4.4. EXERCISES 91
(i) Show that for 2 - D flows, (4.143d) is u
y
v
x
D 0, and is identically satisfied by
introducing a potential u D
x
and v D
y
.
(ii) Assuming that p D c
2
where c is a constant speed of sound, show that the system
reduces to the single equation
2
x
c
2
xx
C 2
x
y
xy
C
2
y
c
2
yy
D 0: (4.144)
(iii) Show that under a Legendre transformation this equation linearizes.
4.7. e governing equations for highly frictional granular materials is given by
@
xx
@x
C
@
xy
@y
D 0; (4.145a)
@
xy
@x
C
@
yy
@y
D g; (4.145b)
xx
yy
2
xy
D 0; (4.145c)
where
xx
,
yy
and
xy
are normal and shear stresses, respectively, the material
density, and g, the gravity. In (4.145), (4.145a) and (4.145b) are the equilibrium
equations, whereas (4.145c) is the constitutive relation for the material [69].
(i) Show that by introducing a potential u, such that
xx
D u
y
;
xy
D u
x
; then
yy
D
u
2
x
u
y
; (4.146)
Eq. (4.145) reduces to
u
2
y
u
xx
2u
x
u
y
u
xy
C u
2
x
u
yy
C gu
2
y
D 0: (4.147)
(ii) Show that under a Hodograph transformation, this linearizes.
4.8. e shallow water wave equations are given by
u
t
C uu
x
C g
x
D 0; (4.148a)
t
C
Œ
u. h.x//
x
D 0; (4.148b)
where u D u.x; t / the velocity, D .x; t/ the free surface, h D h.x/ the bottom
surface, and g is gravity [70].