Zhong Jiaxin, Kirby Ray, Qiu Xiaojun
Centre for Audio, Acoustics and Vibration, Faculty of Engineering and Information Technology, University of Technology Sydney, New South Wales 2007, Australia.
J Acoust Soc Am. 2020 May;147(5):3502. doi: 10.1121/10.0001261.
The existing non-paraxial expression of audio sounds generated by a parametric array loudspeaker (pal) is hard to calculate due to the fivefold integral in it. A rigorous solution of the Westervelt equation under the quasilinear approximation is developed in this paper for circular PALs by using the spherical harmonics expansion, which simplifies the expression into a series of threefold summations with uncoupled angular and radial components. The angular component is determined by Legendre polynomials and the radial one is an integral involving spherical Bessel functions, which converge rapidly. Compared to the direct integration over the whole space, the spherical expansion is rigorous, exact, and can be calculated efficiently. The simulations show the proposed expression can obtain the same accurate results with a speed of at least 15 times faster than the existing one.
由于其中存在五重积分，参数阵列扬声器（PAL）产生的音频声音的现有非傍轴表达式难以计算。本文通过使用球谐展开，针对圆形PAL在准线性近似下对韦斯特维尔特方程进行了严格求解，将表达式简化为一系列具有非耦合角向和径向分量的三重求和。角向分量由勒让德多项式确定，径向分量是一个涉及球贝塞尔函数的积分，收敛速度很快。与在整个空间上的直接积分相比，球谐展开是严格、精确的，并且可以高效计算。仿真结果表明，所提出的表达式能够获得与现有表达式相同的精确结果，且速度至少快15倍。
