A plunge pool is often employed as an energy-dissipating device at the end of a spillway or a pipe culvert. A jet from spillways or pipes frequently generates a scour hole which threatens the stability of the hydraulic structure. Existing scour prediction formulas of plunge pool of spillways or pipe culverts give a wide range of scour depths, and it is, therefore, difficult to accurately predict those scour depths. In this study, a new experimental method and new scour prediction formulas under submerged circular jet for large bed materials with shallow tailwater depths were developed. A major variale, which was not used in previous scour prediction equations, was the ratio of jet size to bed material size. In this study, jet momentum acting on a bed particle and jet diffustion theory were employed to derive scour prediction formulas. Four theoretical formulas were suggested for the two regions of jet diffusion, i.e., the region of flow establishment and the region of established flow. The semi-theoretically developed scour prediction formulas showed close agreement with laboratory experiments performed on a movable bed made of large spherical particles.