. Compendium of meteorology. Meteorology. THE METHOD OF CHARACTERISTICS 423 X3 can be found by solving the two simultaneous equa- tions xs — xi X3 — X2 - a+ ; 7- = a- h - k ts — (2 These values of ^3 can be substituted in the equations (9) and (10): 5i »3 — Ul h - k Eia+)(C2a+ - A) -[B.'^^^'^^ k - k )(C,«+- DO W3 — Wi k - k a+ (CiA - DiCd = 0, B,'^ 'p - E,^- D,) - k - [3,:-^^-=^- -E,a. k - k \{C,a^- D,. k - k a-{CiDo - D1C2) = 0, and these two equations can be solved for Uz and Wz because k and Xz are known. By proceeding in this manner from a line (such as the boundary of a flow, or a
. Compendium of meteorology. Meteorology. THE METHOD OF CHARACTERISTICS 423 X3 can be found by solving the two simultaneous equa- tions xs — xi X3 — X2 - a+ ; 7- = a- h - k ts — (2 These values of ^3 can be substituted in the equations (9) and (10): 5i »3 — Ul h - k Eia+)(C2a+ - A) -[B.'^^^'^^ k - k )(C,«+- DO W3 — Wi k - k a+ (CiA - DiCd = 0, B,'^ 'p - E,^- D,) - k - [3,:-^^-=^- -E,a. k - k \{C,a^- D,. k - k a-{CiDo - D1C2) = 0, and these two equations can be solved for Uz and Wz because k and Xz are known. By proceeding in this manner from a line (such as the boundary of a flow, or an initial state) in the x, t plane, along which u and w are knoAvn, values of u and w over a portion of the {x, t) plane can be found. For instance, in Fig. 1 the. Fig. 1.—The region in which u and w can be computed from values given on L. data between A and B on L allow a computation of the values of u and w in the hatched region R. For a full discussion of such "domains of dependence" and the related "regions of influence" the reader is referred to Courant and Friedrichs [5]. This method of integrating even the most complicated of such equations is easy to derive but is difficult to carry out in practice. Some simpler problems are easier to discuss. Simple Waves. If, in equations (1) and (2), Ei = Ei = 0 and Ai, A2, Bi , B2, etc., are not functions of X ox t but only functions of u and w, then the factor \/dt can be removed from equations (9) and (10). These equations then define two functions of u and w (in implicit form, to be sure): H{u,w) = K+ L{u,w) = K- along C+ , along C-, where in general the constants K+ are different for each C+ characteristic and similarly for K- and C_ . Courant and Friedrichs have shown that a region of constant u and w, say Ua and Wo , is separated from a region of varying u and w by a characteristic. Assume that this is a C+ characteristic. Since C_ characteristics intersect C+ characteristics, some of the C- c
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