A method for compensating sound pressure in a scale model test when sound velocity is mismatched

Abstract

The application relates to a sound velocity mismatched scale-down test sound pressure compensation method, which comprises the following steps: step one: determining a scale-down coefficient N, scaling the underwater acoustic environment geometric parameters of a non-scale-down test environment by 1 / N, and determining the scale-down test parameters without sound velocity mismatch; step two: calculating the sound velocity mismatch amount size Delta c according to the sound velocity profiles of the non-scale-down test environment and the scale-down test environment, and taking the average value c0 of the sound velocity profile in the environment without sound velocity mismatch as a reference sound velocity; and step three: setting a sound velocity mismatch correction coefficient alpha, and calculating a sound pressure field distance phase compensation coefficient beta. The application realizes nonlinear compensation of sound pressure errors caused by linear mismatch of the sound velocity profile by calculating the sound pressure distance phase compensation factor related to the sound velocity mismatch size, can effectively solve the scale-down test sound pressure errors caused by linear mismatch of the sound velocity profile, and provides more accurate sound field data for scale-down test detection performance verification.

Description

Technical fields:

[0001] This invention belongs to the field of underwater acoustics and underwater acoustic signal processing technology, and specifically relates to a method for compensating sound pressure in scaled-down tests when sound velocity mismatch occurs. Background technology:

[0002] Scaled-down testing is an equivalent experimental method for studying the acoustic scattering characteristics of underwater targets and the detection performance of active sonar targets. Using a scaled-down target model, controllable and repeatable scaled-down testing verification studies of full-scale submarine sea trials can be conducted in pools or lakes, reducing the complexity of detection performance verification under actual testing conditions. Scaled-down testing requires ensuring acoustic similarity to obtain sound field propagation laws and target echo characteristics equivalent to those in the actual marine environment. Currently, underwater acoustic scaled-down testing mainly focuses on geometrically scaling down full-scale submarine targets and increasing the transmitted signal frequency to keep ka constant according to the acoustic similarity principle, thereby obtaining the acoustic scattering characteristics of the full-scale submarine target based on the equivalent target's acoustic scattering characteristics. Since the influence of the underwater acoustic channel can be eliminated during the measurement of target acoustic scattering characteristics, the change in sound velocity in scaled-down testing can be ignored.

[0003] Scale-down testing of target detection performance requires a consistent underwater acoustic velocity profile to avoid errors introduced by changes in sound pressure after scaling. However, in actual scale-down testing, it is difficult to ensure that the underwater acoustic velocity profiles in the actual marine environment and lake / laboratory waveguides are completely identical. Changes in the underwater acoustic environment after scaling, especially alterations to the sound velocity profile, will cause errors in the horizontal distance of the sound pressure in the scale-down test, leading to a decrease in the detection performance verification capability dependent on distance propagation loss. Summary of the Invention:

[0004] The technical problem to be solved by this invention is to provide a method for sound pressure compensation in scaled-down tests when sound velocity mismatch occurs. This method proposes a horizontal distance-dimensional sound pressure compensation method based on the normal mode model. By calculating the sound pressure distance phase compensation factor related to the magnitude of sound velocity mismatch, nonlinear compensation for sound pressure error caused by linear mismatch of sound velocity profile is achieved. This method can effectively solve the sound pressure error in scaled-down tests caused by linear mismatch of sound velocity profile, and provide more accurate sound field data for the verification of scaled-down test detection performance.

[0005] The technical solution of this invention is to provide a method for compensating sound pressure in a scaled-down test when there is sound velocity mismatch, comprising the following steps:

[0006] Step 1: Determine the scaling factor N, scale the underwater acoustic environment geometric parameters of the non-scaled test environment by 1 / N, and determine the scaled test parameters without sound velocity mismatch;

[0007] Step 2: Calculate the magnitude of the sound velocity mismatch Δc based on the sound velocity profiles of the non-scaled and scaled-down test environments, and calculate the average value c0 of the sound velocity profile in the environment without sound velocity mismatch as the reference sound velocity.

[0008] Step 3: Set the sound velocity mismatch correction coefficient α, and calculate the sound pressure field distance phase compensation coefficient β;

[0009] Step 4: Determine the distance-phase term of the compensated scaled-down test sound pressure field based on the distance-phase compensation coefficient β. Resample the horizontal distance r′ to obtain a new horizontal distance sample.

