Generalized Closed-Form Solutions for  \(\mathcal{L}\mathcal{P}_3\)

In Chap. 7 , we delve into the solution of the Green’s function and its first spatial derivative for the \(\mathcal{L}\mathcal{P}_2\) . Still employing the strategy used in Chap. 6 for solving the Green’s function of the \(\mathcal{L}\mathcal{P}_1\) , we start with the integral form of the solution. Initially, by changing the integral variable, we rationalize the Rayleigh function in the denominator of the integrand. Then, using the partial fraction expansion of rational fractions, we simplify the integrand into a combination of simple fractions and products of radicals with quadratic or quartic polynomials, referred to as the basic integrals. Finally, we solve these basic integrals one by one to obtain the problem’s generalized closed solution. As shown in Sect. 7.4.3 , compared to integral solutions, the closed-form solution not only significantly enhances computational efficiency but also aids in theoretical analysis of wave field properties, such as the discussion on the excitation mechanism of Rayleigh waves in a half-space in Sect. 7.4.4 . Mathematically, obtaining a closed or generalized closed-form solution is our ultimate goal in solving a boundary value problem; for most such problems, this is unachievable. Fortunately, for Lamb’s problems with relatively simple boundary conditions, this goal is achievable. In fact, up to this point, we have accomplished the ultimate objective for solving the first two types of Lamb’s problems.