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摘要:
由于弹性波传播动力学方程较复杂，若通过材料参数设计控制弹性波的传播路径，难以获得准确的设计结果。通常在对弹性波材料进行设计时，需要根据实际情况(如高频条件)，对动力学方程进行简化或者近似才可以设计出基于弹性波的相应器件。由于这种设计方法的阻抗匹配和介质材料的无损要求难以满足，使得在设计弹性波传播器件过程中出现散射现象，而且往往散射的波形还是频变的信号，因此，在评价弹性波器件设计优劣时，散射现象的大小标志着设计效果的好坏，成为评价设计的手段和工具。但是，散射现象往往仅根据肉眼观察，难以准确地描述，无法给出定量的结果进行说明。为此，本文提出的分数阶Fourier变换(FRFT)对频变信号具有良好的聚焦特性，可利用频率变化的调频率，定量描述散射效应的大小，给出弹性波传播控制散射效应的一种定量的描述方法，降低散射程度认知的盲目性，为简化材料介质参数的设计提供依据。
Abstract:The propagation equation of the elastic wave is more complicated than that of the electromagnetic wave. It is difficult to design medium parameters when controlling the direction of the elastic wave. In order to obtain precise results of elastic wave propagation, the equations were simplified or approximated to achieve the design effect, depending on the actual situation (such as high frequency). Therefore, as impedance matching and lossless dielectric material requirements are difficult to meet, the scattering phenomenon appears in the design of elastic wave propagation in the process of the device. Usually, and the scattering wave is variable frequency signal. It is a way and a tool in the evaluation of elastic wave device design, the size of the scattering phenomenon marks the design effect. The fractional Fourier transform (FRFT), which has good focusing characteristic, is adaptive to analyze frequency variation signal. The frequency change rate provides a quantitative description method of elastic scattering wave propagation control. This reduces the blindness of scattering degree of cognitive, simplifying the dielectric become evaluation of design.
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Key words:
- propagation control /
- scattering effect /
- concentrator /
- fractional Fourier transform
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Overview: The elastic particles are deviated from the equilibrium position under the elastic force. The stress in the particle changes or produces vibration. This change at the same time leads to a stress change and vibration of the surrounding particles. The mechanism of the elastic wave is more complicated than that of the electromagnetic wave, because its propagation relies on medium. This is a typical inverse question. It is difficult to design medium parameters when controlling the direction of the elastic wave. In order to obtain precise results of elastic wave propagation, the equations were simplified or approximated to achieve the design effect, depending on the actual situation (such as high frequency). The elastic is widely used in many domains. In some space, the special demand is presented, such as the cloak. The objects enclosed inside this region cannot be impinged by the illumination light and hence are invisible to the outside observers. The concentrator is also applied to control the energy concentrated in a special zone, in order to highlight this domain. Therefore, as impedance matching and lossless dielectric material requirements are difficult to meet, the scattering phenomenon appears in the design of elastic wave propagation in the process of the device. It is a way and a tool in the evaluation of the elastic wave device design. The size of the scattering phenomenon marks the design effect. In some case, this is judged by our eyes. It is qualitative. Usually, the scattering wave is a variable frequency signal. The fractional Fourier transform (FRFT), which has good focusing characteristic, is adaptive to analyze frequency variation signal. It is a new attempt to transform the time and frequency domain to spatial frequency. The spatial signal in the propagation direction also has the same characters as the signal of time. The result of this transformation has turned the scattering problem into a signal analysis and processing problem. As this, the method of signal can be used to process these questions, such as the wavelet transform, Gabor transform and FRFT. Especially the FRFT, the focusing characters to the chirp signal, which can be used to analyses the spatial signal. The frequency change rate provides a quantitative description method of elastic scattering wave propagation control. With the FRFT, it reduces the blindness of scattering degree of cognitive, simplifying the dielectric become evaluation of design. This method is suitable for the signal of frequency changes, in many cases, the scattering is also complex, and it needs to analyze the reasons and characters of these phenomena and the corresponding method are brought.
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图 1 能量集中器空间变换的映射方法
Figure 1. Mapping method for spatial transformation of energy concentrator
图 2 弹性波集中器变换方法的计算域
Figure 2. The computational domain of the elastic wave concentrator transformation method
图 3 弹性波能量集中器两种设计方法得到的位移和能量分布结果。(a), (b)分别表示空间坐标变换和连续介质坐标法设计的位移分布；(c), (d)则是这两种方法设计的能量密度分布结果
Figure 3. Results of displacement and energy distribution obtained by two design methods of elastic wave energy concentrato. (a), (b) are the displacement distribution that designed by spatial coordinate transformation and continuous medium coordinate method, respectively; (c), (d) are the energy density distribution results of the above two methods, respectively
图 4 多项式的FRFT和短时Fourier变换的拟合效果。(a)设计的原信号；(b)原信号短时Fourier拟合结果；(c)原信号的FRFT拟合
Figure 4. Fitting effect of FRFT and short - time Fourier transform of polynomial. (a) Original signal; (b) Fitting result of short-time Fourier transform; (c) Fitting result of FRF
图 5 设计材料中间位置波形信号。(a)基于空间坐标变换设计的空间波形；(b)法基于连续介质设计的空间波形
Figure 5. Intermediate position signal of the design material. (a) Spatial wave of coordinate transformation; (b) Spatial wave of continuous medium
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