[0010] Step 5: Sampling based on horizontal distance Determine the sound pressure data of the scaled-down test after compensation. The expression is used to calculate the compensated sound pressure data;

[0011] Step 6: Use the normal mode model to generate sound pressure data in the unscaled test environment, and calculate the root mean square error of the sound pressure amplitude before and after compensation and the compensation gain.

[0012] Step 7: Adjust the sound velocity mismatch correction coefficient α according to the distance phase compensation gain to determine whether to recompensate the scaled-down test sound pressure data.

[0013] As a preferred option, step one is performed as follows: assuming the actual target radius is *a*, and ensuring the acoustic parameter *ka* remains constant, the radius of the scaled-down target model is *a′*. The formula for calculating the scaling factor is then:

[0014] N = a / a′,

[0015] The scaled-down test parameters without sound speed mismatch are:

[0016] H′=H / N,

[0017] z′ s =z s / N,

[0018] z′=z / N,

[0019] r′=r / N,

[0020] Wherein, H, z s z and r represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the non-scaled-down test, respectively; H′, z′ s z′ and r′ represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the scaled-down test, respectively.

[0021] As a preferred method, step two is performed as follows: Assuming the sound velocity profile in the scaled-down test environment without sound velocity mismatch is c(z), and the sound velocity profile in the actual scaled-down test environment is c′(z), after smoothing and fitting the sound velocity profile, the magnitude of the sound velocity mismatch is calculated as follows:

[0022] Δc=c′(z)-c(z)

[0023] The formula for calculating the reference speed of sound c0 is:

[0024]

[0025] As a preferred option, step three is specifically operated as follows: In the scaled-down test, under the condition of sound velocity mismatch of Δc, the horizontal wavenumber... Satisfying Relationships

[0026]

[0027] Where ω′ is the angular frequency of the sound source in the scaled-down test. k′ is the horizontal wavenumber of the sound pressure field in the scaled-down test without sound velocity mismatch. zm For the corresponding vertical wavenumber, the sound velocity mismatch degree Δ = Δc / c0; α is the sound velocity mismatch correction coefficient, used to compensate for the error introduced by the linear approximation process of the horizontal wavenumber; when the sound velocity mismatch is small, the initial value of the sound velocity mismatch correction coefficient is set to α = 1; therefore, the sound pressure field distance phase compensation coefficient is calculated as follows:

[0028]

[0029] Preferably, in step four, after determining the distance-phase compensation coefficient β, the distance propagation phase term of the scaled-down test sound pressure under conditions of sound velocity mismatch satisfies the following relationship.

[0030]

[0031] Among them, horizontal distance sampling It is the imaginary part unit of a complex number.

[0032] Preferably, in step five, under the condition of no sound velocity mismatch, the expression for the scaled-down test sound pressure field is:

[0033]

[0034] Ψ′ m (·) represents the depth function of the m-th mode excited in the scaled-down test environment, M is the total number of normal modes excited by the sound source in the scaled-down test environment, and ρ(z′) s The density of the water at the depth of the sound source is usually taken as 1 g / cm³. 3 When sound velocity mismatch exists, the expression for the scaled-down test sound pressure is:

[0035]

[0036] Using the range-phase compensation relationship from step four, βk′ rm r′ is replaced with The expression for the sound pressure level in the scaled-down test after compensation can then be written as:

[0037]

[0038] According to the distance phase term Compensation can offset the sound pressure error at horizontal distance caused by sound speed mismatch.

[0039] Compared with the prior art, the present invention has the following advantages:

[0040] To address the problem of inaccurate sound pressure levels at horizontal distances in scaled-down underwater acoustic tests due to sound velocity mismatch, a sound pressure compensation method for scaled-down tests is proposed. This method utilizes a normal mode model to establish a sound pressure field expression, obtaining the mapping relationship between the magnitude of sound velocity mismatch and the change in horizontal wavenumber in the scaled-down environment. Then, the change in horizontal wavenumber is mapped to the change in the distance-phase term of the sound pressure field at horizontal distances, determining the distance-phase compensation coefficient. Finally, a nonlinear interpolation method is used to compensate for the sound pressure data, thereby obtaining sound pressure data consistent with that in the non-scaled-down test environment. Simulation results show that this invention can obtain satisfactory sound pressure data in both different frequencies and different magnitudes of sound velocity mismatch, proving that this method can reduce the sound pressure error caused by inconsistent sound velocity profiles in scaled-down tests, providing accurate sound pressure data for verifying the detection performance of scaled-down tests. Attached image description:

[0041] Figure 1 This is a flowchart illustrating the implementation process of the present invention.

