# Numerical and Experimental Studies on Inclined Incidence Parametric Sound Propagation

Authors: Haisen Li; Jingxin Ma; Jianjun Zhu; Baowei Chen
Journal: Shock and Vibration
 (2019)

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## Abstract

The Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation has been widely used in the simulation and calculation of nonlinear sound fields. However, the accuracy of KZK equation reduced due to the deflection of the direction of the sound beam when the sound beam is inclined incidence. In this paper, an equivalent sound source model is proposed to make the calculation direction of KZK calculation model consistent with the sound propagation direction after acoustic refraction, so as to improve the accuracy of sound field calculation under the inclined incident conditions. The theoretical research and pool experiment verify the feasibility and effectiveness of the proposed method.


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## Body

## 1. Introduction

Due to the nonlinear effect of the medium, when the parametric acoustic array transmitted two similar frequency primary sound waves, the difference, sum, and harmonic frequency waves will be generated in the radiated sound field [1–3]. The parametric array technology can generate broadband, high directivity, and side-lobe suppressed low-frequency sound beams by using a small aperture array [4–6]. This parametric sound feature makes the parametric sonar more suitable for high-directivity sound beam noninvasive measurement and high-resolution sub-bottom profiling [7–9]. When the parametric sound is obliquely irradiated into different media, the acoustic propagation path changes due to the medium parameters, which has a certain influence on the application of the parametric sound. Such as the direction of sound propagation will be deflected when the high-intensity focused ultrasound used in the noninvasive treatment of solid tumors and inversion error of shallow depth parameter due to inclined seabed.As the most accurate mathematical equation describing the nonlinear sound field, the KZK equation is the most important mathematical tool for the study of acoustic propagation in various media layers [10]. For the problem of parametric sound transmission in near-field, finite-difference frequency-domain method (FDFD) and finite-difference time-domain method (FDTD) were proposed to calculate parametric sound propagation [11–14]. Several numerical and experimental results for the parametric sound field propagation characteristics based on circular parametric array were found [15–18]. However, most of the parametric array equipment used in underwater acoustics detection area is rectangular due to production process conditions. In [19], the KZK equation in the Cartesian coordinate system under the rectangular aperture parameter sound source condition is proposed, and both natural directivity and phase control parametric sound source in free field were numerically simulated. Vander Meulen and Haumesser constructed the model of finite amplitude sound beam propagation for the vertical incident baffle, and the nonlinearity coefficient of the aluminum plate medium has been measured and evaluated by the in-depth analysis of the second harmonic in the sound field [20]. Fujisawa and Asada numerically and experimentally examined the parametric sound field when the sound beams through different fluid layers vertically and proposed a nonlinear parametric sound enhancement method by placing a highly nonlinear coefficient of fluid layer in front of the parametric array [21]. However, the numerical model of the parametric sound beam through the medium with different angles research is rarely carried out.In this paper, under a rectangular aperture parameter source, based on the KZK equation, a theoretical model of the inclined incidence parametric sound field is proposed and constructed. The parametric sound field after inclined incidence of aluminum plates and different media is studied, respectively. Finally, the results of the aluminum plate were verified by the pool test.


## 2. Theory Equation and Geometry Model

### 2.1. Governing Equation

As an extended form of the Burgers model, the KZK equation is the most accurate equation model for calculating the paraxial region of a nonlinear sound field. The KZK equation defined in the Cartesian rectangular coordinates can be described as follows [19]:(1)∂2p∂z∂t′=c02∂2∂x2+∂2∂y2p+δ2c03∂3p∂t′3+β2ρ0c03∂2p2∂t′2,where p is the sound pressure, c0 is the sound velocity, ρ0 is the medium density, δ is the diffusivity of sound, β is the nonlinearity coefficient, and t′=t−z/c0 is the time delay at z with t being the time.By using the Lee and Hamilton method [12], equation (1) can be transformed into nondimensional form by coordinate transformation to improve the computational efficiency. The transformation parameters are as follows:(2)σ=zR0,ux=xa1+σ,uy=yb1+σ,P=p1+σp0,τ=ω0t′−k0x2+y22R01+σ.The dimensionless form of the KZK equation can be written as(3)∂P∂σ=Rπ1+σ2∫−∞τ∂2P∂ux2+∂2P∂uy2dτ′+α0R0∂2P∂τ2+NP1+σ∂P∂τ,where α0=δω02/2c03 is the absorption coefficient, R0=4ab/λ is the Rayleigh distance, N=βp0ω0R0/ρ0c03 is the nonlinear coefficient in dimensionless form, p0 is input sound pressure, ω0 is the angular frequency, k0 is the wave number, R = b/a is the aspect ratio, and the parametric source dimension is 2a×2b (a≥b).


