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2.6 Classification of Conservation Equations 67

disturbances in a boundary layer on a flat plate by means of a vibrating ribbon
held parallel to the plate and normal to the freestream. The measured neutral
stability curve, critical Reynolds number and amplification rates, discussed in
Chapter 8, were found to agree well with the predictions of the linear stability
theory.

2.6 Classification of Conservation Equations

To understand some of the basic ideas underlying the classification of conserva-
tion equations, we examine the general linear second-order partial-differential
equation in two independent variables:

2
2
2
d u _ d u d u Bu du „ 7 / , /rt ^ ^
LU + b + + d + + /U HX V) (2 6 1}
" °&* ^dy °W d~x % " > " -
Rotation of the xy-plane into the x^-plane by the coordinate transformation,
x = x cos 6 + y sin #, y — — x sin 6 + y cos 6 (2.6.2)
allows Eq. (2.6.1) to be transformed to
2
2
- _d u _d u -du _du p / n / l o .

This removal of the mixed derivative term by choosing 6 so that b = 0 is similar
to the transformation applied to the general quadratic equation; the new coor-
dinate axes are aligned with the principal axes of the conic section. Indeed, we
adopt the terminology for the conies to classify the differential equations.
Thus if ac > 0 (i.e., a and c have the same sign), we call Eq. (2.6.3) an
elliptic equation; if ac < 0, we call Eq. (2.6.3) a hyperbolic equation; and finally
if ac — 0 (but both do not vanish), we call Eq. (2.6.3) a parabolic equation. As
2 2
with the conies, we note that b — 4ac — b — 4ac is an invariant of the rotation.
2
Thus we say that Eq. (2.6.1) is elliptic, hyperbolic, or parabolic if b — 4ac < 0,
> 0, or = 0, respectively.
In the elliptic case we can stretch or compress the coordinates by introducing

x=-=, V=^f (2.6.4)

and we get finally

2
2
d u d u =du =du
Lu d
= 7^2 2 + ~^2 + ^E + ^ + fu = h (2.6.5)
<9I 9 ^ OS
The leading term here is the well-known Laplacian in the xp-plane. So we see
that, by a simple rotation and stretching of the coordinate system, any elliptic
equation in the plane can be reduced to the Laplacian, plus lower-order terms.