Narrow Results

The problem examined is that of a localized energy source which undergoes planar motion along the surface of a reactive-diffusive medium. This is representative of a laser beam that is moving across the flat surface of a combustible material during a cutting, welding or heat treating process. The mathematical model for this situation is a heat equation in two-dimensions with a nonlinear source term, which is localized around a reference point that is allowed to move. Results are derived that indicate the roles played by the size, strength and motion of the localized source in determining whether or not a blow-up occurs.

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.

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 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.