### 2.2. Equivalent Source Incident Numerical Model

When the parametric sound field is incident obliquely into another medium, the beam direction will deviate due to the acoustic refraction effect. At this time, an equivalent parametric sound source coordinate system is established such that the acoustic refraction parametric sound field is considered to be generated by the equivalent parametric sound source. As shown in Figure1(b), the new transformed coordinate system (ux1,uy1,σ1) is introduced in region Ω1 to be consistent with the direction of sound line propagation. The coordinates of the center position of the inclined interface are (0, σ00=z0/R0) and (0, σ01), respectively, in the two coordinate systems.Figure 1
Schematic diagram of nonlinear acoustic propagation under inclined incidence conditions. (a) Cartesian rectangular coordinates. (b) Dimensionless transformed coordinate system.
(a)
(b)Under the transformed coordinate system constructed under the condition of equation (2), the coordinate parameter scale is related to the Rayleigh distance. Therefore, not only the acoustic parameters are different in new media but also the dimensionless coordinate system’s coordinate scale. The relationship between the two coordinate systems in the axial direction is as follows:(4)Δσ1σ1=Δσ0R0/R1cosαR0/R1σ0−σ00cosα+σ01,where R1 is the Rayleigh distance in the new media and α=θ0−arcsinc1/c0⋅sinθ0 is the angle of acoustic refraction deflection.As shown in Figure1(b), any point x,y,z on the inclined plane can be transformed into two nondiffusion coordinate systems with equation (2), respectively, which are(5)ux0=−R0tanθ⋅σ0−σ00a1+σ0,ux1=−R1tanθ−α⋅σ1−σ01a1+σ1.Analyzing the relationship ofux0 and ux1 by equation (4), we can get(6)ux1ux0=R1tanθR0tanθ−α⋅σ1−σ01σ0−σ00⋅1+σ01+σ1=R1tanθR0tanθ−α⋅1+σ0R1/R01+σ01cosα−σ00+σ0.To simplify the complexity of the model coordinate system transformation, it is assumed that the coordinates of the points on inclined plane are linear in the two coordinate systems, in which the value ofσ01 can be set as(7)σ01=R0R1cosα1+σ00−1.The relationship between the coordinate-scale transformation in the new coordinate system and the original coordinate system on inclined plane is(8)Δux1Δuy1Δσ1=R1tanθR0tanθ−α⋅Δux0R1tanθR0tanθ−α⋅Δuy0R0R1cosα⋅Δσ0.The transformation relationship between the two coordinate systems on inclined plane is(9)ux1uy1σ1=R1tanθR0tanθ−α⋅ux0R1tanθR0tanθ−α⋅uy0R0/R1σ0−σ00cosα+σ01.Therefore, the coordinate system (ux1,uy1,σ1) in Figure 1(b) can be transformed back to the Cartesian coordinate system (x, y, z) in Figure 1(a) with(10)xyz=acosα0−R1sinα0b0sinα0R1cosα1+σ1ux11+σ1uy1σ1−σ01+00z0.


