Page 643 radio acoustic ranging 6826 



computing the distances to the R.A.R. stations and then plotting by conventional 

 methods. The principle involved in the solution of this problem is that the locus of 

 a point which moves so that the difference of distances from two fixed points is a 

 constant, is a hyperbola. The two fixed points are the R.A.R. stations, and the sign 

 of the difference determines the branch of the hyperbola on which the point is found. 

 It follows then that where there are three R.A.R. stations, the R.A.R. position is at 

 the intersection of three hyperbolas, each one of which satisfies the difference condi- 

 tions for a pair of R.A.R. stations. One of the tlu-ee hyperbolas merely serves as a 

 check on the position as determined from the other two and will not be considered 

 further. The two hyperbolas giving the best angle of intersection should be the 

 ones used. 



The velocity of sound must be known so that the time differences can be con- 

 verted into meters; othei-wise the problem is insoluble. Also, the identity of the two 

 R.A.R. stations to which any given time difference applies and the sign of the differ- 

 ence must be laiown. If the signs of the two differences were unknown, there would 

 be, theoretically at least, four possible intersections, since each hyperbola has two 

 branches. 



It must be borne in mind that a given R.A.R. position is not as strongly fixed by 

 the method of differences as by the conventional methods using total distances. The 

 strength of position in the former case depends on the angle of intersection of the 

 hyperbolas, which angle is always more acute than the angle of intersection of the 

 distance arcs, for any given position. 



For the methods explamed below, assume thi-ee R.A.R. stations. A, B, and C, 

 whose geographic positions are known and which are plotted on a base sheet, and let 

 P denote the unknown position sought. Assume further that the radio returns from 

 the three stations are received in the above order, and let p and q represent the known 

 distance differences, so that PB—PA=p, and PC—PB=q. Then the three actual 

 distances are PA=x, PB=x-{-p, and PC=x-\-p-{-q, in which x is the unknown con- 

 stant to be added to the distance differences for plotting by conventional R.A.R. 

 methods. The problem, therefore, is one of determining the unknown distance x. 



There are several methods of accomplishing this which are described briefly 

 under the following headings: 



A. GRAPHIC METHOD 



(a) On the base sheet at B scribe a circle with radius p and at C another circle with radius (p + q)- 

 The problem is to find a point P which is equidistant from A and the circles with radii p and {p -\- q). 

 Make a transparent overlay on which is drawn a series of conceiitric circles large enough to satisfy 

 the conditions. The transparency should resemble the Odessey R.A.R. protractor (4537), and ad- 

 jacent circles should be close enough to each other to permit accurate interpolation by eye. The 

 transparency need have no scale. Lay the transparency over the base sheet and move it around 

 until a circle with radius x is found which passes through A and is tangent to the circles scribed at 

 B and C. The desired point P is then at the center of the series of concentric circles on the overlay. 



(6) In this method a series of concentric circles similar to the Odessey R.A.R. protractor must be 

 drawn on the base sheet with each R.A.R. station as a center. These circles must be constructed so 

 that circles of equal radii are easily identifiable. On a transparent overlay draw two concentric 

 circles, with radii p and (p -\- q). Lay the transparency over the base sheet and move it around 

 until their common center and the two circles are at equal distances from the R.A.R. stations at 

 A, B, and C, respectively; that is, until the center of the transparency is on a circle from A with radius 

 X, the p circle is tangent to a circle from B with radius x, and the {p -\- q) circle is tangent to a circle 

 from C with radius x. When the above conditions are satisfied the center of the overlay is at the 

 desired point P. Plotters will find this procedure awkward at first as they are not accustomed to this 