[0042] Figure 2 This is a schematic diagram of a scaled-down test under the sound speed mismatch condition of the present invention and a simulated sound speed profile of the non-scaled test environment.

[0043] Figure 3 This invention provides a comparative calculation of horizontal wavenumbers in scaled-down test environments under conditions of mismatched sound speeds at different sound source frequencies.

[0044] Figure 4 This invention provides a comparative analysis of the horizontal wavenumber conversion in scaled-down test environments under different sound velocity mismatch conditions.

[0045] Figure 5 This is a comparison of compensation results under the condition of a sound source frequency of 300Hz and a sound velocity mismatch of 3m / s in an embodiment of the present invention.

[0046] Figure 6 This is a comparison of compensation results under the condition of a sound source frequency of 400Hz and a sound velocity mismatch of 3m / s in an embodiment of the present invention.

[0047] Figure 7 This is a comparison of compensation results under the condition of a sound source frequency of 300Hz and a sound velocity mismatch of 5m / s in an embodiment of the present invention.

[0048] Figure 8 This is a comparison of compensation results under the condition of a sound source frequency of 400Hz and a sound velocity mismatch of 5m / s in an embodiment of the present invention.

[0049] Figure 9 This is a comparison of compensation gain under different sound speed mismatch conditions in the embodiments of the present invention. Detailed implementation method:

[0050] The present invention will be further described below with reference to the accompanying drawings and specific embodiments:

[0051] A method for compensating sound pressure in scaled-down tests under sound velocity mismatch, such as Figure 1 As shown, the specific implementation process is as follows:

[0052] 1) Determine the scaling factor N, scale the underwater acoustic environment geometric parameters of the non-scaled test environment by 1 / N, and determine the scaled test parameters without sound velocity mismatch.

[0053] Assuming the actual target radius is *a*, and keeping the acoustic parameter *ka* constant, the radius of the scaled-down target model is *a′*. The formula for calculating the scaling factor is:

[0054] N = a / a′

[0055] The scaled-down test parameters without sound speed mismatch are:

[0056] H′=H / N

[0057] z′ s =z s / N

[0058] z′=z / N

[0059] r′=r / N

[0060] Wherein, H, z s z and r represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the non-scaled-down test, respectively; H′, z′ s z′ and r′ represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the scaled-down test, respectively.

[0061] 2) Calculate the magnitude of the sound velocity mismatch Δc based on the sound velocity profiles of the non-scaled test environment and the scaled test environment, and calculate the average value c0 of the sound velocity profile in the environment without sound velocity mismatch as the reference sound velocity.

[0062] Assuming the sound velocity profile in a scaled-down test environment without sound velocity mismatch is c(z), and the sound velocity profile in the actual scaled-down test environment is c′(z), after smoothing and fitting the sound velocity profiles, the magnitude of the sound velocity mismatch is calculated as follows:

[0063] Δc=c(z)-c(z)

[0064] The formula for calculating the reference speed of sound c0 is:

[0065]

[0066] 3) Set the sound velocity mismatch correction coefficient α and calculate the sound pressure field distance phase compensation coefficient β.

[0067] In a scaled-down test with a sound velocity mismatch of Δc, the horizontal wavenumber... Satisfying Relationships

[0068]

[0069] Where ω′ is the angular frequency of the sound source in the scaled-down test. k′ is the horizontal wavenumber of the sound pressure field in the scaled-down test without sound velocity mismatch. zm For the corresponding vertical wavenumber, the sound velocity mismatch degree Δ = Δc / c0. α is the sound velocity mismatch correction coefficient, used to compensate for the error introduced by the linear approximation process of the horizontal wavenumber. When the sound velocity mismatch is small, the initial value of the sound velocity mismatch correction coefficient is set to α = 1. Therefore, the sound pressure field distance phase compensation coefficient is calculated as follows:

[0070]

[0071] 4) Determine the distance-phase term of the compensated scaled-down test sound pressure field based on the distance-phase compensation coefficient β. Resample the horizontal distance r′ to obtain a new horizontal distance sample.