### 2.3. Boundary Condition

Figure1 shows the sound field computational domain, which is set as 10 × 10 × 100 in the ux, uy, and σ directions, respectively. Around the computational domain, an absorbing boundary condition is imposed to minimize the acoustic reflections from the edges. The entire computational sound field is divided into two water areas (Ω0 and Ω1) by the inclined plane. The inclined plane section is perpendicular to the x-o-z plane and is at an angle of θ to the plane x-o-y, and the center of the section is at a horizontal distance z0 from the parametric source.For the sound pressure calculation, in natural parametric sound field, the sound source onz = 0 is used as an initial condition prescribed by two different primary frequencies as follows:(11)P0=p0cos2πf1t+cos2πf2t,where f1=f0−fd/2 and f2=f0+fd/2 are the two primary frequencies of parametric source, f0 and fd are the center and difference frequencies of the primary waves, respectively, and t is the time. After the parameter transformation, two different primary frequencies in simulation calculation are as follows:(12)P0ux,uy=p0cos2πf1τ+πux2+uy24+cos2πf2τ+πux2+uy24⋅Hux,uy,where(13)Hux,uy=1,ux≤1,uy≤R,0,other.Since the KZK equation describes only the unidirectional propagation of the nonlinear parametric sound field, the reflected sound beam is not considered in the actual calculation. At the water-ethanol boundary, there is a thin rubber film with the same acoustic impedance as water to prevent any mixing of ethanol and water. And, the sound pressure transmission coefficient of the water-ethanol boundary and the water-aluminum boundary, respectively, is as follows:(14)TWE=2ρEcE/cosθρEcE/cosθ+ρWcW/cosθ−α,TWA=2ZWZA2ZWZAcoskAlcosθA+jZW2+ZA2sinkAlcosθA,where ZW=ρWcW/cosθ, ZE=ρEcE/cosθ−α, θA is the refraction angle in the aluminum plate, kA is the wave number in aluminum, and l is the thickness of the thin aluminum plate. The subscripts E, A, and W represent ethanol, aluminum, and water, respectively. The parameters used in this study are given in Table 1.Table 1
Density and sound speed values.
Material
Water
Ethanol
Aluminum
Densityρ (kg/m3)
1000
785
2700
Speedc (m/s)
1473
1168
6300
Zmat (Rayls)
1.47
×
10
6
9.17
×
10
5
1.7
×
10
7
Nonlinear coefficientβ
3.5
6.9
—
Absorption coefficient (10−13f2dB/m)
2.17
4.78
—


### 2.4. Numerical Algorithm

In the nonlinear sound field numerical analysis, the KZK equation was solved with the finite-difference time-domain method (FDTD) for the acoustic field from a rectangular aperture parameter array source. The staggered grid system at inclined plane boundary is described in Figure2. The four terms on the right side of equation (1) represent diffraction in ux, uy axial directions, attenuation, and nonlinearity, respectively. Based on an orthogonal spatial grid, each term of the equation can be calculated as independent equation. The discretized forms of these equations are expressed as(15)Pm,nj,k+1−Pm,nj,kΔσk=Rπ1+σ2∫−∞τPm+1,nj,k+1−2Pm,nj,k+1+Pm−1,nj,k+12Δux2dτ′,(16)Pm,nj,k+1−Pm,nj,kΔσk=Rπ1+σ2∫−∞τPm,n+1j,k+1−2Pm,nj,k+1+Pm,n−1j,k+12Δuy2dτ′,(17)Pm,nj,k+1−Pm,nj,kΔσk=α0R0Pm,nj+1,k+1−2Pm,nj,k+1+Pm,n−1j−1,k+1Δτ2,(18)Pm,nj,k+1=Pm,nj,k1−NPm,nj+1,k−Pm,nj,k/Δτln1+Δσk/1+σk,Pm,nj,k≥0,Pm,nj,k1−NPm,nj,k−Pm,nj−1,k/Δτln1+Δσk/1+σk,Pm,nj,k<0,where m, n, and k are the spatial coordinate indexes in the ux, uy, and σ axial directions, respectively, and j is the time step. For example, Pm,nj,k=Pm⋅Δux,n⋅Δuy,σk,j⋅Δτ.Figure 2
Finite-difference method staggered grid system. (a) Staggered grid system at inclined boundary. (b) Staggered grid system in x-directions represent diffraction.Consider the effect of the slope boundary on the calculation of the discretization equation, as shown in Figure2. The calculation parameters are different on both sides of the slope boundary in equations (16)–(18), but the calculation conditions are consistent with the free field. However, in equation (15), the inclined boundary divides the ux-direction scattering calculation into two parts, as shown in Figure 2(b), and the approximate inclined boundary is(19)Pm,nj,k−Pm,nj,k−1Δσk=Rπ1+σ2∫−∞τPm+1,nj,k/T−2Pm,nj,k+Pm−1,nj,k2Δux2dτ′.When theux-direction diffraction calculation is completed, the initial condition of sound pressure Pmj,k at inclined incidence boundary, as shown in Figure 2(a), can be defined as follows:(20)Pmj,k=Pmj,k⋅T.


