Modal perturbation analysis instead of nonlinear radiation pressure to derive the area sensitivity function for resonance tuning in an axisymmetric duct with variable cross-section

Axisymmetric ducts with variable cross-section are of importance in many acoustic problems ranging from horn theory to vocal tract acoustics. Webster's equation is commonly used to describe their performance in the plane wave propagation regime. In some problems, mostly related to voice generation, one is interested in modifying the area of the duct cross-sections to adjust the frequency of a resonance. For instance, one may want to increase its value, or to bring a group of resonances closer together, to emulate effects that occur in natural voice production. To that goal, an optimization iterative process can be followed in which the cross sections are subsequently changed, according to an area sensitivity function, until the resonances of the duct are placed at the target position. Traditionally, the area sensitivity functions have been derived from the non-linear radiation pressure inside the duct. In this work we demonstrate there is no need to resort to such non-linear phenomenon because the same result can be deduced from a first order modal perturbation analysis of the duct eigenfrequencies. After proving that, we present some simulations in the framework of expressive vowel production.