The near field, Westervelt far field, and inverse-law far field of the audio sound generated by parametric array loudspeakers

Abstract

The near and far fields of traditional loudspeakers are differentiated by whether the sound pressure amplitude is inversely proportional to the propagating distance. However, the audio sound field generated by a parametric array loudspeaker (PAL) is more complicated, and in this article it is proposed to be divided into 3 regions: near field, Westervelt far field, and inverselaw far field. In the near field, the audio sound experiences strong local effects and an efficient quasilinear solution is presented. In the Westervelt far field, local effects are negligible so that the Westervelt equation is used, and in the inverse-law far field a simpler solution is adopted. It is found that the boundary between the near and Westervelt far fields for audio sound lies at approximately a 2 /λ -λ/4, where a is transducer radius and λ is ultrasonic wavelength. At large transducer radii and high ultrasonic frequencies, the boundary moves close to the PAL and can be estimated by a closed-form formula. The inverse-law holds for audio sound in the inverselaw far field and is more than 10 meters away from the PAL in most cases. With the proposed classification, it is convenient to apply appropriate prediction models to different regions.

A cylindrical expansion of the audio sound for a steerable parametric array loudspeaker

“…14,15 It has also been demonstrated that the predictions obtained using the Westervelt equation are accurate enough except for observation points close to the radiation surface of the PAL. 14,15 After using the successive method, 16 the quasilinear solution of the audio sound based on the Westervelt equation can be seen as the radiation from a virtual volume source with the source density proportional to the product of the ultrasound pressure. 17,18 The expression of the solution is therefore a threefold integral over the three dimensional space.…”

“…8,11 However, the ultrasound beams are assumed to be exponentially attenuated in each direction, which is not true in reality because of the complexity of the ultrasound beams in the near field, where the majority of the nonlinear interactions take place. Furthermore, the far field is found to be more than 10 meters away from a PAL when its size is larger than 0.04 m, 15,28 which is too far when compared to real applications. Therefore, differences between predictions and measurements continue to be observed, even for the sound pressure 4 meters away from the PAL.…”

“…2 This process can be modelled using a second-order nonlinear wave equation if cubic and higher order terms are neglected. [13][14][15] After neglecting the Lagrangian density, the equation can be further simplified to give the Westervelt equation characterizing the local effects, which is often used instead because it is easier to solve. 14,15 It has also been demonstrated that the predictions obtained using the Westervelt equation are accurate enough except for observation points close to the radiation surface of the PAL.…”

“…[13][14][15] After neglecting the Lagrangian density, the equation can be further simplified to give the Westervelt equation characterizing the local effects, which is often used instead because it is easier to solve. 14,15 It has also been demonstrated that the predictions obtained using the Westervelt equation are accurate enough except for observation points close to the radiation surface of the PAL. 14,15 After using the successive method, 16 the quasilinear solution of the audio sound based on the Westervelt equation can be seen as the radiation from a virtual volume source with the source density proportional to the product of the ultrasound pressure.…”

“…17,19,23,24 However, since the paraxial approximation for the ultrasound is retained in the non-paraxial model, it is still inaccurate when the sound beams of a phased array PAL are steered to a larger angle. Recently, a spherical expansion has been developed to give a rigorous simplification of the fivefold integral into a threefold series consisting of uncoupled radial and angular components, 15,18 which converges more than 100 times faster. Although no paraxial approximations are assumed in the spherical expansion, it can only be applied to the PAL with a velocity profile that is axisymmetric about the radiation axis.…”