## 2.1. Governing Equation

As an extended form of the Burgers model, the KZK equation is the most accurate equation model for calculating the paraxial region of a nonlinear sound field. The KZK equation defined in the Cartesian rectangular coordinates can be described as follows [19]:(1)∂2p∂z∂t′=c02∂2∂x2+∂2∂y2p+δ2c03∂3p∂t′3+β2ρ0c03∂2p2∂t′2,where p is the sound pressure, c0 is the sound velocity, ρ0 is the medium density, δ is the diffusivity of sound, β is the nonlinearity coefficient, and t′=t−z/c0 is the time delay at z with t being the time.By using the Lee and Hamilton method [12], equation (1) can be transformed into nondimensional form by coordinate transformation to improve the computational efficiency. The transformation parameters are as follows:(2)σ=zR0,ux=xa1+σ,uy=yb1+σ,P=p1+σp0,τ=ω0t′−k0x2+y22R01+σ.The dimensionless form of the KZK equation can be written as(3)∂P∂σ=Rπ1+σ2∫−∞τ∂2P∂ux2+∂2P∂uy2dτ′+α0R0∂2P∂τ2+NP1+σ∂P∂τ,where α0=δω02/2c03 is the absorption coefficient, R0=4ab/λ is the Rayleigh distance, N=βp0ω0R0/ρ0c03 is the nonlinear coefficient in dimensionless form, p0 is input sound pressure, ω0 is the angular frequency, k0 is the wave number, R = b/a is the aspect ratio, and the parametric source dimension is 2a×2b (a≥b).


## 2.2. Equivalent Source Incident Numerical Model

When the parametric sound field is incident obliquely into another medium, the beam direction will deviate due to the acoustic refraction effect. At this time, an equivalent parametric sound source coordinate system is established such that the acoustic refraction parametric sound field is considered to be generated by the equivalent parametric sound source. As shown in Figure1(b), the new transformed coordinate system (ux1,uy1,σ1) is introduced in region Ω1 to be consistent with the direction of sound line propagation. The coordinates of the center position of the inclined interface are (0, σ00=z0/R0) and (0, σ01), respectively, in the two coordinate systems.Figure 1
Schematic diagram of nonlinear acoustic propagation under inclined incidence conditions. (a) Cartesian rectangular coordinates. (b) Dimensionless transformed coordinate system.
(a)
(b)Under the transformed coordinate system constructed under the condition of equation (2), the coordinate parameter scale is related to the Rayleigh distance. Therefore, not only the acoustic parameters are different in new media but also the dimensionless coordinate system’s coordinate scale. The relationship between the two coordinate systems in the axial direction is as follows:(4)Δσ1σ1=Δσ0R0/R1cosαR0/R1σ0−σ00cosα+σ01,where R1 is the Rayleigh distance in the new media and α=θ0−arcsinc1/c0⋅sinθ0 is the angle of acoustic refraction deflection.As shown in Figure1(b), any point x,y,z on the inclined plane can be transformed into two nondiffusion coordinate systems with equation (2), respectively, which are(5)ux0=−R0tanθ⋅σ0−σ00a1+σ0,ux1=−R1tanθ−α⋅σ1−σ01a1+σ1.Analyzing the relationship ofux0 and ux1 by equation (4), we can get(6)ux1ux0=R1tanθR0tanθ−α⋅σ1−σ01σ0−σ00⋅1+σ01+σ1=R1tanθR0tanθ−α⋅1+σ0R1/R01+σ01cosα−σ00+σ0.To simplify the complexity of the model coordinate system transformation, it is assumed that the coordinates of the points on inclined plane are linear in the two coordinate systems, in which the value ofσ01 can be set as(7)σ01=R0R1cosα1+σ00−1.The relationship between the coordinate-scale transformation in the new coordinate system and the original coordinate system on inclined plane is(8)Δux1Δuy1Δσ1=R1tanθR0tanθ−α⋅Δux0R1tanθR0tanθ−α⋅Δuy0R0R1cosα⋅Δσ0.The transformation relationship between the two coordinate systems on inclined plane is(9)ux1uy1σ1=R1tanθR0tanθ−α⋅ux0R1tanθR0tanθ−α⋅uy0R0/R1σ0−σ00cosα+σ01.Therefore, the coordinate system (ux1,uy1,σ1) in Figure 1(b) can be transformed back to the Cartesian coordinate system (x, y, z) in Figure 1(a) with(10)xyz=acosα0−R1sinα0b0sinα0R1cosα1+σ1ux11+σ1uy1σ1−σ01+00z0.


