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amplitude applied approximation arc of trajectory assumed assumption asymptotic boundary conditions boundary layer bounds calculation Cauchy relations characteristic coefficients coeﬂicients components computation consider constant convergence coordinates corresponding curve deﬁned deﬁnition denote derivatives determined diagonalizing operator differential equation discussed distribution elastic expression ﬁeld ﬁgures ﬁlm ﬁnal ﬁnd ﬁnite ﬁrst ﬁxed ﬂow formula G. N. Watson given heat Hence hodograph IIIIII incompressible ﬂuid inequality inﬁnite integral integral curves limiting limiting Mach linear Mach number Mach wave Math mathematical matrix melting method noise normal obtained parameter plane plastic potential pressure problem quantities reﬂected waves respectively saddle point satisﬁed shown in Fig signiﬁcance singular points solution speciﬁc strain stream function streamline stress subsonic substitution supersonic surface symmetric temperature theorem theory transformation values vanishes variables vector velocity velocity potential viscous wall zero