旁轴

In order to solve the problem,We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential manifold. The travel time are expressed as polynomials of the horizontal offset between the two points, and the single-square-root operator in frequency-wavenumber domain are expressed as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddle-point method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量(地面点到地下相点的水平距离)的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

In order to solve the problem, We proposed a simple formula for computing paraxial travel time of single-way wave operator. The formula is based on the forward and inverse transform between time-space domain to frequency-wavenumber domain and from vector field to exponential as polynomials of wavenumber. Coefficients of travel time polynomials and that of single-square-root operator are related each other and calculated by Lie algebraic integrand, exponential map and the saddlepoint method.

针对此，基于时间空间域到频率波数域和向量场到指数流形上的正反变换，提出了计算单程波算子旁轴走时的简便公式，将走时表示成空间变量（地面点到地下相点的水平距离）的多项式，将频率波数域单平方根算子表示成波数的多项式，运用Lie代数积分、指数映射和鞍点法将走时多项式的系数与单平方根算子的系数联系起来，运用单平方根算子的系数计算走时多项式的系数。

Paraxial approximation absorption boundary condition based on acoustic wave equation can obtain better absorption results by increasing paraxial approximation order of Taylor series. However, this kind of operation will increase the complexity of difference computation, and result in instability of computation process.

基于声波波动方程的旁轴近似吸收边界条件可以通过提高泰勒级数展开的旁轴近似阶数获得更好的吸收效果，但这样不仅增加了差分计算的复杂性，同时还导致计算过程不稳定。

The hyperbolic and parabolic travel time of the reflection in layer media are presented in expression of matrix with paraxial ray theory.

利用旁轴射线理论给出了层状介质反射波的双曲线旅行时和抛物线旅行时的矩阵表达式。

The derivation of the relationship between the paraxial wave equation and the complex source points is given and proved.

推导并证明了旁轴波动方程的解与复源点方法之间的一致性关系。

Comparing to conventional expansion, our paraxial travel time expansion increases asymmetric terms of the horizontal offset in the expression.

该展开式与通常的旁轴展开式相比，增加了关于空间坐标的非对称项。

The method is easily realized with small computation amount,and its absorption effect is better than conventional paraxial approximation absorption boundary condition.

该方法简单易行，计算量小，而吸收效果比常规旁轴近似吸收边界条件更好。

Comparing to conventional expansion, our paraxial travel time expansion increases a symmetric terms of the horizontal offset in the expression.

该展开式与通常的旁轴展开式相比，增加了关于空间坐标的非对称项。

The full parallax CGHs are generated by the point source method and the wave-oriented method without paraxial approximation.

这种全视差计算全息图是由点基元方法和平面波导向法生成并且没有经过旁轴近似。

Comparing with common side hybrid laser-arc welding, coaxially combined laser beam and arc can provide a symmetrical circular thermal source on the workpiece surface, which is convenient for 3-D welding.

与目前常用的旁轴激光-电弧复合焊相比，激光-电弧同轴复合可以在工件表面提供对称热源，焊接质量不受焊接方向影响而适于三维焊接。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。