∂λ
∂χ
¼
υ
υ λ 0 , 0
ð
Þ
1
λ
2
:
If the latter equation is written in terms of the normalized specific volume of the
ambient air, namely, υ 0 ¼ 1/ρ 0 , such that, e υ ¼ υ=υ 0 and e υ λ 0 , 0
ð
Þ¼ υ λ 0 , 0
ð
Þ=υ 0 , we
have
e υ ¼ e υ λ 0 , 0
ð
Þλ
2 ∂λ
∂χ
,
and differentiating this with respect to τ yields,
∂e υ
∂τ
¼ e υ λ 0 , 0
ð
Þλ 2e u
∂λ
∂χ
þ λ
∂e u
∂χ
,
ð6:20Þ
which is an alternative form of the continuity equation.
6.2.3 Energy Equation
The energy equation (Eq. 4.12) in Lagrangian form with artificial viscosity included
is
∂p
∂t
¼
1
ρ
γp þ γ À 1
ð
Þq
½
Š
∂ρ
∂t
,
and it is straightforward to show that it reduces to
∂e p
∂τ
¼
1
e ρ
γe p þ γ À 1
ð
Þe q
½
Š
∂e ρ
∂τ
,
where all quantities are now written in dimensionless units. When the latter equation
is written in terms of the specific volume rather than the density it is easy to show that
it becomes;
γe p þ γ À 1
ð
Þe q
½
Š
∂e υ
∂τ
þ e υ
∂e p
∂τ
¼ 0:
ð6:21Þ
6.2 Lagrangian Equations in Spherical Geometry
287