## 2.3. Boundary Condition

Figure1 shows the sound field computational domain, which is set as 10 × 10 × 100 in the ux, uy, and σ directions, respectively. Around the computational domain, an absorbing boundary condition is imposed to minimize the acoustic reflections from the edges. The entire computational sound field is divided into two water areas (Ω0 and Ω1) by the inclined plane. The inclined plane section is perpendicular to the x-o-z plane and is at an angle of θ to the plane x-o-y, and the center of the section is at a horizontal distance z0 from the parametric source.For the sound pressure calculation, in natural parametric sound field, the sound source onz = 0 is used as an initial condition prescribed by two different primary frequencies as follows:(11)P0=p0cos2πf1t+cos2πf2t,where f1=f0−fd/2 and f2=f0+fd/2 are the two primary frequencies of parametric source, f0 and fd are the center and difference frequencies of the primary waves, respectively, and t is the time. After the parameter transformation, two different primary frequencies in simulation calculation are as follows:(12)P0ux,uy=p0cos2πf1τ+πux2+uy24+cos2πf2τ+πux2+uy24⋅Hux,uy,where(13)Hux,uy=1,ux≤1,uy≤R,0,other.Since the KZK equation describes only the unidirectional propagation of the nonlinear parametric sound field, the reflected sound beam is not considered in the actual calculation. At the water-ethanol boundary, there is a thin rubber film with the same acoustic impedance as water to prevent any mixing of ethanol and water. And, the sound pressure transmission coefficient of the water-ethanol boundary and the water-aluminum boundary, respectively, is as follows:(14)TWE=2ρEcE/cosθρEcE/cosθ+ρWcW/cosθ−α,TWA=2ZWZA2ZWZAcoskAlcosθA+jZW2+ZA2sinkAlcosθA,where ZW=ρWcW/cosθ, ZE=ρEcE/cosθ−α, θA is the refraction angle in the aluminum plate, kA is the wave number in aluminum, and l is the thickness of the thin aluminum plate. The subscripts E, A, and W represent ethanol, aluminum, and water, respectively. The parameters used in this study are given in Table 1.Table 1
Density and sound speed values.
Material
Water
Ethanol
Aluminum
Densityρ (kg/m3)
1000
785
2700
Speedc (m/s)
1473
1168
6300
Zmat (Rayls)
1.47
×
10
6
9.17
×
10
5
1.7
×
10
7
Nonlinear coefficientβ
3.5
6.9
—
Absorption coefficient (10−13f2dB/m)
2.17
4.78
—


