Nomenclature M= Mach number n = total number of shock waves in the system /?, p 0 = static, stagnation pressure S = ratio of the overall total pressure ratio for an /z-shock-wave system to the total pressure ratio for case with a single shock j8 = shock-wave inclination 8= flow deflection f, = ratio of static pressures, p l + l lpi Subscripts i'(l, 2, . . /,* , n) -number of the shock in the system = number of iteration step = limiting value for n -> o°I ntroduction T HE effectiveness of supersonic inlets is strongly influenced by the unavoidable presence of shock waves. At high supersonic speeds it is desirable to design inlets with multiple-shock configuration in order to maximize the pressure recovery. The theory of shock waves is based on the assumption of discontinuity in flow parameters across the shock surface and the requirement that the principles of conservation of mass, momentum, and energy across this surface be satisfied. This approach results in shock relations, which when combined with a pressure recovery criteria, yields values of the shock parameters corresponding to the optimal shock configuration. In the principal work by Oswatitsch 1 the optimization problem is formulated for ^-shock-wave system n -I of these being oblique shocks and the last one a normal shock wave. The task set forth is to determine, for a given n and an upstream Mach number M l9 the shock configuration producing the maximum overall total pressure ratio p 0 /i + i/PoiThe solution to this problem yields 6,-. It is possible to formulate the optimization problem for n oblique shock waves

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presented previously. 1 ' 2 For some applications, the method of Ref. 2 might be preferred because with it the velocity vector of the moving object can be found without knowing the coordinates of the listening stations. However, at least three such stations are needed. The present Note gives an alternate method of computation in which only two observation posts are required. This can be advantageous in some cases.Let t be the time at which the shock wave arrives at a particular point. Finite difference approximations for dt/dx, dt/dy and dt/dz can be obtained by having two or more microphones along the x, y and z axes through the point, respectively. This gives Vt, the gradient of t. But the shock wave travels with the speed of sound, C, in the direction perpendicular to the wave front. Hence Vt is in the direction of the vector C, and its magnitude is 1/C. That is,(2) If the speed of sound is known, then only two of the partial derivatives of t need be measured. The third one can be calculated, except for sign, by means of Eq. (2). Thus a cluster of microphones can serve as an observation post to determine Vt, and therefore C. Fig. 1 The object at B 9 moving with velocity V greater than C, creates a shock cone as shown.Refer now to Fig. 1. It shows a portion of the shock cone generated by an object moving along path A 1 A 2 B with velocity V. Observers are at P 1 and P 2 . The shock wave has just arrived at P i and Q 2 . The time will be designated as t 19 and the arrival time at P 2 will be called t 2 . Hence Q 2 = P 2 -C 2 (; 2 -gThis gives the point Q 2 . Let i\ be the time required for the shock wave to travel from A l to P v and i 2 the time it spends going from A 2 to Q 2 . Then V(T 1 -T 2 ) = C 1 T 1 + Q 2 -P 1 -C 2 i 2 (4) But the cosine of the angle between V and C is equal to C/V. Therefore V C = C 2 . So dotting Eq. (4) with C 2 and then C 1 gives (C'-CVC^^C^-PJ (5) (G 2 -C 1 -C 2 )i 2 = C 1 -(P 1 -Q 2 ) (6) These equations give r 1 and i 2 unless C t = C 2 . If (C 1 + C 2 )-(Q 2 -P x ) is not zero, then T I and T 2 will be different, and V can be obtained from Eq. (4). Also, A^P^C^ (7) Thus the point A 1 on the path of the moving object, and the time t 1 -i 1 when it was there, are also known.

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