Wave Momentum as Real Physics: From Linear to Areal Dynamics

This paper proposes a generalized definition of momentum grounded in closed surface dynamics. Conventional physics employs the linear formula p=mvp = mvp=mv, based on point particles moving along imaginary coordinate axes. However, points and axes are abstractions, not real physical entities. Real objects are extended closed surfaces, and their physical behavior emerges from the time variation of surface area. We argue that volume is not a simultaneous three-dimensional block but an accumulation of surfaces with infinitesimal temporal phase shifts, similar to sequential frames in a film. By replacing linear displacement dxdxdx with surface area SSS, momentum is reformulated as p = I = Iω where rrr is the surface radius, ω\omegaω the angular frequency of phase change, and III a surface inertial parameter. This areal momentum unifies wave phenomena with classical mechanics, showing that waves are closed surfaces with complementary areas oscillating in time. The approach highlights surfaces as the fundamental entities of real physics, while points and axes remain mathematical fictions. Applications include electron scattering, string oscillations, and general wave mechanics, offering a consistent geometric framework that bridges classical and quantum domains.