based on research conducted as a doctoral candidate at Texas A&M University.
frequency and its second-order harmonic components, since the perturbation method retains up to the second-order acoustic variables; () The application of this NNAH algorithm is limited to monofrequency source cases, since a multi-frequency source can generate mixed frequency components that include both linear and nonlinear components 13,14 ; (3) The effects of bended wave rays induced by transverse acoustic particle velocities 10,11 cannot be considered in the WWE-based procedure; (4) Only acoustic pressure fields can be reconstructed by using this WWE-based NNAH procedure. For the purpose of addressing the aforementioned limitations with the WWE-based NNAH algorithm, a transient, nonlinear acoustical holography algorithm is derived in this article by applying the temporal and spatial Fourier Transforms (FTs) to a nonlinear governing equation, Kuznetsov Wave Equation (KWE) that is represented in terms of an acoustic particle velocity potential 15. Then, transient, 3-D acoustic potential fields are reconstructed by applying inverse FTs to angular frequency and wavenumber spectral solutions represented in terms of the acoustic pressure spectrum measured on a -D measurement surface. In addition to the acoustic pressure fields, acoustic particle velocity fields can be reconstructed by applying the spatial derivatives to the reconstructed acoustic potential fields, while the acoustic particle velocity fields cannot be reconstructed by using the WWE-based algorithm. Since the proposed procedure is based on a transient signal processing technique, it can be applied to multi-frequency sources. The effects of the nonlinear bended wave rays induced by transverse particle velocities can be also considered in the proposed procedure. To the best knowledge of the authors, it is the first acoustical holography algorithm that can be used to reconstruct both nonlinear acoustic pressure and particle velocity fields with bended wave rays even under multi-frequency source conditions.. THEORY The governing nonlinear wave equation for the proposed nonlinear acoustical holography algorithm is the Kuznetsov Wave Equation (KWE) 15 : i.e., 0 0 0 0 1 b 1 1 c t c t c t c t x y z, (1a) where c 0 is the speed of sound, b is the sound diffusivity, ϕ is the acoustic particle velocity potential, and β is the nonlinearity parameter of a fluid medium. Acoustic particle velocity vector, u can be then related with the acoustic velocity potential as Acoustic pressure can be also obtained from the velocity potential: i.e., u. (1b) p where ρ 0 is the density of fluid medium. In an ideal gas such as air, the sound diffusivity, b is small enough to be negligible, and thus the quadratic nonlinear terms in Eq. (1a) are dominant over the thermoviscous dissipation term. For ultrasound wave propagations in water or human tissues, however, the dissipation in the fluid medium is not negligible. The right hand side (RHS) of Eq. (1a) contains four quadratic nonlinear source terms. In order to solve Eq. (1a), the velocity potential can be replaced by the inverse temporal and spatial Fourier Transforms (FTs) of its frequency-wavenumber spectrum, and an inhomogeneous ordinary differential equation (ODE) can be obtained: i.e., 0 t, (1c)
