Abstract The analysis of planar wave resonances in ducts of variable cross-section is of importance in many areas of acoustics such as horn theory, the design of musical instruments or the numerical generation of voice. An interesting problem is that of modifying the tube geometry and physical parameters to move its resonances to desired target values. In vowel production, for instance, one can properly adjust the length, cross-sectional area and wall admittance of the vocal tract (VT) to change the frequency of a resonance or to bring together or set apart a group of them for achieving some natural voice effects. An optimization strategy can be implemented to that purpose, which induces small sequential variations of the VT parameters according to sensitivity functions, until one obtains the desired resonance values. The sensitivity functions are usually derived from the work done by variations in the non-linear radiation pressure within the duct. In this paper we will show that, elegant as it may be, there is in fact no need of non-linear phenomena to determine appropriate sensitivity functions. These can readily be obtained from first order modal perturbation analysis. To demonstrate that, we transform the three-dimensional (3D) problem of voice generation into a one-dimensional one (1D), driven by a Webster-type equation. Despite of the latter being 1D, we discretize it using the finite element method (FEM) to benefit from its weak formulation to deduce expressions for the sensitivity functions. The latter are recovered from a perturbation modal analysis of the discrete Webster equation for VT area, length and wall admittance variations. Several examples concerning vowel to vowel transformations are presented to illustrate the potential and limitations of the method. The herein proposed 1D approach for resonance tuning could be easily coupled to 3D FEM codes for expressive vowel production in the future.

中文翻译：

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来自模态扰动分析而不是非线性辐射压力的声道声学共振调谐

摘要 变截面管道中平面波共振的分析在声学的许多领域都很重要，例如喇叭理论、乐器设计或声音的数值生成。一个有趣的问题是修改管几何形状和物理参数以将其共振移动到所需的目标值。例如，在元音的产生中，可以适当调整声道的长度、截面积和壁导纳（VT）来改变共振的频率，或者将一组它们聚集或分开，以达到一些自然的效果。语音效果。可以为此目的实施优化策略，该策略根据灵敏度函数引起 VT 参数的小的连续变化，直到获得所需的谐振值。灵敏度函数通常来自管道内非线性辐射压力变化所做的功。在本文中，我们将表明，尽管可能很优雅，但实际上不需要非线性现象来确定适当的灵敏度函数。这些可以很容易地从一阶模态扰动分析中获得。为了证明这一点，我们将语音生成的三维 (3D) 问题转换为由韦氏方程驱动的一维 (1D) 问题。尽管后者是一维的，但我们使用有限元方法 (FEM) 将其离散化，以从其弱公式中受益，以推导出灵敏度函数的表达式。后者是从 VT 面积、长度和壁导纳变化的离散韦伯斯特方程的微扰模态分析中恢复的。介绍了有关元音到元音转换的几个示例，以说明该方法的潜力和局限性。本文提出的用于共振调谐的 1D 方法可以很容易地耦合到 3D FEM 代码，以便将来产生富有表现力的元音。