We derive an explicit solution representation for the Lamb's problem. The solution formula is given in terms of the fundamental solutions of the d'Alembert wave equations, the Kirchhoff's formula in 3-D and the Hadamard's formula in 2-D, when the Poisson ratio $\frac{1}{2} \lambda /(\lambda+\mu)$ is less than or equal to a critical value $\sigma^*$. The critical Poisson ratio value $\sigma^*$ is derived in terms of the Lamé constants $(\lambda,\mu)$. Our solution formula yields rich surface wave patterns as a consequence of the coupling of 2D and 3D wave structures. These surface wave patterns are much richer than the Rayleigh waves discussed by the pioneers of the field, [19, 8]. Our analysis originates from the notions of master relationship in [11], Laplace-Fourier path in [12], and the determinant for surface waves in [13]. We use these notions to form an LY algorithm and apply this algorithm to solve the Lamb's problem completely.