where d kx ky z dz,,, k k, k, z, F k, k, z, z x y x y, (a) 0 0 z c x y 0 i b k k k, (b) i 1 d d F kx kx ky ky c i b c dz dz Φ is the frequency-wavenumber spectrum of the acoustic velocity potential, ω is the angular frequency, and k x, k y, k z are the wavenumbers in x, y, and z directions, respectively. In Eq. (c), the symbol of represents the convolution integral in the frequency and wavenumber domain. The inhomogeneous source term, F includes the four convolution integrals of the frequencywavenumber spectrum in the domain of (k x,k y,ω). The nonlinear effects induced by the mixing of frequency and wavenumber components are thus included in Eq. (c). Here, the plane at z = 0 is defined as the hologram plane where the acoustic velocity potential in the frequency and wavenumber domain is written as Φ 0. By applying the method of variation of parameters and considering only positive-propagating wave components in a free field, Eq. (a) can be directly solved: i.e., z,(c) 1 z ikzz ikzz ikzz z 0e e e F z dz. (3) ik 0 The velocity potential at any z-location can be thus obtained by using Eq. (3). Since F(z) includes the velocity potential terms as shown in Eq. (c), the velocity potential solution in Eq. (3) is implicit. The integral in Eq. (3) can then be solved iteratively or numerically 16. The projected acoustic pressure and particle velocity fields can be obtained from Eqs. (1b) and (1c) as 1 ikx x ky y t p x, y, z, t i 3 0 z e dkxdkyd, (4a) 8 1 ikx x ky y t ux x, y, z, t ik 3 x z e dkxdk yd, (4b) 8 1 ikx x ky y t uy x, y, z, t ik 3 y z e dkxdk yd, (4c) 8 1 ikx x ky y t uz x, y, z, t ik 3 z z e dkxdk yd. (4d) 8 3. NUMERICAL VALIDATION In order to validate the proposed nonlinear acoustical holography procedure, numerical simulations are performed with four types of source configurations: 1) an infinite-size panel with mono-frequency source excitation; ) a finite-size panel with mono-frequency source excitation; 3) a pulsating sphere with bi-frequency source excitation; 4) an ultrasound transducer with transient source excitation. The sampling time step is set to include the twentieth harmonic of the maximum fundamental excitation frequency for each simulation case.
A. Simulation Setups Figure 1(a) shows the numerical simulation setup for the infinite-size panel with mono-frequency excitation at 1 khz. The excitation amplitude is set to 1.6 kpa (i.e., 158 db referenced at 0 Pa) on the panel surface. The Fubini solution 13 is used for calculating acoustic fields radiated from this panel. In this simulation, the array of acoustic pressure sensors is set to have the size of 41 41 with the sampling intervals of x = y = 0.05 m that are determined from the minimum wavelength at the maximum frequency. The array covers the measurement aperture of (x,y) = ([0,],[0,]) m. The panel is tilted by an angle of 0 degrees around the y-axis and the hologram height is 0.1 m as shown in Fig. 1(a). Figure 1(b) shows the numerical simulation setup for the finite-size panel with the freefree boundary condition at its ends on the x-z plane. This panel has the width of L = m and the infinite length in the y-direction. Perturbation and modified renormalization procedures are used for this simulation to generate the bended wave rays induced by transverse particle velocities. The surface normal velocity of the plate is set to V 0 = 30 m/s with the excitation frequency of 1 khz. The size of the acoustic pressure sensor array, the sampling intervals, and the hologram height are identical with the infinite-size panel case in Fig. 1(a). (a) (b) (c) (d) Figure 1: Sketch of numerical simulation setups: (a) Infinite-size panel simulation with monofrequency excitation at 1 khz, (b) Finite-size panel simulation with free-free boundary condition and mono-frequency excitation at 1 khz, (c) Pulsating sphere simulation with bi-frequency excitation at 0.8 and 1. khz, and (d) Ultrasound transducer simulation with transient excitation at center frequency of 1 MHz.
