# Preprint 2008-029

# 3-D kinematical Conservation Laws: Equations of Evolution of a Surface

## K. R. Arun and Phoolan Prasad

**Abstract:**
3-D KCL are equations of evolution of a propagating
surface
Ω_{t} in 3-space dimensions
and were first derived by Giles, Prasad and Ravindran in 1995
assuming the motion of the surface to be isotropic.
Here we discuss various properties of these 3-D KCL.
These are the most general equations in conservation form,
governing the evolution of
Ω_{t} with
singularities which we call kinks
and which are curves across which the normal
\mathbf{n}
to Ω_{t}
and amplitude *w* on
Ω_{t}
are discontinuous.
From KCL we derive a system of six differential equations
and show that the KCL system is equivalent to the ray equations for
Ω_{t}.
The six independent equations and an energy transport equation
(for small amplitude waves in a polytropic gas)
involving an amplitude
*w* (which is related to the normal velocity *m* of
Ω_{t})
form a completely determined system of seven equations.
We have determined eigenvalues of the system by a very novel method
and find that the system has two distinct nonzero eigenvalues
and five zero eigenvalues
and the dimension of the eigenspace associated with the multiple
eigenvalue 0 is only 4.
For an appropriately defined *m*,
the two nonzero eigenvalues are real when
m > 1 and pure imaginary when
m < 1.
Finally we have shown how to use the theory to concrete examples.