Numerical methods generally need analytical solutions as test cases and validations in simplified problems. This work provides Laplace domain explicit analytic solutions for fluid-structure interaction (FSI) water-hammer waves within a pipe. Rather than applying the Transfer Matrix Method (TMM) to the FSI four-equations, it is transposed to the equivalent two-wave propagating problem considered instead. Using the classical wave matrix diagonalisation approach permits to decouple waves propagation, whilst, at the same time, coupling boundary conditions in the diagonal base. This approach permits to provide the transfer matrix for coupled waves boundary conditions so as to obtain a Laplace domain solution for the pressure/stress vector solution. This solution is written in a general framework which can be adapted for general applied boundary conditions for a single pipe. Three sets of boundary conditions are considered as examples and illustrations from solving the inverse Laplace transform of the considered explicit solutions. Consistent results with recently proposed time-domain solutions are found, and a one-to-one mapping between Laplace domain and time domain approaches is also established. This permits to find the discrete spectrum of FSI water-hammer waves mode decomposition from TMM solutions.