The combined wave superposition method (combined WSM) is an approximation method, and the accuracy of the approximation is needed to be discussed. It approximates not only the normal mode order of the Green’s function in sea, but also the process of inverting the virtual source strength. The ray theory is also an approximate acoustic propagation model in the acoustic positive problem. In the following sections, we will analyze the error caused by the local approximation in combined WSM, and some characteristics of the whole method in numerical calculation.
3.1. Accuracy Analysis of Green’s Function in Channel
The basis of the wave superposition method is the Green’s function. The accuracy of the Green’s function, especially in the near field, directly determines the accuracy of the transfer matrix. The Green’s function is expressed as the sum of the normal modes. The high-order normal mode has little contribution to the far field; however, it has a great influence on the near field. Therefore, the normal order determines the near field accuracy of the Green’s function. When the order is too small, the result of the Green’s function is inaccurate in the near field, but gradually approaches the exact value as the distance increases.
In order to illustrate the near field accuracy of the Green’s function, a case is selected for analysis. In the shallow sea of 200 m depth, the sea surface is the Dirichlet boundary and the sea floor is the Neumann boundary. A point source with a frequency of 20 Hz is located at a depth of 25 m. According to the boundary conditions, the following constraints exist in Equation (5) as shown as:
The eigen function
that satisfies the constraints is
. The specific form of the Green function can be obtained according to the constraint conditions, which is the analytical solution under boundary condition as shown as:
where,
shows the source depth,
indicates the depth of the sea, and
and
represent the vertical and horizontal components of the wave number
, respectively. The parameter
is the horizontal distance, and
is the normal order.
The model is built using finite element software COMSOL and compared with the Green’s function method when the mesh density is sufficiently dense, and the results are shown in
Figure 2.
It can be seen that under far field conditions, the calculation result of the Green’s function is nearly consistent with the finite element method. In fact, the situation is different under near field conditions, which is what needs to be discussed next.
We assume the error threshold 1 dB, that is, the accuracy of the near field calculation is characterized by the horizontal distance when the absolute error between the transmission loss of the Green’s function method and the finite element method is less than 1 dB. The further the distance is, the more accurate are the results of the near field of the Green’s function we obtain.
Figure 3 below shows the near field accuracy of the Green’s function after changing the normal mode order of the Green’s function at different frequencies or different depths, while the sea environment and point source parameters are consistent with the above case.
The overall trend is that if the order is larger, the results of the near field of the Green’s function are more accurate. In order to ensure that the wave superposition method can effectively reverse the acoustic field information, it needs to increase the order and to ensure that the horizontal distances from the equivalent sources to the positions of vibration velocity point are not too close. As the frequency increases, the energy of the acoustic field concentrates on the low-order normal modes, and loss due to the lack of energy of the high-order is also reduced. However, in order to get the accuracy of the near-field, it is necessary to keep the order large enough. In terms of greater depth, the requirement of the normal order is higher in this method. Therefore, theoretically the accuracy of the wave superposition method can be guaranteed with a small depth.
3.2. Applicability of Rays Superimposed
For the Green’s function requiring high-order superposition, the calculation time of each virtual source propagating the acoustic field increases rapidly with the increase of the calculated range and order, making the calculation efficiency low. The large number of virtual sources inversion by the wave superposition method makes the superposition with the Green’s function not the most ideal choice. The ray theory performs ray tracing in the entire acoustic field and extracts the eigenray, which is less computationally expensive. Since the ray theory is derived under the high frequency approximation, it is necessary to verify that the ray solution can meet the accuracy requirements of the acoustic field superposition at different frequencies and depths.
The channel is a typical shallow sea waveguide (i.e., the sea surface boundary is the Dirichlet boundary; the sea bottom boundary is the Neumann boundary), and the six point sources are located at depths of 25 to 30 m at intervals of 1 m. The depth of water and frequency are changed, and the acoustic field is superimposed by the ray theory and the Green’s function method, respectively. The field comparison chart after superposition is shown in
Figure 4 as:
The ray solution will have a slight oscillation near the true value at low frequency, and the superposition of multiple point sources will amplify the amplitude of the oscillation. A large number of virtual sources can cause an increase in the amplitude of the oscillation. The amplitude of the oscillations is different at different frequencies and depths. The error caused by oscillation is characterized by mean square error (MSE).
where
represents the number of measurement points, and
and
are the transmission losses obtained by ray theory and the Green’s function methods respectively. The table below gives the MSE for certain frequencies and depths.
In
Table 1, in the case where the frequency is too low or the depth is too small, there is a large error in ray theory. At high frequency, the solution of the ray theory is very accurate. The field of virtual source can be directly superimposed, and the result is reliable. As a whole, the oscillation of the ray theory has a limited influence on the calculation results. The performance at high frequency is superior, and the application range of the wave superposition method can be expanded.
