A rigorous solution based on the method of L. Cagniard has been obtained for the transmission of a pulse from a point source of pressure in a fluid through a solid plate imbedded in the fluid. On the normal from the source to the plate the pressure response is given by a closed form expression requiring no integration. Numerical results on the normal agree in essential features with experiment. The formal solution on the normal can be expressed in terms of reflection and transmission coefficients for compressional and shear waves at the two boundaries. In this form the solution is easily generalized to a many layer problem. [A preliminary report was given at the American Physical Society meeting held on Febraury 26 and 27, 1954, at Austin, Texas. See Phys. Rev.94, 3, 812 (1954).]