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.

中文翻译：

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球形扩展，用于圆形参量阵列扬声器生成的音频声音。

参数阵列扬声器（pal）生成的音频声音的现有非旁轴表达）由于其中有五重积分而难以计算。通过使用球谐函数展开，针对圆形PAL，在准线性近似下，为Westervelt方程开发了严格的解，从而简化了表达式，将其简化为一系列具有角和径向分量不耦合的三重和。角分量由勒让德多项式确定，而径向分量是一个包含球形贝塞尔函数的积分，并且会迅速收敛。与在整个空间上进行直接积分相比，球面膨胀是严格，精确且可以有效计算的。仿真表明，所提出的表达式可以以比现有表达式至少快15倍的速度获得相同的准确结果。