Theory of Characteristics Page: 4 of 29
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NADA TM 1242l
will be considered (fig. I.) The characteristic family of curves (2) may
be intersected by another family of curves.
9 (x,y) = Constant (3)
No further data are given concerning this second family of curves; for
hyperbolic differential equations where two families of characteristics
appear, the second family of characteristic base curves will be
selected as t - family. This is mentioned only incidentally; for the
immediately following considerations only q = Constant are assumed as
characteristic base curves which are intersected by the
curves I = Constant. Thus one has as coordinate along a characteristic
curve, I, as transverse coordinate, 11. The derivative of a
function f(x, y) with respect to 9 along a characteristic base
curve (as which r = 0 will be selected below) will be called "interior
derivative"; to attain it, nothing but the course of the function within
the considered - region on n = 0 is needed. Derivatives with
respect to q require knowledge of the behavior of the function to be
derived outside of the characteristic curve; they will be called
"exterior derivatives". It is obvious that the conceptions of interior
and exterior derivatives are very closely connected with the conceptions
of tangential and normal erivatives.
After these preliminary statements the announced definitions of
characteristics are set up.
3. Characteristics as Loci of Discontinuities
of the Second Order
The integral function X and its first derivatives are to remain
continuous when the characteristic T = 0 is transversed. Disconti-
nuities in the second derivatives - the highest ones occurring in the
differential equation (1) - are to be permissible but with the
restriction that at least the interior or tangential derivatives still
remain continuous. The permitted discontinuities concern at most the
exterior derivatives of the derivatives of the first order.
These properties are best formulated in the a, r - coordinate
system. Obviously this is permissible since the geometrical
interpretations of the conceptions used for definition of characteristics
are independent of any coordinate system. x is the height of the
integral surface above the base plane, the first derivatives of X3
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Tollmien, W. Theory of Characteristics, report, September 1949; (https://digital.library.unt.edu/ark:/67531/metadc64520/m1/4/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.