The Helmholtz Equation Least Squares Method

Sound and vibration are intimately related to each other, yet they are two very different physical phenomena. In physics, sound is defined as the disturbance that travels in a compressible medium such as air in terms of a pressure (also called the longitudinal or compressional) wave. In physiology, sound is defined as the sensation of this pressure wave perceived by the brain auditory system of a human being.

All acoustic radiation problems can be boiled down to solving the wave equation subject to certain initial and boundary conditions. For a constant frequency case, the problem reduces to solving the Helmholtz equation [40], \( {\nabla}^2\widehat{p}+{k}^2\widehat{p}=0 \), subject to certain boundary conditions on the source surface. This sounds simple but in reality the analytic solution to the Helmholtz equation exists only for certain types of source geometry that the Helmholtz equation is separable. In most engineering applications the source geometry is arbitrary, so the analytic solution to the Helmholtz equation cannot be found. In these circumstances numerical or approximate solutions are sought.

In engineering applications, most vibrating surfaces are of arbitrary shapes. Moreover, the environments are often nonideal such that the radiated acoustic pressure field cannot be solved by any analytic methods, including expansion theories. Therefore, approximate solutions are sought. The Helmholtz equation least-squares (HELS) method [37, 38] offers such approximate solutions to a wide variety of acoustic radiation problems encountered in practice. Note that HELS can not only be used to reconstruct but also to predict the radiated acoustic field emitted by an arbitrarily shaped vibrating body.

The validity challenges came at the joint meetings of the 136th Meeting of the Acoustical Society of America (ASA), the 2nd Convention of the European Acoustics Association (EAA), and the 25th German Annual Conference on Acoustics (DAGA) held in Berlin, Germany, 1999 [58]. The major questions were as follows: “How can the acoustic field on the surface of any non-spherical structure be described by the spherical wave functions?” “Is this a Rayleigh hypothesis in NAH that pushes a solution formulation beyond its region of validity?”

In this chapter we present the general guidelines for setting up measurement to get the desired accuracy and spatial resolution for the HELS-based NAH. There are several parameters that may influence the reconstruction results such as the number of measurement points, standoff distance, measurement aperture size versus source surface area, microphone spacing, and SNR. These parameters are generic for all NAH applications. The strategies for setting up the optimal measurement scheme are basically the same.

Although the HELS method has exhibited a great promise in reconstructing the acoustic fields in both exterior and interior regions, the accuracy in reconstruction for an arbitrarily shaped structure can be unsatisfactory. This is because the expansion based on the spherical waves for an acoustic field generated by on an arbitrarily shaped surface is incomplete.

All traditional NAH techniques are limited to handle cases where sound sources are only on one side of an array of microphones. The reality is much more complicated however. A typical example is the analysis of noise radiation from a vehicle stationed on the chassis dynamometers inside a semi-anechoic chamber. For safety and durability concerns, the surfaces of the chamber cannot be made as acoustically absorptive as they should be. Consequently, the measured acoustic pressures consist of both direct and reflected waves. Also, the dynamometer is generating its own noise, making NAH application and analysis very difficult. To date, vehicle noise is still analyzed by measuring transfer functions between a source and receiver, or by sweeping an intensity probe over a target source surface at close range. The information obtained is often isolated and valid at the measurement locations. These traditional noise diagnosis and analysis processes cannot reveal much insightful information of the root causes of noise and structural vibrations.

In an effort to reduce the overall measurement points associated with BEM-based NAH, Jeon and Ih [111] explore the use of an equivalent source method where the field acoustic pressures are regenerated by point sources distributed inside the real source surface. To this end, Jeon and Ih reformulate the HELS formulations by expanding the spherical Hankel functions and spherical harmonics with respect to multiple points distributed in the interior region of the source surface. Contributions from all equivalent sources are determined by matching the assumed-form solution to the boundary conditions specified on the source surface [112–114] or to the acoustic pressures on the hologram surface [115–122]. The equivalent sources locations can be optimized by using either the natural algorithm [123] or EfI method [124]. The optimal number of expansion terms is obtained by using a spatial filter and regularization scheme. Once the expansion coefficients are specified, the field acoustic pressures are regenerated and taken as the input data to BEM codes, just like CHELS algorithms. In this way, the overall measurement points are greatly reduced.

Most vibrating structures are subject to impulsive or transient force excitations in practice. Oftentimes transient excitations are unknown and therefore the resultant acoustic field cannot be predicted. Even if the excitations are given, prediction of a transient acoustic field produced by an arbitrarily shaped source is very difficult. The scarcity in literature on predicting, not to mention reconstructing a transient acoustic field, is the testimony of how challenging this problem is.

In this chapter we show how to use the HELS method to assess the relative contributions of individual panels of a complex vibrating structure toward SPL at any field point, for example, in diagnosing vehicle interior noise or reducing noise emission from any vibrating machinery. Being able to identify the major contributors of acoustic emission is the first step toward an effective noise reduction of a vibrating structure in engineering applications.