Manual on the Code of the Book Entitled “Acoustic Waves Generated by Parametric Array Loudspeakers”

1 Introduction

This document introduces the usage of the code package, which is a supplementary material for the book “Acoustic Waves Generated by Parametric Array Loudspeakers”. All demos and functions were tested by MATLAB R2022b installed on a personal computer with an AMD Ryzen Threadripper 3960X central processing unit (CPU) with 256 GB of random access memory (RAM).

1.1 Installation

Steps:

1. Download all codes from GitHub: JiaxinZhong/AWPAL

2. Run the script AWPAL.m at first to add subfolders to the path.

2 Demo Scripts and Core Functions

2.1 Direct Integration Method (DIM)

2.1.1 Ultrasound Field

2.1.1.1 General Solution

Function: DIM3D.m

The calculation utilizes the Rayleigh integral reading that [1, Eq. (2.39)]

$$p_i(𝐫)=\frac{{\rho }_0{\omega }_i}{2\mathrm{\pi }\mathrm{i}}{\iint }_{-\mathrm{\infty }}^{\mathrm{\infty }}{v}_{i,z}({𝐫}_{\mathrm{s}})\frac{{\mathrm{e}}^{\mathrm{i}{k}_i\left|𝐫-{𝐫}_{\mathrm{s}}\right|}}{\left|𝐫-{𝐫}_{\mathrm{s}}\right|}{d}^2{𝐫}_{\mathrm{s}}.$$ (1)

2.1.2 Audio Sound Field

2.1.2.1 General Solution

2.1.3 Function: PalDIM3D.m

$$p_{\mathrm{a}}(𝐫)=-\frac{\beta {\omega }_{\mathrm{a}}^2}{4\mathrm{\pi }{\rho }_0{c}_0^4}{\iiint }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{p_1^{\ast }({𝐫}_{\mathrm{v}})p_2({𝐫}_{\mathrm{v}})}{\left|𝐫-{𝐫}_{\mathrm{v}}\right|}{\mathrm{e}}^{\mathrm{i}{k}_{\mathrm{a}}\left|𝐫-{𝐫}_{\mathrm{v}}\right|}{d}^3{𝐫}_{\mathrm{v}}.$$ (2)

2.1.3.1 On-Axis

Demo: PalDIM3D_CircSrc_Axis_Demo.m

Figure 1 is generated by this demo file. The data is stored in file PalDIM3D_CircSrc_Axis_Demo_Uniform.mat.

Figure 1: On-axial SPL (dB) as a function of the axial distance ($z$, m) [2, Fig. 2(d)]. The PAL has a circular uniform profile with a radius of $a=0.1\mathrm{m}$.

Figure 2 is generated by this demo file. The data is stored in file PalDIM3D_CircSrc_Axis_Demo_Focus.mat.

Figure 2: On-axial SPL (dB) as a function of the axial distance ($z$, m) [2, Fig. 2(e)]. The PAL has a focusing profile with a focal distance of $r_{\mathrm{f}}=0.2m$. The radius of the PAL is $a=0.1\mathrm{m}$.

Function: PalDIM3D_CircSrc_Axis.m

Calculate the audio sound field on the axis $\rho =0$ using the DIM. The source profile is assumed to be axisymmetric in the azimuthal direction, i.e., $v_{i,z}({𝐫}_{\mathrm{s}})$ is independent of ${\phi }_{\mathrm{s}}$. The formula used in this function is

$$p_{\mathrm{a}}(\rho =0,\phi ,z)=-\frac{\beta {\omega }_{\mathrm{a}}^2}{2{\rho }_0{c}_0^4}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\int }_0^{\mathrm{\infty }}\frac{p_1^{\ast }({𝐫}_{\mathrm{v}})p_2({𝐫}_{\mathrm{v}})}{\sqrt{{\rho }_{\mathrm{v}}^2+{(z-z_{\mathrm{v}})}^2}}{\mathrm{e}}^{\mathrm{i}{k}_{\mathrm{a}}\sqrt{{\rho }_{\mathrm{v}}^2+{(z-z_{\mathrm{v}})}^2}}{\rho }_{\mathrm{v}}d{\rho }_{\mathrm{v}}d{z}_{\mathrm{v}}$$ (3)

2.2 Spherical Wave Expansion (SWE)

2.2.1 Function: SWE3D_RadialInt.m

Calculate

(4)

where and $r_{>}=\max (r,r_{\mathrm{s}})$.

3 Summary of Equations

3.1 Source Profile

The focusing profile reads

$$v_{i,z}({𝐫}_{\mathrm{s}})=v_0\exp \left(-\mathrm{i}\mathrm{\Re }(k_i)\left|{𝐫}_{\mathrm{f}}-{𝐫}_{\mathrm{s}}\right|\right).$$ (5)

where ${𝐫}_{\mathrm{f}}$ denotes the location of the focal point.

4 Known Issues