Technical Memorandum 527249322-3

Sound power determination using Fourier-transform representations of Rayleigh’s integrals
Publication date
2026-03-06
Document type
Bericht
Author
Organisational unit
Language
English
DDC Class
530 Physik
534 Schall und verwandte Schwingungen
621 Angewandte Physik
Keyword
Rayleigh integral
Fourier Transform
Radiation of sound
Sound intensity
Free-field Green's function
Transform representation
Hankel Transform
Integral representations
Weyl's identity
Abstract
For the project Near-field Acoustical Holography - a new sensor concept for methods of active noise reduction, it is necessary to formulate the radiated sound power of a planar source in the transform domain. The transform representation is commonly derived by applying the Fourier transform to the inhomogeneous Helmholtz equation. After all, one ends up with Weyl’s identity, which is derived here first. Second, sound-power formulations based on Fourier transformed particle velocities and sound pressures are derived.
Description
Commonly, the sound power of an acoustic source in free field or in half space can be determined by integrating the time-averaged sound intensity over a surface which encloses the source. Therefore, the acoustic quantities particle velocity and sound pressure need to be measured, which can be troublesome. In a different approach just one acoustic quantity is measured, and the other is calculated using an acoustic mathematical model. The easiest model is the plane-wave model, which can be applied to sound-pressure measurements in the far field to obtain the particle velocity. This technique requires an appropriate facility like an anechoic chamber. For a vibrating plate in a rigid baffle, the structural velocity—assumed equal to the normal surface particle velocity—is measured, and the sound pressure is determined using Rayleigh's velocity-based integral equation. The main advantage of this technique is that measurements can be conducted in the near field of the planar sound-radiating structure without the need of any microphone. Nevertheless, free-field conditions are not necessary, if the effect of the fluid on the structure is negligible. In a further step, Rayleigh’s integral, as the acoustic model in use, can be replaced by its Fourier-transform representation. This requires the free-field Green’s function to be reformulated. For a two-dimensional source the resulting integral equation is fourfold. One may wonder why this method is preferred over the traditional Greens function in this context. The reasoning behind this choice is the connection to the Fraunhofer diffraction. The Fourier transform of a source function of a plane source field is proportional to the sound pressure radiated into the far field. When the radiated sound pressure in the far field is known, the sound power can be determined. Thus, an implementation using a Fast Fourier Transform algorithm to determine the radiated sound pressure and the radiated sound power is possible.
Version
Author's original
Access right on openHSU
Open access