Predicting the frequency spectrum from sources on non-circular trajectories is challenging because closed-form representations are scarce. This study derives Mathieu-function-based solutions for harmonic sources moving on elliptical paths. First, the two-dimensional sound field of line sources traveling along an ellipse is analyzed for a single revolution and for periodic motion over infinitely many revolutions. The resulting frequency spectrum is expressed in terms of Mathieu functions. As the eccentricity of the ellipse decreases and the orbit approaches circularity, the solution converges to that of the corresponding circular orbit. The discrete line spectrum of the periodic case coincides with the continuous spectrum obtained for a single revolution. In the second part, a monopole moving along an elliptical helix is investigated. The three-dimensional spectral sound field generated by the source is represented by a series expansion in Mathieu functions. The series solution shows good agreement with that derived from the integral formulation of the Cartesian Convolution Integral over a range of source frequencies and angular velocities. As the cross section of the elliptical helix approaches a circular shape, the computed sound field converges to the known solution for the circular helix. The results enable sound-field prediction for imperfect rotating sources and propellers.

In a translation-invariant environment, the three-dimensional sound field can be determined through spatial Fourier transform by superimposing two-dimensional sound fields. This technique is commonly referred to as the 2.5D method, due to the dimensional reduction that takes place. If the sound source is not stationary but moves along the axis of invariance, the calculation of the sound field generally becomes more complex. However, if a harmonically radiating point source moves uniformly at a constant speed along the invariance axis, the opposite is true, and the calculation is significantly simplified. Motivated by the form of the Green’s function in the free field, the so-called separation of variables, or product approach, reduces the problem to a purely two-dimensional one, the general solution of which is referred to in this work as the Product-Doppler formula. Constructing a Fourier integral over the wavenumber domain along the invariance axis is no longer necessary. It is shown that the Product-Doppler formula can be used to solve both interior and exterior problems. The sound field generated by a moving source inside a cylindrical tunnel, and the sound generated by an exterior moving source and scattered from an absorbing cylinder are analyzed. The complex problem of sound diffraction caused by a source moving along the edge of a wedge or a screen is studied in detail. A comparison with results from the literature shows strong agreement.

This paper presents a boundary element method for calculating the sound pressure level on and above a ground-board placed on an infinite ground plane with constant impedance. This method uses the Green's function for the Helmholtz equation in a half-space bounded by an infinite impedance plane and therefore avoids the complications which occur when modeling the ground as a finite impedance plane (which results in diffraction from the edges of the ground plane). Two formulations are presented: the first one models a finite thickness ground-board, whilst the second one models a zero-thickness ground-board with the upper surface coplanar with the ground plane. The method is validated against an equivalent finite element method simulation and example calculations are presented for typical ground-board designs used for outdoor noise measurements on different ground impedance planes and incidence angles. It is shown that the sound pressure level on the upper surface of a ground-board placed on a non-rigid ground can vary significantly and that this variation is dependant on the ground impedance, angle of incidence and frequency. These findings are consistent with previous investigations and highlight the fact that noise measurements made using ground-board mounted microphones on a non-rigid ground are dependant on the impedance of the surrounding ground surface. The method presented in this paper could be used to develop corrections to straightforwardly account for this effect.

Wavefields that are related to a source point with complex coordinates, which are not all real numbers, meanwhile have a long history. They have first been applied in electromagnetic theory, and subsequently also in acoustics. For example, such wavefields exhibit a directivity whose degree can be controlled by the imaginary part of the source point. The wavefields in question are obtained if in the fundamental solution for a real source point this point is replaced by its complex counterpart. With respect to the complex source point, sometimes it is mystically spoken about the associated complex delta function. In a rigorous way, however, only an equivalent source, which is defined on the real space, could be derived. In the present paper, we show that a wavefield, which is related to a complex source point and solves the Helmholtz equation, can be extended to an ultradistributional solution of the Helmholtz equation with the complex delta function as right-hand side. This result demonstrates that the connection between those wavefields and the complex delta function is not only a formal one. Our analysis is performed in one, two and three space dimensions.

The sound field of a harmonically radiating monopole moving along arbitrary trajectories in three-dimensional space is studied in the frequency range and represented as a convolution integral in Cartesian coordinates. The observation time of the motion of the source can be finite or infinite. This convolution integral is referred to as the "Cartesian Convolution Integral " and is applied to point sources moving on special orbits described by circular and conical helices. For such helical orbits, an alternative approach in cylindrical or spherical coordinates leads to closed form expressions in the form of infinite series for the frequency spectra. The comparison of the results of the Cartesian Convolution Integral with those of the series expansion shows a very good agreement. Finally, the Cartesian Convolution Integral is used to calculate the spectral sound field of a source moving along an elliptical helix.

In this paper, the sound field due to a line source moving above a homogeneous semi-infinite fluid medium emitting an arbitrary time signal is deduced. The source moves with a constant velocity parallel to the flat interface at a fixed height. The expression for the velocity potential is obtained from the impulse response of the stationary line source by including the motion of the source in the source density function. The resulting sound field is expressed in terms of convolution integrals. The correctness of the formulas is verified by comparing them with the results known for the case of a continuous harmonic line source. Two examples, the sound field due to a delta pulse and a Gaussian pulse, are computed and presented. Finally, the relation between the solutions obtained here and the ones obtained when the source moves above an absorbing impedance surface is shown.

It was shown in previous publications that the impulse response of an infinite line source, as well as a point source in the three-dimensional space above an absorbing plane, can be expressed in closed form. However, line sources of a finite length are much more practical. In this study, the impulse response of a finite line source above the absorbing ground is derived and presented in a closed form, which is the main result of the present work. The corresponding analytical solution is composed of three distinct signals. Those are the direct signal, the mirrored signal, and the "pressure tail." The direct signal is the line-of-sight signal. The mirrored signal is the signal emitted from the image source. It is weighted with the reflection coefficient of the infinite line source, which is a surprising mathematical result. The direct signal and the mirrored signal are both of finite duration. However, the third signal-the pressure tail-starts at the end of the mirrored signal and continues indefinitely for a finite surface impedance. In general, this last signal is relatively weak; it can be considered as the reverberation of the source caused by the infinite plane.

Journal of Theoretical and Computational AcousticsVol. 27, No. 01, 1902001 (2019) No AccessEditorial: Special Issue on Advances in Finite and Boundary Element MethodsManfred Kaltenbacher, Steffen Marburg, Martin Ochmann, and Rafael PiscoyaManfred KaltenbacherVienna University of Technology, Austria, Steffen MarburgTechnical University of Munich, Germany, Martin OchmannBeuth University of Applied Sciences, Germany, and Rafael PiscoyaBeuth University of Applied Sciences, Germanyhttps://doi.org/10.1142/S2591728519020018Cited by:0 Next This article is part of the issue: Special Issue on Advances in Finite and Boundary Element Methods (II); Edited by M. Kaltenbacher, S. Marburg, M. Ochmann and R. Piscoya AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail FiguresReferencesRelatedDetails Recommended Vol. 27, No. 01 Metrics History PDF download