Some Integrated Properties of Solutions of the Wave Equation with Non-planar Boundaries* By

Integrated properties of solutions of the wave equation with non-planar boundaries are found and applied to three dimensional supersonic flow problems and two dimensional diffraction problems. For the problem of supersonic flow outside a cylindrical surface with generators parallel to the flow direction, a theorem is proved concerning the integrated properties of the linearized pressure distribution and the prescribed normal velocity on the surface. The theorem is a generalization of the integral relationships obtained previously and is useful in the evaluation of total lift and drag of wing-body combinations when the linear dimensions of the cross section of the body are not small as compared to the chord length. For the diffraction of a pulse or a weak shock over a rectangular notch, a pressure integral theorem is obtained. Its usefulness is demonstrated in reducing the labor of obtaining the mean pressure distribution along any depth inside the notch at different instants for various width-height ratios of the notch. Introduction. Theorems concerning certain integrated properties of the linearized pressure field due to planar source distributions in a supersonic stream have been presented by Lagerstrom and Van Dyke [1] and by Bleviss [2]. Extension of the theorems and their applications to a class of three dimensional problems involving biplanes or cruciform wing arrangements were presented by Ferri [3] and by Ferri and Clarke [4], Ferri, Clarke and Ting [5] obtained a theorem regarding the pressure integral along the line of intersection of a forward Mach plane with a specified non-planar surface which represents a prismatic body of rectangular cross section mounted on a planar wing with supersonic edges. The pressure integral is related to the integral of the source distribution which in turn is related to the integral of the prescribed normal velocity on the top (or bottom) surface of the body and that on the wing surface. A similar relationship can be established if the normal velocity is prescribed on the side walls of the body. The wing surface together with the surface of the body above (or below) the wing represents a cylindrical surface in the form of a "single" step with generator parallel to the direction of the undisturbed supersonic flow. By a further extension of the theorem in [5], it can be shown that a similar integral relationship is valid if the cylindrical surface is of the form of a "stairway" with a finite number of steps. As the number of steps becomes infinite and the size of each step infinitesimal, then a sequence of "stairways" is obtained whose limit can approximate the shape of any given cylindrical surface. Hence, the generalized integral relationship to be presented in this paper is conjectured. However, the argument which leads to the conjecture cannot be accepted as a proof of the validity of the relationship unless it can be shown that the limit of the sequence of the corresponding disturbance pressures or potentials exists and equals the corresponding value for the given cylindrical surface. •Received November 12, 1957. 374 LU TING [Vol. XVI, No. 4 Such a convergence proof is by no means simple. An attempt to verify the relationship directly by the same procedure used in the preceding investigations has not been successful owing to the difficulty of finding the proper source distribution which corresponds to the prescribed boundary condition on the cylindrical surface. The fact that the velocity potential is a solution of the wave equation is expressed implicitly in the preceding investigations through the relationship between the integral of the pressure distribution and that of the source distribution. In the present paper the integral relationship between the pressure distribution and the prescribed normal velocity is verified by observing the fact that the velocity potential obeys the wave equation. Thus, the necessity of finding the proper source distribution is avoided. Generalization of integral relationship. As shown in Fig. 1, a cylindrical surface, y = F(z), is placed in a supersonic stream with its generator parallel to the x-axis, Cylindrical turfac*, y • F(z), abcdnma Mach plan*, x + By« M/9 *hll* Plan* x»0, bcgfb Plan* z«K|(£) ckgc Plan* z • Kt(fl) bjfb Domain of turfac* integral, I*. *had*d ar*a Path of lin* int*gral,T, 11kI