[0072] After determining the distance-phase compensation coefficient β, the distance propagation phase terms of the scaled-down test sound pressure under conditions of sound velocity mismatch satisfy the following relationship.

[0073]

[0074] Among them, horizontal distance sampling It is the imaginary part unit of a complex number.

[0075] 5) Sampling based on horizontal distance Determine the sound pressure data of the scaled-down test after compensation. The expression is used to calculate the compensated sound pressure data.

[0076] Under conditions of no sound velocity mismatch, the expression for the sound pressure field in the scaled-down test is:

[0077]

[0078] Among them, Ψ′ m (·) represents the depth function of the m-th mode excited in the scaled-down test environment, M is the total number of normal modes excited by the sound source in the scaled-down test environment, and ρ(z′) s The density of the water at the depth of the sound source is usually taken as 1 g / cm³. 3 When sound velocity mismatch exists, the expression for the scaled-down test sound pressure is:

[0079]

[0080] Using the range-phase compensation relationship in step 4), βk′ rm r′ is replaced with The expression for the sound pressure level in the scaled-down test after compensation can then be written as:

[0081]

[0082] Therefore, according to the distance phase term Compensation can offset the sound pressure error at horizontal distance caused by sound velocity mismatch. This invention uses a nonlinear interpolation method to perform distance-phase compensation on the horizontal distance of the scaled-down test sound pressure p′, obtaining the compensated scaled-down test sound pressure.

[0083] 6) Use the normal mode model to generate sound pressure data in the unscaled test environment, and calculate the root mean square error of the sound pressure amplitude before and after compensation and the compensation gain.

[0084] The sound pressure data p is generated based on the non-scaled test environment parameters, and then the amplitude is reduced by N times to obtain the sound pressure under the scaled-down test environment. The formula for calculating the root mean square error of the sound pressure amplitude in the scaled-down test after distance and phase compensation is as follows:

[0085]

[0086] By Replacing it with p′ allows us to calculate the root mean square error of the sound pressure amplitude e(z) in the scaled-down test before compensation. The formula for calculating the distance phase compensation gain G is then:

[0087]

[0088] 7) Adjust the sound velocity mismatch correction coefficient α according to the distance phase compensation gain to determine whether to recompensate the scaled-down test sound pressure data.

[0089] Determine whether to correct α based on the distance-phase compensation gain G. When G < 0 dB, adjust the sound velocity mismatch correction factor to α = 1 + Δ / 10, recalculate the sound pressure field horizontal wavenumber variation factor β, and repeat steps 3) to 7) to calculate the sound pressure data after distance-phase compensation. To reduce the impact of sound velocity mismatch and provide accurate sound pressure data for verifying the detection performance of scaled-down tests.

[0090] Simulation and sea trial data test results

[0091] Based on the above implementation process, the processing results of this invention are given through simulation. The simulation processing is as follows: Figures 2-9 As shown.

[0092] Computer Simulation 1: Using the Kraken acoustic simulation tool, sound pressure data was generated under conditions of sound velocity mismatch. The simulation environment was as follows: Figure 2 As shown. The simulation parameters are as follows: for the non-scale test environment, the water depth is 106m and the water density is 1g / cm³. 3 sound speed profile as Figure 2 As shown in (a), beneath the water body lies a 20m thick sedimentary layer with a density of 1.5g / cm³. 3 The sound velocity in the sediment layer is 1800 m / s, with an attenuation of 0.5 dB / λ. The substrate is an acoustic half-space bottom with a density of 2.6 g / cm³. 3 The sound velocity is 5200 m / s, and the attenuation is 1.8 dB / λ. The sound source depth is 40 m, the receiving depth is 50 m, and the horizontal receiving distance is set at 1 m intervals, up to 5 km. The sound source frequencies are 300 Hz and 400 Hz. Figure 2 (b) A scaled-down test environment with linear mismatch in sound velocity profile is given, with a sound velocity mismatch of Δc = 3 m / s and a scaled-down factor N = 5. In the scaled-down test environment, the water depth is changed to 21.2 m, the sound source depth is changed to 8 m, the receiving depth is changed to 10 m, and the horizontal receiving distance is changed to intervals of 0.2 m, up to 1 km. Figure 3 The results show a comparison of the changes in the horizontal wavenumber of the sound pressure field at source frequencies of 300Hz and 400Hz under the mismatch condition of Δc=3m / s. It can be seen that after N=5 times the conversion, the horizontal wavenumber error caused by the sound velocity mismatch increases when the source frequency increases, which leads to an increase in the sound pressure field error. Figure 4 The results show a comparison of the changes in horizontal wavenumber when the sound source frequency is 300Hz and the sound velocity mismatch Δc = 3m / s and Δc = 5m / s. It can be seen that the larger the sound velocity mismatch, the greater the resulting horizontal wavenumber error. Simulation results indicate that both the sound source frequency and the magnitude of the sound velocity mismatch will cause errors in the sound pressure level of the scaled-down test when the sound velocity mismatch is present. The sound pressure error is mainly introduced by the change in the horizontal wavenumber.