In addition to the vibrating panel simulations with mono-frequency source excitations, Fig. 1(c) shows the simulation setup for the pulsating sphere with a bi-frequency excitation. The bi-frequency excitation is at 0.8 khz and 1. khz with the excitation amplitudes of 5 kpa (i.e., 168 db) and 3 kpa (i.e., 163.5 db), respectively. The Fenlon s bi-frequency solution is applied to this simulation 13,14. The sphere with the undisturbed radius of 0.5 m is centered at (x,y,z) = (0.75,0.75,-0.5) m. The array of 31 31 acoustic pressure sensors with the sampling intervals of x = y = 0.05 m is placed at the hologram height of 0.1 m. The array covers the measurement aperture of (x,y) = ([0,1.5],[0,1.5]) m. In this pulsating sphere simulation, the frequency components of hologram data contain both linear and nonlinear components, since all the frequency components are mixed during the nonlinear wave propagation as indicated in the Fenlon s solution 13,14. Figure 1(d) shows the simulation setup for the ultrasound transducer with the size of L x = L y = 46.5 mm. This ultrasonic transducer generates transient acoustic fields with a transient excitation at the center frequency of 1 MHz. The array of 94 94 acoustic pressure sensors is placed at z = 0 mm with the sampling intervals of x = y = 0.5 mm, which covers the measurement aperture of (x,y) = ([0,46.5],[0,46.5]) mm. The rectangular ultrasonic transducer has the focal length of 60 mm: i.e., the focal plane is at z = 55 mm, which is used as a plane for validating the proposed nonlinear holographic procedure. An open source software package, Abersim 17,18 is used to generate the transient acoustic pressure fields on the hologram plane at z = 0 mm and the validation plane at z = 55 mm. Here, the fluid medium is distilled water with the sound speed of 148 m/s and the density of 998 kg/m 3, while the simulations in Figs. 1(a) 1(c) use air as the fluid medium with the sound speed of 340 m/s and the density of 1.04 kg/m 3. B. Projected Acoustic Fields Figure shows the directly-calculated and nonlinearly-projected acoustic pressure fields and the corresponding reconstruction errors for the infinite-size panel simulation. At the second and third harmonics of the fundamental frequency of 1 khz, the KWE-based method can be used to reconstruct the acoustic pressure fields correctly: i.e., as shown in Figs. (c) and (f), the maximum reconstruction errors are 0.8 db and. db for the second and third harmonic fields, respectively. The reconstruction errors can be caused by the hologram aperture edge truncations, the ghost imaging effects induced by the periodicity of the spatial FFT 4, and the numerical integration errors for solving Eq. (3). On the same x-z plane at y = 1 m, particle velocity fields are also reconstructed by using the proposed algorithm at the second and third harmonics of the fundamental frequency. Figure 3 shows the z-direction particle velocity fields. The nonlinearly-projected particle velocity fields are obtained by using Eq. (4d). Since the acoustic waves radiated from the infinite-size panel can be assumed to be plane waves, the exact particle velocity fields can be directly calculated by using the following equation: u z = pcos(θ)/(ρ 0 c 0 ), where θ is the tilting angle of the panel. The projected particle velocity fields match well with the directly-calculated ones. Similarly, the particle velocity fields at the other two directions can be projected by using Eqs. (4b) and (4c).
(a) [Pa] (b) [Pa] (c) (d) [Pa] (e) [Pa] (f) Figure : Directly-calculated and nonlinearly-reconstructed acoustic pressure fields, on x-z plane at y = 1 m, radiated from infinite-size panel : (a) Directly-calculated at khz, (b) Reconstructed at khz, (c) Reconstruction error at khz, (d) Directly-calculated at 3 khz, (e) Reconstructed at 3 khz, and (f) Reconstruction error at 3 khz. (a) [m/s] (b) [m/s] (c) (d) [m/s] (e) [m/s] (f) Figure 3: Directly-calculated and nonlinearly-reconstructed z-direction particle velocity fields, on x-z plane at y = 1 m, radiated from infinite-size panel : (a) Directly-calculated at khz, (b) Reconstructed at khz, (c) Reconstruction error at khz, (d) Directly-calculated at 3 khz, (e) Reconstructed at 3 khz, and (f) Reconstruction error at 3 khz. Figure 4 shows the directly-calculated and nonlinearly-projected acoustic pressure fields at the second harmonic of khz for the finite-size panel simulation case. It is shown that the bended wave rays can be reconstructed accurately by using the KWE-based procedure with the maximum reconstruction error of db. Figure 5 shows the directly-calculated and nonlinearly-projected acoustic pressure fields at the frequencies of 1.6 khz, khz, and.4 khz for the bifrequency, pulsating sphere
