| Acoustic wave problems widely exist in many fields such as mechanics,machinery,architecture,and aerospace.Although numerical methods such as finite element method（FEM）and meshless method have been successfully applied in the analysis of acoustic wave propagation problems,they still face some challenges,especially for high wave number problems.For example,the finite element method needs to set more than ten nodes in a wavelength to obtain an effective solution,otherwise there will be a significant phase difference between the numerical solution and the exact solution.Therefore,for high wave number problems,extremely fine grids must be used,resulting in a huge amount of computation,even unacceptable.This thesis focuses on the challenging issue of quantitative solution for high wave number acoustic problems.Based on the wavelet multiresolution interpolation format that can efficiently approximate oscillation functions,multiresolution solution formats for frequency domain and time domain acoustic equations are established,respectively.The effectiveness of the method and its superiority over existing numerical methods such as FEM are verified through classic test examples.Firstly,the multiresolution solution scheme of variable coefficient Helmholtz equation with the third kind of boundary conditions is established by combining the wavelet multiresolution interpolation scheme and the variational principle.This method does not require grids（including background grids）and can highly refine the local node distribution to analyze the problem of wave number height changes,and can directly apply essential boundary conditions like the FEM.At the same time,for the variable wave number problem,the generation of the stiffness matrix still only needs to calculate the integral values exactly the same as those in the solution of the constant wave number problem,and these integral values can be efficiently obtained from the basic data unrelated to the problem without numerical integration.The numerical results show that the accuracy and efficiency of this method are significantly better than existing numerical methods such as the FEM.For example,for the one-dimensional constant wave number problem,when only 8 nodes are set at each wavelength,the absolute error is less than 10-3,while the maximum absolute error is about 8×10-2 when the FEM uses 32 nodes at each wavelength.In addition,a modified technique based on the second order wavelet method for the constant coefficient Helmholtz equation is proposed,which can effectively suppress the phase difference between the numerical solution and the exact solution,and hardly increase the amount of computation.For example,when 16 nodes are used at each wavelength,the absolute error of the modified wavelet method can be as low as 7×10-5,while that of the FEM can be as high as 3×10-1,showing excellent computational accuracy and efficiency for high wave number acoustic problems.Finally,a wavelet solution format for time-domain acoustic problems was presented by using wavelet multi resolution method for spatial discretization.The numerical test results show that the established method has good computational accuracy,efficiency,and stability.For example,for two-dimensional Gaussian pulsed acoustic wave problems,a wavelet solution that is highly consistent with the reference solution can be obtained when using the sparse nodes with node spacing h=0.25 and time step（35）t=0.01.At the same time,in the numerical simulation of complex acoustic problems such as acoustic wave propagation in homogeneous and heterogeneous media,the wavelet solutions obtained are also highly consistent with the physical laws. |
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