The possibility of travelling reaction–diffusion waves developing in the chemical system governed by the quadratic autocatalytic or branching reaction A + B → 2B (rate
k
1
ab
) coupled with the decay or termination step B → C (rate
k
2
b
) is examined. Two simple solutions are obtained first, namely the well-stirred analogue of the spatially inhomogeneous problem and the solution for small input of the reactant B. Both of these indicate that the criterion for the existence of a travelling wave is that
k
2
<
k
1
a
0
, where
a
0
is the initial concentration of reactant A. The equations governing the fully developed travelling waves are then discussed and it is shown that these possess a solution only if this criterion is satisfied, i. e. only if
k
=
k
2
/
k
1
a
0
< 1. Further properties of these waves are also established and, in particular, it is shown that the concentration of A increases monotonically from its fully reacted state at the rear of the wave to its unreacted state at the front, while the concentration of B has a single hump form. Numerical solutions of the full initial value problem are then obtained and these do confirm that travelling waves are possible only if
k
< 1 and suggest that, when this condition holds, these waves travel with the uniform speed
v
0
= 2√ (1 –
k
). This last result is established by a large time analysis of the full initial value problem that reveals that ahead of the reaction–diffusion front is a very weak diffusion-controlled region into which an exponentially small amount of B must diffuse before the reaction can be initiated. Finally, the behaviour of the travelling waves in the two asymptotic limits
k
→ 0 and
k
→ 1 are treated. In the first case the solution approaches that for the previously discussed
k
= 0 case on the length scale associated with the reaction–diffusion front, with the difference being seen on a much longer,
O
(
k
–1
), length scale. In the latter case we find that the concentration of A is 1 +
O
(1 –
k
) and that of B is
O
((1 –
k
)
2
), with the thickness of the reaction–diffusion front being of
O
((1 –
k
)
½
).
Research Article. Geodesic equations and their numerical solutions in geodetic and Cartesian coordinates on an oblate spheroid