simulation case. These selected frequencies are the harmonics, summations, and differences of the two excitation frequencies. The acoustic pressure fields are reconstructed on the x-z plane at y = 0.75 m. The projection results agree well with directly-calculated ones with the reasonably small reconstruction errors at the maximum error of 1 db. Thus, it is concluded that the KWEbased algorithm can be also used to reconstruct acoustic fields with multi-frequency source excitations. (a) [Pa] (b) [Pa] (c) Figure 4: Directly-calculated and nonlinearly-reconstructed acoustic pressure fields radiated from finite-size panel on x-z plane at y = 1 m: (a) Directly-calculated at khz, (b) Reconstructed at khz, and (c) Reconstruction error at khz. (a) [Pa] (b) [Pa] (c) (d) [Pa] (e) [Pa] (f) (g) [Pa] (h) [Pa] (k) Figure 5: Directly-calculated and nonlinearly-reconstructed acoustic pressure fields, on x-z plane at y = 0.75 m, radiated from pulsating sphere : (a) Directly-calculated at 1.6 khz, (b) Reconstructed at 1.6 khz, (c) Reconstruction error at 1.6 khz, (d) Directly-calculated at khz, (e) Reconstructed at khz, (f) Reconstruction error at khz, (g) Directly-calculated at.4 khz, (h) Reconstructed at.4 khz, and (k) Reconstruction error at.4 khz.
All of the acoustic fields in Figs. 5 are presented at the several selected peak frequencies for the steady-state excitations. In Fig. 6, transient acoustic pressure fields reconstructed on the focal plane at z = 55 mm are presented for the ultrasound transducer simulation case. Figures 6(a) and 6(b) show the acoustic pressure fields on the x-t domain at y = 3.5 mm and z = 55 mm. The nonlinearly-reconstructed acoustic pulse signal in Fig. 6(b) agrees well with the directly-calculated one in Fig. 6(a). At the center location of the focal plane, the nonlinearly-reconstructed and directly-calculated acoustic pressure time data show a good agreement (see Fig. 6(c)). Figures 6(d) and 6(e) show the directly-calculated and nonlinearlyreconstructed, time-averaged SPLs, respectively, on the x-y plane at z = 55 mm. The maximum reconstruction error is 1. db as shown in Fig. 6(f). Thus, the KWE-based algorithm is also validated for this transient simulation case. (a) [MPa] (b) [MPa] (c) (d) (e) (f) Figure 6: Acoustic pressure fields radiated from ultrasound transducer at z = 55 mm (i.e., focal plane): (a) Directly-calculated on x-t plane at y = 3.5 mm, (b) Nonlinearly-reconstructed on x-t plane at y = 3.5 mm, (c) Directly-calculated and nonlinearly-reconstructed time data at the center location of focal plane, (d) Directly-calculated, time-averaged SPL (ref: 1 Pa) on focal plane, (e) Nonlinearly-reconstructed, time-averaged SPL on focal plane, and (f) Reconstruction error. 4. CONCLUSIONS In this paper, the nonlinear acoustical holography algorithm based on the Kuznetsov Wave Equation (KWE) is proposed and validated numerically. When compared to the WWE-based NNAH algorithm proposed by the authors of this article in Ref. 9, the current transient KWEbased algorithm has the remarkable improvements to make it possible to reconstruct: 1) acoustic particle velocity fields in addition to acoustic pressure fields, ) acoustic fields generated from multi-frequency sources, 3) nonlinear harmonic components higher than the second-order harmonic component, 4) acoustic fields with bended wave rays, and 5) transient acoustic fields. The four types of the numerical simulations are performed to validate the KWE-based algorithm. These simulations can be used to successfully validate the proposed algorithm. For example, both the acoustic pressure and z-direction particle velocity fields generated from the infinite-size panel at the second and third harmonics of the fundamental frequency at 1 khz are well reconstructed with the maximum reconstruction errors of 0.8 and. db, respectively. The acoustic pressure field, with the bended wave rays, radiated from the finite-size panel is projected correctly with the maximum reconstruction error of db. The acoustic pressure fields
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