Acoustic holography LMS Test.Lab Rev 12A Copyright LMS International 2012
Table of Contents Chapter 1 Introduction... 5 Chapter 2... 7 Section 2.1 Temporal and spatial frequency... 7 Section 2.2 Time domain... 8 Section 2.3 Spatial domain... 8 Section 2.4 (Back) propagating to other planes... 12 Section 2.5 The Wiener filter and the AdHoc window... 13 Section 2.6 Derivation of other acoustic quantities... 14 Rev 12A 3
Chapter 1 Introduction Chapter 1 Introduction Acoustic holography allows you to accurately localize noise sources. It therefore helps in both the reduction of unwanted vibro-acoustic noise and optimization of noise levels. It : estimates the acoustic power and the spectral content emitted by the object under examination. maps sound pressure, velocity and intensity on the measurement plane and on all parallel planes. The mapping of these acoustical quantities outside the measurement plane is done through acoustical holography (near field - far field). estimates the acoustic level of the principal sources, including contribution analysis. This document describes the principles of taking acoustic measurements and the subsequent analysis of acoustic holography data, for both stationary and transient measurements. Basic principles In performing acoustic holography, you need to measure cross spectra between a set of reference transducers and the hologram microphones. From these measurements you can derive sound intensity, particle velocity and sound power values. A basic assumption is that you are operating in free field conditions and that the energy flow is coming directly from the source. Measurements need to be taken close to the source. It provides you with an accurate 3D characterization of the sound field and the source with a higher spatial resolution than is possible with conventional intensity measurements. Rev 12A 5
Chapter 2 In This Chapter Temporal and spatial frequency... 7 Time domain... 8 Spatial domain... 8 (Back) propagating to other planes... 12 The Wiener filter and the AdHoc window... 13 Derivation of other acoustic quantities... 14 The principle of acoustic holography is to decompose the measured pressure field in plane waves, by using a spatial Fourier transform. With the frequency being fixed, we can calculate how each of these plane waves propagates, and by adding them we can find the pressure field on any plane which is parallel to the measurement plane. Consider an acoustic wave. Measuring the pressure on a plane means cutting the wavefronts by the measurement plane: The goal is to determine the whole acoustic wavefront from the known pressure on the measurement plane. Each microphone in the array measures the complex pressure (amplitude and phase). Section 2.1 Temporal and spatial frequency In considering how to do this we will compare the time and the spatial domain. Rev 12A 7
Section 2.2 Time domain When considering measurements in the time domain, then the position from the sound source (m) is fixed and we obtain a measure of the pressure variation as a function of time. The transformation from the time to the frequency domain is achieved using the Fourier Transform given below Section 2.3 Spatial domain If we now consider measurements where time is fixed and pressure varies as a function of distance, we can obtain a measure of energy flow. The spatial frequency of this function or wavenumber (k0) is defined as : If we fix the temporal frequency, this means that the acoustic wavelength is fixed too. 8 LMS Test.Lab Acoustic holography
The complex pressure as a function of the space is called the pressure image at the specified frequency. Conversion from the spatial domain is also done using a Fourier transform. In Acoustic holography pressure is measured in two dimensions (x and y for example), so a 2-dimensional transformation is performed. where S (kx, ky) is the spatial transform of the measured pressure field to the wavenumber (kx and ky) domain resulting in the 2-D hologram pressure field. A measured pressure (sound) wave with a particular temporal frequency can propagate in a number of directions, so the wavenumber vector (k) will have a number of components. The appearance of these vectors depends on the plane on which you are looking at them. The aim is to find the components of these vectors in the 2 dimensions that define the plane and to do this projections of the vectors in the plane are made. Summation of plane waves Rev 12A 9
The spatial Fourier transform implies that a measured pressure field can be considered as a sum of sinusoidal functions. Each of these sinusoidal functions can be understood as the result of cutting the wavefronts of a plane wave by the measurement plane. There is a coincidence between the nodes of the sinusoidal function and the wavefronts. In effect, decomposing the pressure field into a sum of sinusoidal functions means decomposing the real acoustic wave into a sum of plane waves. Whatever the angle of incidence, the spatial periodicity must be greater than the wavelength (l). Propagating and evanescent waves There are two kinds of plane waves : propagating waves - whose level remains the same as they propagate but who undergo a phase shift. evanescent waves - whose level decreases as they propagate. Propagating waves represent the sound field that is propagated away from the near towards the far field. Evanescent waves describe the complex sound field 10 LMS Test.Lab Acoustic holography
