Wu S.F. The Helmholtz Equation Least Squares Method. For Reconstructing and Predicting Acoustic Radiation

Springer, 2015. — 243 pp.This is the first and only book on the HELS (Helmholtz equation least-squares) method. While the original contract with Springer to write this book was signed in 2003, it took 10 years for me to actually sit down and complete the writing. This is because during the past decade I have been heavily involved in research projects and teaching, which has constantly distracted me from fulfilling my obligation with the publisher. On the other hand, we have seen tremendous growth and expansion in the HELS theory. Its applications have been extended to many areas that have not been explored such as hybrid near-field acoustical holography (NAH), transient NAH, and NAH-based panel acoustic contributions analyses. Hence, in this sense it was good that I did not write this book 10 years ago. Of course, the HELS method is being further developed and expanded to new frontiers, including reconstruction of the aerodynamically generated sound field generated by an aircraft jet engine and realization of super resolution in discerning acoustic sources by taking input data in space at a rate less than the Nyquist sampling requirement. These new developments will be included in the second edition of this book.
What makes the HELS method unique is its simplicity in mathematical form, efficiency in numerical computation, and flexibility in engineering applications. The idea of using an expansion of certain basis functions to approximate the acoustic field can be traced back to the beginning of the last century. The most famous example was given by Lord Rayleigh to depict the acoustic field scattered from a corrugated surface. The differences and interrelationships between the Rayleigh series and the HELS method are explained in great detail in this book.
The underlying principles of the HELS method are strikingly different from the traditional Fourier acoustics and boundary element method (BEM)-based NAH. The Fourier acoustics-based NAH relies on the Fourier transforms and requires the source surface to contain a level of constant coordinate such as an infinite plate, an infinite cylinder, and a sphere. Moreover, the source must be in free space without the presence of any other source or boundary surface. Although the BEM-based NAH is suitable for arbitrarily shaped surfaces, it also requires the source to be in a source-free region. In addition, both of them require that the hologram surface enclose the entire source surface. If these conditions are not met, then they are invalid theoretically. This makes it difficult for these methods to be adopted in engineering applications because a source-free region is nonexistent and oftentimes a source surface cannot be enclosed by a measurement surface in reality.
In contrast to the traditional NAH implementations, the HELS method does not seek an analytic solution to the acoustic field produced by an arbitrarily shaped structure that cannot be found anyway. Rather, it attempts to obtain the best approximation of an acoustic field through the expansion of certain basis functions. Therefore it significantly simplifies the complexities of the reconstruction process, yet still enables one to acquire a good understanding of the root causes of different noise and vibration problems that involve arbitrarily shaped surfaces in non-free space using much fewer measurement points than both Fourier acoustics and BEM-based NAH do. The examples given in this book illustrate that the HELS method may potentially become a practical and versatile tool for engineers to tackle a variety of complex noise and vibration issues in engineering applications.
Since 2001, I have developed a new course on ME7460 Advanced Acoustic Radiation for graduate students in the Department of Mechanical Engineering at Wayne State University. The main objective of this course is for students to learn the state-of-the-art technology, namely, NAH to diagnose various noise and vibration problems encountered in practice. The major parts of this book are based on my class notes plus new developments in the HELS method accumulated over the past decade. While attending various acoustics conferences sponsored by professional societies such as the Acoustical Society of America, American Society for Mechanical Engineers, and Society for Automobile Engineering, I often have people asking me questions about the HELS method and its implementation. I am happy to report that finally there is a formal textbook on this subject that outlines in great detail this methodology, its implementation steps, and guidelines in practice. In particular, I have provided many examples on how to reconstruct and predict the acoustic fields emitted from different types of sources, and illustrated the intermediate steps in the derivations of various formulations. I sincerely hope that this textbook can serve as a resourceful reference, helpful guidance, and valuable tool for students, engineers, practitioners, and users to understand the HELS-based NAH, how it can be implemented in practice, and why.The Spherical Wave Functions
The Helmholtz Equation Least-Squares Method
Validity of the HELS Method
Implementation of the HELS-Based NAH
Combined Helmholtz Equation Least-Squares (CHELS) Method
Hybrid NAH
Equivalent Sources Using HELS
Transient HELS
Panel Acoustic Contribution Analysis Using HELS