[0093] Computer Simulation 2: Calculate the distance-phase compensation coefficient β and the corresponding horizontal distance resampling based on the magnitude of the sound speed mismatch Δc. A one-dimensional nonlinear interpolation method was used to perform distance-phase compensation for the sound pressure data. Based on the sound velocity profile in the unscaled test environment, the reference sound velocity c0 = 1533.9 m / s was calculated. When the sound velocity mismatch Δc = 3 m / s and Δc = 5 m / s, the correction factor α = 1 was taken, and the distance-phase compensation coefficients β = 0.9980 and β = 0.9968. Figure 5 A comparison of compensation results is presented for a sound source frequency of 300Hz and a sound velocity mismatch Δc = 3m / s. It can be seen that, with sound velocity mismatch, there are significant errors in the horizontal distance between the scaled-down and unscaled-down sound pressure fields. After distance-phase compensation, the changes in sound pressure after compensation are consistent with those in the unscaled-down test, indicating a good compensation effect. Figure 6 A comparison of compensation results is presented for a sound source frequency of 400Hz and a sound velocity mismatch Δc = 3m / s. It can be seen that the sound pressure data from the scaled-down test after distance and phase compensation are consistent with the sound pressure results in the non-scaled-down test environment, proving that the method proposed in this invention is effective at different sound source frequencies. Figure 7 and Figure 8 Comparisons of compensation results for scaled-down experimental sound pressure data corresponding to different sound source frequencies under a sound velocity mismatch Δc = 5 m / s are presented. Figure 5 and Figure 6 The results show that, at the same sound source frequency, the distance-phase compensation method proposed in this invention is effective under different sound velocity mismatches, and can obtain sound pressure data consistent with those in the non-scaled test environment. These results demonstrate that the method of this invention can obtain satisfactory sound pressure field data under different sound source frequencies and different sound velocity mismatches, reducing the impact of sound pressure field errors caused by sound velocity mismatches in scaled-down tests.

[0094] Computer Simulation 3: The sound velocity mismatch magnitude Δc is varied from 0 to 8 m / s. The distance-phase compensation gain G is calculated under different Δc conditions. The sound source frequency is taken as 300 Hz and 400 Hz respectively. The results are as follows: Figure 9 As shown, when Δc = 0 m / s, the compensation gain is 0 dB at both sound source frequencies. In this case, there is no sound velocity mismatch in the scaled-down test environment, and the method of this invention is applicable to scaled-down tests without sound velocity mismatch. When Δc ≠ 0, the compensation gain after compensating the sound pressure data of the scaled-down test using the method of this invention can reach above 0 dB, and the compensation gain increases accordingly as the sound velocity mismatch increases, proving the effectiveness of the method of this invention.

[0095] The simulation analysis results above show that the sound field compensation method for scaled-down tests under sound velocity mismatch proposed in this invention can effectively reduce the sound pressure field error caused by sound velocity mismatch in scaled-down tests. Furthermore, the compensated sound pressure data obtained by the proposed compensation method under different sound source frequencies and different sound velocity mismatch magnitudes are consistent with the sound pressure data in the non-scaled-down test environment. This invention provides an effective method to combat the influence of sound velocity mismatch in underwater acoustic scaled-down test signal processing applications, which can obtain accurate scaled-down test sound pressure data and has good application prospects.

[0096] The above description only illustrates preferred embodiments of the present invention and should not be construed as limiting the scope of the claims. Any equivalent procedural modifications made using this specification are included within the patent protection scope of this invention.

