Vanishing Waves on Semi-closed Space Intervals and Applications in Mathematical Physics

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Abstract

Test-functions (which differ to zero only on a limited interval and have continuous derivatives of any order on the whole real axis) are widely used in the mathematical theory. Yet less attention was given to intuitive aspects on dynamics of such test functions or of similar functions considered as possible solution of certain equations in mathematical physics (as wave equation). This study will show that the use of wave equation on small space interval considered around the point of space where the sources of the generated field are situated can be mathematically represented by vanishing waves corresponding to a superposition of travelling test functions. As an important consequence, some directions for propagating the generated wave appears and the possibility of reverse radiation being rejected. Specific applications for other phenomena involving wave generation (as the Lorentz formulae describing the generation of a wave with different features after the interaction with the observer’s material medium are also presented.