## 2.4. Numerical Algorithm

In the nonlinear sound field numerical analysis, the KZK equation was solved with the finite-difference time-domain method (FDTD) for the acoustic field from a rectangular aperture parameter array source. The staggered grid system at inclined plane boundary is described in Figure2. The four terms on the right side of equation (1) represent diffraction in ux, uy axial directions, attenuation, and nonlinearity, respectively. Based on an orthogonal spatial grid, each term of the equation can be calculated as independent equation. The discretized forms of these equations are expressed as(15)Pm,nj,k+1−Pm,nj,kΔσk=Rπ1+σ2∫−∞τPm+1,nj,k+1−2Pm,nj,k+1+Pm−1,nj,k+12Δux2dτ′,(16)Pm,nj,k+1−Pm,nj,kΔσk=Rπ1+σ2∫−∞τPm,n+1j,k+1−2Pm,nj,k+1+Pm,n−1j,k+12Δuy2dτ′,(17)Pm,nj,k+1−Pm,nj,kΔσk=α0R0Pm,nj+1,k+1−2Pm,nj,k+1+Pm,n−1j−1,k+1Δτ2,(18)Pm,nj,k+1=Pm,nj,k1−NPm,nj+1,k−Pm,nj,k/Δτln1+Δσk/1+σk,Pm,nj,k≥0,Pm,nj,k1−NPm,nj,k−Pm,nj−1,k/Δτln1+Δσk/1+σk,Pm,nj,k<0,where m, n, and k are the spatial coordinate indexes in the ux, uy, and σ axial directions, respectively, and j is the time step. For example, Pm,nj,k=Pm⋅Δux,n⋅Δuy,σk,j⋅Δτ.Figure 2
Finite-difference method staggered grid system. (a) Staggered grid system at inclined boundary. (b) Staggered grid system in x-directions represent diffraction.Consider the effect of the slope boundary on the calculation of the discretization equation, as shown in Figure2. The calculation parameters are different on both sides of the slope boundary in equations (16)–(18), but the calculation conditions are consistent with the free field. However, in equation (15), the inclined boundary divides the ux-direction scattering calculation into two parts, as shown in Figure 2(b), and the approximate inclined boundary is(19)Pm,nj,k−Pm,nj,k−1Δσk=Rπ1+σ2∫−∞τPm+1,nj,k/T−2Pm,nj,k+Pm−1,nj,k2Δux2dτ′.When theux-direction diffraction calculation is completed, the initial condition of sound pressure Pmj,k at inclined incidence boundary, as shown in Figure 2(a), can be defined as follows:(20)Pmj,k=Pmj,k⋅T.


## 3. Simulation Results and Discussion

The distance between the center of the inclined plane in Figure1 and the parametric sound source is z0 = 3 m. p0=1×10−6Pa and SL = 239 dB. The rectangular parametric array was used as a sound source (the aperture is 0.22 m × 0.22 m). The transducer was excited by two different primary frequencies at f1 = 95 kHz and f2 = 110 kHz with the transmittance time of the signal 0.267 ms. The parameters used in this study are given in Table 1.


### 3.1. Sound Pressure Distribution

In order to better understand the inclined incidence mechanism of the parametric sound field, the numerical simulation of the acoustic field propagation of the parametric sound field under normal incidence and 30° angle incidence is carried out. The numerical calculation results of the water-aluminum boundary model are shown in Figure3, and the water-ethanol boundary model results are shown in Figure 4.Figure 3
Simulated sound pressure distribution with the different angle incident water-aluminum boundary models. (a)f=95kHz, θ=0∘; (b) f=95kHz, θ=30∘; (c) f=205kHz, θ=0∘; (d) f=205kHz, θ=30∘; (e) f=15kHz, θ=0∘; (f) f=15kHz, θ=30∘.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Simulated sound pressure distribution with the different angle incident water-ethanol boundary models. (a)f=95kHz, θ=0∘; (b) f=95kHz, θ=30∘; (c) f=205kHz, θ=0∘; (d) f=205kHz, θ=30∘; (e) f=15kHz, θ=0∘; (f) f=15kHz, θ=30∘.
(a)
(b)
(c)
(d)
(e)
(f)Compared to the aluminum plate boundary in Figure3, higher nonlinear parameters make the difference frequency sound in the ethanol medium have a distinct sound enhancement process. In Figures 4(b), 4(d), and 4(f), the sound propagation direction in ethanol medium deflects 6.66° due to the refraction effect. The difference frequency and the sum frequency sound still maintain good symmetry in the direction of sound propagation.


