General approach to travelling-wave solutions

The behaviour we are expecting to emerge from this physico-chemical model is that of a steady wave of reaction moving from left to right in Fig. 11.2 into the region of unreacted A. By steady, we really mean that the wavefront should maintain its shape as it moves with a constant speed. It is this shape and speed which we seek to determine (and express in terms of the rate constant, diffusion coefficient, etc.). 

we assume that there actually is a steady wave and that it has a constant velocity c = dx/dr. (We can identify a particular point on the wavefront, e.g. the point at which the reaction is half-way to completion, a = p = 0.5, and c is then the speed at which this point is moving to the right.) 

A useful technique which can be applied in cases where constant velocity solutions arise is that of changing from a fixed coordinate system (the present x coordinate has an origin whose position is fixed in space) to travelling-wave coordinates. With the latter, the origin moves from left to right with the front at the same constant velocity c we are constantly adjusting our frame of reference so that within the frame the reaction front appears stationary. This new coordinate z is defined in terms of x and t by 

We must now write our reaction-diffusion equation in terms of the z coordinate. For this we need the identities 

Equation (11.15) is a highly significant development from the original equations. In stages, we have come from two equations to one and have now manipulated that partial differential form in x and t into an ordinary differential equation in z. 

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