ABSTRACT
Let's apply the concepts presented in Sections I1I.3 and TIL7 to determine the characteristics of the system of two coupled convection equations, Eqs. (11.65) and (11.66), where c has been replaced by a to model acoustic wave propagation:
It + agx = 0 gt +alx = 0
Applying the chain rule to the continuous functions f(x, t) and g(x, t) yields
(11.84) (11.85)
- - -
df = It dt + Ix dx and (11.86)
(11.87)
(11.88)
(11.89)
[}, I ! ~ J[f,J= [~J o 0 dt dx gx dg
The characteristics of Eqs. (11.84) and (11.85) are determined by setting the determinant of the coefficient matrix of Eq. (11.87) equal to zero. This gives the characteristic equation:
(1)[-(dx)2] + dt(a2 dt) = 0 Solving Eq. (11.88) for dx/dt gives
I :~±a I Equation (11.89) shows that there are two real distinct roots associated with the characteristic equation. The physical speed of information propagation c along the characteristic curves is
dx c =-= ±a
dt (11.90)
Consequently, information propagates in both the positive and negative x directions at the wave speed a.