in the near field of the source. To understand why we must take evanescent plane waves into account, let us consider our decomposition of the pressure field into sinusoidal functions. If the spatial periodicity of a sinusoidal function is shorter than the wavelength, it cannot be the result of cutting a propagating plane wave by the measurement plane : Whatever the direction of the propagating plane wave may be, there is no possible coincidence between the nodes of the sinusoidal function and the wavefronts. Therefore, this sinusoidal function must be understood as the intersection between an evanescent wave (which can have a smaller spatial periodicity than propagating waves) and the measurement plane. A mathematical interpretation of the evanescent waves is based on the value of kz which is the component perpendicular to the measurement directions in the wave number domain. kz can be determined from the wave number k0 and the known values of kx and ky from the transformation. Rev 12A 11
kz is real when (the spatial periodicity is greater than the wavelength). This means that the waves lie in the circle defined by the radius w/c in the wave number domain. kz is imaginary outside of this region. When kz is imaginary, the propagation factor becomes a damped exponential function (e-jkz z) meaning that a propagated wave undergoes an amplitude modification while the phase is not changed. Section 2.4 (Back) propagating to other planes Pressure levels at other planes can be found using Raleigh's integral Equation with Dirichlet's Green function : where the Green function Gd can be thought of as the transformation function to transform the sound pressure field from one plane to another. We can use wave domain properties (k) to predict the pressure at a different spatial position (z). The practical computation of Raleigh's equation is: where z' is the measurement plane and z is the position of the required plane. The green function is given by The final step is to perform an inverse transformation back to the temporal domain. 12 LMS Test.Lab Acoustic holography
Section 2.5 The Wiener filter and the AdHoc window As mentioned above, evanescent waves undergo a change in amplitude when propagating. Propagating towards the source implies an amplification of the signal that is a function of kz. Evanescent waves that lie far away from the unit circle have a large kz, therefore their amplitude is amplified significantly when propagating to the source. The contribution of these evanescent waves results in an increase of spatial resolution. Note that the inclusion of evanescent waves is only appropriate when propagating towards the source. Propagating away from the source, the evanescent waves decrease so rapidly in amplitude that their contribution to the spatial resolution becomes negligible. However the further away a wave is located from the circle, the less accurate the amplitude estimate becomes so that at a certain point noise is propagated and at that point the propagated image starts to blur. When propagating towards the source, a Wiener filter can be used to include a certain number of evanescent waves to improve the resolution. Taking a higher number of waves taken into account may result in the amplification becoming unstable. This depends on a parameter of the Wiener filter known as the Signal to Noise Ratio (SNR). When the SNR value is greater than 15dB, then the amplification will become unstable as the number of evanescent waves included increases. Using an low SNR value (5dB for example) means that the evanescent waves are taken into account but they are so attenuated that the improvement in resolution is negligible. The default value of 15dB provides the best compromise in terms of resolution and amplification. When the Wiener filter is used, the pressure image needs to be multiplied by a two-dimensional window. As is the case with a single FFT, the observed pressure must be 'periodic' within the observed hologram. If this is not the case, then truncation errors occur as with a single FFT. These truncation errors manifest themselves as ghost sources at the borders of the observed area. Two windows are used: The rectangular window, which does not modify the pressure image. In case of a rectangular window, only propagating waves are included in the calculations resulting in a resolution equivalent to an intensity measurement. The so-called Ad Hoc window For a time signal, the FFT algorithm takes the time signal and duplicates it from minus to plus infinity. If the amplitude of the measured time signal differs Rev 12A 13
between the start and the end of the window, a discontinuity occurs during this multiplication introducing an error in the FFT algorithm. This can be corrected using a Hanning window. Holography used a double FFT so the AdHoc window is used, which is basically a two dimensional Hanning window thus removing discontinuities in the both the x and y directions. The one-dimension Ad Hoc window (W) would be: Section 2.6 Derivation of other acoustic quantities If we know how the plane waves propagate, we can calculate the pressure field in any parallel plane, by adding the contributions of all plane waves. This will be correct only if all acoustic sources are on the same side of both planes : Knowing the pressure field on the parallel plane, it is possible to calculate the particle velocity and eventually the intensity on this plane. The particle velocity (V) will be known if the pressure differential can be determined -which is the case with Acoustic holography since the pressure can be measured at r and (r + dr) 14 LMS Test.Lab Acoustic holography
Once the pressure and the velocity are known then the intensity is just the product of the two. Rev 12A 15
( Index (Back) propagating to other planes 13 A 7 D Derivation of other acoustic quantities 15 I Introduction 5 S Spatial domain 8 T Temporal and spatial frequency 7 The Wiener filter and the AdHoc window 14 Time domain 7 Rev 12A 17