Supplementary notes on the Gaussian beam expansion

Abstract

The letter provides alternatively a simple way of computing the Fresnel field integral, a further extension to the Gaussian-beam expansion. The zeroth-order Bessel function of the first kind is expanded into an approximate sum of Gaussian functions. The field integral is then expressible in terms of these simple functions. The approach is useful in treatment of the field radiation problem for a large and important group of piston sources in acoustics. As examples, the calculation results for the uniform and the simply-supported piston sources are presented, in a good agreement with those evaluated by numerical integration.

A Novel Algorithm for the Sound Field of Elliptically Shaped Transducers

“…( 8) and ( 9) as the known results and apply them to treat again the field problem for the uniform elliptical transducer. Similarly in the previous work, [22,25] if the cosine function appearing in the integrand of Eq. ( 6) is expressible in terms of Gaussian functions, then Eq.…”

“…Much advantage of our approach is illustrated by the above and other examples. [22,25] It is worth pointing out that the present method may be applied to the other circumstances. For example, the weakly focused sound field by an acoustical lens may be included as a general case of the above examples.…”

A Novel Algorithm for the Sound Field of Rectangular-Shaped Transducers

“…It avoids the onerous computation in computer optimization procedure. A similar deduction [23] obtains a Gaussian expansion of the cosine function in the form…”

“…( 12) while do very well for the other functions. [3,10,[23][24][25]27] Actually, the 'worst' among these four sets leads to a very great error of 0.5 (1/2). However, carefully we observe that this error of 0.5 is extremely uniformly distributed and nearly independent of the variable 𝑥 in the interval from 𝑥 = 0 to about 25.…”

“…( 10), into Eq. ( 6), we obtain the expansion function 𝐺 𝑛 or 𝐺(𝜈) of the field [23,27] 𝐺 𝑛 (𝜉, 𝜂; 𝐵 𝑘 , 𝐵 𝑗 ) = 𝐺(𝜉, 𝜂; 𝐵 𝑘 , 𝐵 𝑗 ; 𝜈)…”