Waves on the surface of the ocean are a dramatic and beautiful phenomena that impact every aspect of life on the planet. At small length scales, ripples driven by surface tension on the surface of these “water waves” affect remote sensing of underwater obstacles. At intermediate scales, waves on the surface and the interface between internal layers of water of differing densities affect shipping, coastal morphology, and near–shore navigation. At larger lengths, tsunamis and hurricane–generated waves can cause devastation on a global scale. Additionally, water waves play a crucial role at all length scales in the exchange of momentum and thermal energy between the ocean and atmosphere which, in turn, affect the global weather system and climate. From a mathematical viewpoint, the water wave equations pose severe challenges for rigorous analysis, modeling, and numerical simulation. The traditional water wave problems were mostly solved under the assumption of harmonic oscillation in time. This assumption can provide a lot of advantages in the analysis. However, there are situations when the initial value problem approaches are necessary. The most famous one must be Cauchy-Poisson problem in water waves. Since this approach needs initial condition for the initial elevation and initial value of the potential, the delta function was adopted for the initial impulse. Therefore, the present approach is considered to be based on physics. The main purpose of this paper is to study such a problem which seems to be important and interesting from a mathematical as well as a physical point of view. The problem is solved by the joint Laplace and Henkel transforms and the integral solution is obtained.