### 3.2. Simulation Results of Sound Pressure Distribution

In the sound field calculation units at the inclined boundary sound field area, the whole sound field is divided intoΩ0 and Ω1 areas by the inclined plane, where areas Ω0 and Ω1, respectively, are before and after the boundary region of the inclined plane. Figure 5 shows the 2D sound pressure distribution at σ0 away from the sound source in which the sound pressure amplitude is normalized to the sound source’s primary frequency value. All sound beams are symmetric in the y direction, and the sound pressure in Ω1 is significantly smaller than the sound pressure in Ω0 due to the inclined plane’s sound pressure sound transmission coefficient. The acoustic beam of the primary frequency, as shown in Figure 5(a), forms distinct side lobes in the horizontal and vertical directions, while the difference and sum frequency sound beam has no side lobes.Figure 5
2D sound pressure distribution at the center of water-ethanol boundary position (z0=σ0R0=3m). (a) Primary frequency; (b) difference frequency; (c) sum frequency.
(a)
(b)
(c)


## 3.1. Sound Pressure Distribution

In order to better understand the inclined incidence mechanism of the parametric sound field, the numerical simulation of the acoustic field propagation of the parametric sound field under normal incidence and 30° angle incidence is carried out. The numerical calculation results of the water-aluminum boundary model are shown in Figure3, and the water-ethanol boundary model results are shown in Figure 4.Figure 3
Simulated sound pressure distribution with the different angle incident water-aluminum boundary models. (a)f=95kHz, θ=0∘; (b) f=95kHz, θ=30∘; (c) f=205kHz, θ=0∘; (d) f=205kHz, θ=30∘; (e) f=15kHz, θ=0∘; (f) f=15kHz, θ=30∘.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Simulated sound pressure distribution with the different angle incident water-ethanol boundary models. (a)f=95kHz, θ=0∘; (b) f=95kHz, θ=30∘; (c) f=205kHz, θ=0∘; (d) f=205kHz, θ=30∘; (e) f=15kHz, θ=0∘; (f) f=15kHz, θ=30∘.
(a)
(b)
(c)
(d)
(e)
(f)Compared to the aluminum plate boundary in Figure3, higher nonlinear parameters make the difference frequency sound in the ethanol medium have a distinct sound enhancement process. In Figures 4(b), 4(d), and 4(f), the sound propagation direction in ethanol medium deflects 6.66° due to the refraction effect. The difference frequency and the sum frequency sound still maintain good symmetry in the direction of sound propagation.


## 3.2. Simulation Results of Sound Pressure Distribution

In the sound field calculation units at the inclined boundary sound field area, the whole sound field is divided intoΩ0 and Ω1 areas by the inclined plane, where areas Ω0 and Ω1, respectively, are before and after the boundary region of the inclined plane. Figure 5 shows the 2D sound pressure distribution at σ0 away from the sound source in which the sound pressure amplitude is normalized to the sound source’s primary frequency value. All sound beams are symmetric in the y direction, and the sound pressure in Ω1 is significantly smaller than the sound pressure in Ω0 due to the inclined plane’s sound pressure sound transmission coefficient. The acoustic beam of the primary frequency, as shown in Figure 5(a), forms distinct side lobes in the horizontal and vertical directions, while the difference and sum frequency sound beam has no side lobes.Figure 5
2D sound pressure distribution at the center of water-ethanol boundary position (z0=σ0R0=3m). (a) Primary frequency; (b) difference frequency; (c) sum frequency.
(a)
(b)
(c)


## 4. Experimental Results and Discussion

### 4.1. Experimental Setup

To confirm the numerical results, the sound pressure distributions at primary, sum, and difference frequencies through inclined plane were measured using a TC4038 hydrophone, as shown in Figure6. The water pool used in the experiments was 20 m long, 6 m wide, and 5 m deep (with the sound-absorbing tip around except the surface and bottom). The German SES2000 parametric array (0.22 m × 0.22 m) was used as a rectangular sound source, and the signal is collected by Tuopu’s multichannel dynamic signal acquisition analyzer. In the experiment, the hydrophone was rigidly connected to the pool driving and the speed was set to 2 cm/s. A thin aluminum plate having a size of 2 m × 2 m and l = 2.0 mm thickness is used as the inclined transmission plane.Figure 6
Experimental setup for the measurement of sound propagation through inclined plane.


