phenomenon is called Huygens' principle. It is true for odd numbers of space dimension, where for one dimension the integration is performed over the boundary of an interval with respect to the Dirac measure. It is not satisfied in even space dimensions. The phenomenon of lacunas has been extensively investigated in Atiyah, Bott and Gårding (1970, 1973). Scalar wave equation in two space dimensions In two space dimensions, the wave equation is We can use the three-dimensional theory to solve this problem if we regard as a function in three dimensions that is independent of the third dimension. If then the three-dimensional solution formula becomes where and are the first two coordinates on the unit sphere, and is the area element on the sphere. This integral may be rewritten as a double integral over the disc with center and radius It is apparent that the solution at depends not only on the data on the light cone where but also on data that are interior to that cone. Scalar wave equation in general dimension and Kirchhoff's formulae We want to find solutions to for with and . See Evans for more details. Odd dimensions Assume is an odd integer, and , for . Let and let Then , in , , . Even dimensions Assume is an even integer and , , for . Let and let then in Problems with boundaries One space dimension Reflection and transmission at the boundary of two media For an incident wave traveling from one medium (where the wave speed is ) to another medium (where the wave speed is ), one part of the wave will transmit into the second medium, while another part reflects back into the other direction and stays in the first medium. The amplitude of the transmitted wave and the reflected wave can be calculated by using the continuity condition at the boundary. Consider the component of the incident wave with an angular frequency of , which has the waveform At , the incident reaches the boundary between the two media at . Therefore, the corresponding reflected wave and the transmitted wave will have the waveforms The continuity condition at the boundary is This gives the equations and we have the reflectivity and transmissivity When , the reflected wave has a reflection phase change of 180°, since . The energy conservation can be verified by The above discussion holds