Claims

1. A method for compensating sound pressure in a scaled-down test under sound velocity mismatch, characterized in that: Includes the following steps, Step 1: Determine the scaling factor N, scale the underwater acoustic environment geometric parameters of the non-scaled test environment by 1 / N, and determine the scaled test parameters without sound velocity mismatch; Step 2: Calculate the magnitude of the sound velocity mismatch Δc based on the sound velocity profiles of the non-scaled and scaled-down test environments, and calculate the average value c0 of the sound velocity profile in the environment without sound velocity mismatch as the reference sound velocity. Step 3: Set the sound velocity mismatch correction coefficient α, and calculate the sound pressure field distance phase compensation coefficient β; Step 4: Determine the distance-phase term of the compensated scaled-down test sound pressure field based on the distance-phase compensation coefficient β. Resample the horizontal distance r′ to obtain a new horizontal distance sample. Step 5: Sampling based on horizontal distance Determine the sound pressure data of the scaled-down test after compensation. The expression is used to calculate the compensated sound pressure data; Step 6: Use the normal mode model to generate sound pressure data in the unscaled test environment, and calculate the root mean square error of the sound pressure amplitude before and after compensation and the compensation gain. Step 7: Adjust the sound velocity mismatch correction coefficient α according to the distance phase compensation gain to determine whether to recompensate the scaled-down test sound pressure data.

2. The method for compensating sound pressure in a scaled-down test under sound velocity mismatch as described in claim 1, characterized in that: Step one is operated as follows: Assuming the actual target radius is *a*, and ensuring the acoustic parameter *ka* remains constant, the radius of the scaled-down target model is *a′*. The formula for calculating the scaling factor is: N = a / a′, The scaled-down test parameters without sound speed mismatch are: H′=H / N, With' s =z s / N, z′=z / N, r′＝r / N, Wherein, H, z s z and r represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the non-scaled-down test, respectively; H′, z′ s z′ and r′ represent the water depth, sound source depth, receiving depth, and receiving horizontal distance in the scaled-down test, respectively.

3. The method for compensating sound pressure in a scaled-down test under sound velocity mismatch according to claim 1, characterized in that: Step two is performed as follows: Assume the sound velocity profile in the scaled-down test environment without sound velocity mismatch is c(z), and the sound velocity profile in the actual scaled-down test environment is c′(z). After smoothing and fitting the sound velocity profiles, the magnitude of the sound velocity mismatch is calculated as follows: Δc=c′(z)-c(z) The formula for calculating the reference speed of sound c0 is:

4. The method for compensating sound pressure in a scaled-down test under sound velocity mismatch according to claim 1, characterized in that: Step three is operated as follows: In the scaled-down test, under the condition of sound velocity mismatch of Δc, the horizontal wavenumber is... Satisfying Relationships Where ω′ is the angular frequency of the sound source in the scaled-down test. k′ is the horizontal wavenumber of the sound pressure field in the scaled-down test without sound velocity mismatch. zm For the corresponding vertical wavenumber, the sound velocity mismatch degree Δ = Δc / c0; α is the sound velocity mismatch correction coefficient, used to compensate for the error introduced by the linear approximation process of the horizontal wavenumber; when the sound velocity mismatch is small, the initial value of the sound velocity mismatch correction coefficient is set to α = 1; therefore, the sound pressure field distance phase compensation coefficient is calculated as follows:

5. The method for compensating sound pressure in a scaled-down test under sound velocity mismatch according to claim 1, characterized in that: In step four, after determining the distance-phase compensation coefficient β, the distance propagation phase term of the scaled-down test sound pressure under conditions of sound velocity mismatch satisfies the following relationship. Among them, horizontal distance sampling It is the imaginary part unit of a complex number.

6. The method for compensating sound pressure in a scaled-down test under sound velocity mismatch according to claim 1, characterized in that: In step five, under the condition of no sound velocity mismatch, the expression for the sound pressure field in the scaled-down test is: Ψ′ m (·) represents the depth function of the m-th mode excited in the scaled-down test environment, M is the total number of normal modes excited by the sound source in the scaled-down test environment, and ρ(z′) s The density of the water at the depth of the sound source is usually taken as 1 g / cm³. 3 When sound velocity mismatch exists, the expression for the scaled-down test sound pressure is: Using the range-phase compensation relationship from step four, βk′ rm r′ is replaced with The expression for the sound pressure level in the scaled-down test after compensation can then be written as: According to the distance phase term Compensation can offset the sound pressure error at horizontal distance caused by sound speed mismatch.

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