### 4.2. Inclined Plate Experimental Validations

The experimental study on the acoustic propagation characteristics of the parametric sound field under vertical and inclined incidence conditions is carried out. The primary frequency and difference frequency directivities measured atz = 4 m (Figure 7) away from the sound source are measured and compared with the numerical calculation. The results show that there was no significant difference between the sound pressure distributions through thin aluminum plate within a 10 degree. And, the beam width on the difference frequency component in Figure 7(c) is wider than the theoretical prediction because the acoustic transmission coefficient of thin aluminum plates varies with frequency.Figure 7
The directivity of sound pressure at 4 m away from the sound source with aluminum plate. (a) The primary sound pressuref  =  95 kHz; (b) the primary sound pressure f  =  110 kHz; (c) the difference frequency sound pressure f  =  15 kHz; (d) the sum frequency sound pressure f  =  205 kHz.
(a)
(b)
(c)
(d)Figure8 shows axial sound pressure distribution of parametric sound field through the thin aluminum plate. The difference frequency sound pressure measured at the plate position reduced much less than the primary frequency sound pressure because the actual transmission coefficient of the aluminum plate changes with the frequency. However, the difference frequency sound pressure begun to decrease, and the attenuation speed decreases as the distance increases.Figure 8
The axial sound pressure distribution of sound pressure. (a) The primary sound pressuref  =  95 kHz. (b) The difference frequency sound pressure f  =  15 kHz.
(a)
(b)


## 4.1. Experimental Setup

To confirm the numerical results, the sound pressure distributions at primary, sum, and difference frequencies through inclined plane were measured using a TC4038 hydrophone, as shown in Figure6. The water pool used in the experiments was 20 m long, 6 m wide, and 5 m deep (with the sound-absorbing tip around except the surface and bottom). The German SES2000 parametric array (0.22 m × 0.22 m) was used as a rectangular sound source, and the signal is collected by Tuopu’s multichannel dynamic signal acquisition analyzer. In the experiment, the hydrophone was rigidly connected to the pool driving and the speed was set to 2 cm/s. A thin aluminum plate having a size of 2 m × 2 m and l = 2.0 mm thickness is used as the inclined transmission plane.Figure 6
Experimental setup for the measurement of sound propagation through inclined plane.


## 4.2. Inclined Plate Experimental Validations

The experimental study on the acoustic propagation characteristics of the parametric sound field under vertical and inclined incidence conditions is carried out. The primary frequency and difference frequency directivities measured atz = 4 m (Figure 7) away from the sound source are measured and compared with the numerical calculation. The results show that there was no significant difference between the sound pressure distributions through thin aluminum plate within a 10 degree. And, the beam width on the difference frequency component in Figure 7(c) is wider than the theoretical prediction because the acoustic transmission coefficient of thin aluminum plates varies with frequency.Figure 7
The directivity of sound pressure at 4 m away from the sound source with aluminum plate. (a) The primary sound pressuref  =  95 kHz; (b) the primary sound pressure f  =  110 kHz; (c) the difference frequency sound pressure f  =  15 kHz; (d) the sum frequency sound pressure f  =  205 kHz.
(a)
(b)
(c)
(d)Figure8 shows axial sound pressure distribution of parametric sound field through the thin aluminum plate. The difference frequency sound pressure measured at the plate position reduced much less than the primary frequency sound pressure because the actual transmission coefficient of the aluminum plate changes with the frequency. However, the difference frequency sound pressure begun to decrease, and the attenuation speed decreases as the distance increases.Figure 8
The axial sound pressure distribution of sound pressure. (a) The primary sound pressuref  =  95 kHz. (b) The difference frequency sound pressure f  =  15 kHz.
(a)
(b)


## 5. Conclusion

In this paper, the existing normal incidence KZK model is generalized to the 3D inclined incident model. The sound field with the transmittance inclined plane boundary condition is numerically simulated by solving the transformed beam KZK equation with FDTD method, and the sound pressure distribution of the parametric beam at the center of the sound is given out. The pool experiment verifies the correctness of theoretical analysis and numerical calculation. The proposed transform KZK model can also be used to calculate and analyze the parametric sound field propagation characteristics under inclined incident multilayer media.



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*Source: 2984191-2019-11-20.xml*