3.3. Calculation of Acoustic Radiation Field by Combined Method
The key to the combined method is the solution of the acoustic inverse problem. The results of the virtual source inversion have a significant impact on the calculation of the acoustic field [
19]. Research shows that the accuracy of wave superposition inversion has a great relationship with virtual source configuration. These researches are based on free space or half space Green’s function, but the Green’s function in sea is a cylinder functions, which has its own particularity. In order to illustrate the calculation accuracy of the method proposed in this paper and analyze the influence of different virtual source configuration methods, an excited spherical shell is selected to calculate its far field pressure using the proposed method and compared with the finite element model in COMSOL software. The reason why this axisymmetric structure was selected is that the calculation distance of the three-dimensional field by COMSOL is limited by the performance of the computer. The selected spherical shell has a radius of 2 m and a thickness of 0.1 m. The Material elastic Young’s modulus is
and Poisson’s ratio is
; density is
. The outside medium is water, in which the density is
, and the speed of sound is
. The inside of the spherical shell is a vacuum. The center of the spherical shell is located at a water depth of 25 m. A normal force of 1 N with a frequency of 100 Hz is applied at the bottom of the spherical shell. The depth of the sea is 200 m, and the sea surface and seabed boundary conditions satisfy typical channel conditions. The schematic diagram of the COMSOL model is shown in
Figure 5. The sound pressure level (SPL) of the radiated field of this model is calculated using COMSOL. As the spherical shell is circumferentially symmetrical, the vibration velocity of the two-dimensional structure surface calculated by COMSOL can be scanned every 36 degrees in the circumferential direction to obtain the three-dimensional structure surface vibration velocity distribution. We also calculate SPL of the radiated field using combined WSM based on 3D vibration velocity distribution data.
The virtual sources are distributed on a surface that has the same shape as the surface of the structure, which is an appropriate virtual source configuration. The virtual source surface is generally obtained by shrinking the surface of the structure inwardly. Here the radius ratio is chosen to be 0.5. Except for the upper and lower vertices, the virtual source surface is divided into 10 and 6 intervals in the circumferential and vertical directions respectively, that is, a total of 72 virtual source points (
virtual sources) are created, as shown in
Figure 6.
The sound pressure level is calculated at different depths as shown in
Figure 7.
The SPL is calculated at the received depth of 25 m, 50 m, 100 m, and 150 m. The results of combined WSM are compared with the finite element calculation results. Further, the relative error is defined as:
where,
is the number of measurement points. The
and
are the sound pressures generated by combined WSM and FEM, at the
ith measurement point, respectively. The
and
virtual sources (excluding the upper and lower vertices) are selected respectively, to form a virtual source surface, as shown in
Figure 8, and analyzed the effect of the radius ratio
on the calculation result at the receive depth of 25 m, as shown in
Figure 9.
Theoretically, when the number of virtual sources is large, the dispersion degree of the volume source will be higher, and the inversion of the sound field will be more accurate. In fact, for the situation analyzed in this section, after the number of virtual sources meets the basic requirements of the calculation, it does have some impact on the accuracy of the calculation when the radius ratio is in the range of 0.7 to 1. The impact of this factor is far greater than the impact of the number of virtual sources.
Figure 9 shows that in the case where the number of virtual sources is constant, the radius ratio is too small or too large, which leads to an increase in error. When the radius ratio approaches 1, because the Green’s function is a cylindrical function, the horizontal distance between the virtual source and the surface node of the structure becomes too small, and the calculated
matrix singularity increases. When the radius ratio is too small, the difference in surface vibration velocity is not sufficiently reflected on the virtual source. In a word, the appropriate radius ratio plays a critical role in the accuracy of the calculation. For the spherical shell, the optimum ratio of the radius ratio is in the range of 0.5 to 0.9. When the shape or other parameters of the structure change, the configuration of the virtual sources should be adjusted.
The
virtual sources (excluding the upper and lower vertices) and a radius ratio of 0.5 are selected, which is same with the model in
Figure 7 to get a two-dimensional color figure of SPL in 100 Hz using combined WSM and FEM respectively as shown in
Figure 10.
The field interference fringes in
Figure 10a are clear, and the results are similar to the finite element results in
Figure 10b. Same as in
Figure 10b,
Figure 10a can also describe the phenomenon caused by the interaction of radiated sound with the sea surface and the sea bottom during the propagation of the sound, that is, the sound rays reflect back and forth between the sea surface and the sea bottom. However, the difference is that compared with
Figure 10b, the peak and trough amplitudes of
Figure 10a are not clear enough, which is also reflected in
Figure 7.
It should be noted that in order for combined WSM to accurately predict the radiated field of the structure, the environment of the sea needs to meet the applicable range of ray theory. The following relationships can be used as a guide for high-frequency approximation of ray theory [
1] as shown as:
where
is the frequency,
is sound speed in water, and
is the depth of the water. This relationship can also be used as guidance for the applicable conditions of combined WSM. Although the guidance given by this relationship is not necessarily accurate, it can indeed reflect the potential of combined WSM for the calculation of high-frequency problems.