IL OF REGIONAL SCIENCE 男,1891 WAVES OF SPATIAL DIFFUSION RECONSIDERED Herbert G. Kariel and Michael J. va Two stimulating and innovative papers by Morrill were concerned with waves of spatial diffusion.(Morrill,[1, 2 )The purpose of the first was to propose some insights and interpretations which can be obtained by characteriz ive-like phend Morrill scopic and theoretical implications of such a conceptualization. He was concerned with both time and distance, and noted that in earlier conceptualizations more emphasis had been placed on de ecay re elated to time than on that related to distance. He therefore hypothesized that the number of innovations would de- crease not only over time but also with distance from the original source; that is in areas farther from the source the final proportion of adopters would be smaller than in areas closer to it. This notion seemed to him compatible with wave henomena in general, where the size and force of the wave decrease over both time and space as it moves outward from the source Using standard wave equations, he showed by means of a diagram the relationship between adoption and distance. (Figure 1. )1 He also proposes that space and time are symmetrical in that they can be substituted for each other in the model In order to test this model, in the second paper he analyzed the actual diffusion of three phenomena He concluded that for these data space and time were not good substitutes for each other, because at the source of the innovation the peak of acceptance came well after the beginning of the diffusion proces, whereas according to the model it should have come at the beginning. He therefore suggested two modifications of the original equation, one for distance(Figure 2)and one for time(Figure 3) e propose to show that by extending the x-axis so as to include a zero oint, Morrill's original formulation is more accurate than he had realized. We also have expanded the model so as to incorporate the notion of energy which can be considered to exist in a system where spatial diffusion is occurring We first observed that in Morrill's original model the origin was not at the zero point, (Figure 1)whereas by convention a wave starts with an amplitude of zero. When he tested it, however, the origin was at the zero point(Figures 4 and Associate Professor and Student, Department of Geography, University of Calgary All figures are from Morrill, [1, 2] Figure 1= Figure 10 in [ u] Figure 2= Figure 10 in [2], Figure 3= Figure l1 in [2] Figure 4= Figur res 6,7, and 8 in [2]
JOURNAL OF REGIONAL SCIENCE, VOL. 13, NO. 2. 1973 5). We then noticed that when the curves constructed for testing the model were ally shifted to the right, the fit of the model to the data seemed to be fairly Such a shift would also yield a more realistic view of the diffusion process, the rate of adoption starts at zero, increases to an inflection point, and then decreases until some saturation level is reached, resulting in the well-known logistic growth curve. Because we had considered it possible to conceptualize the diffusion process in terms of energy inputs into the system, we wondered if it would be possible to Distance(or Time) FIGURE 1: Changing Form of a Single Wave with Time and Distance Interchangeable T Distance FIGURE 2: anging Form of an Innovation Wave, Acceptance Over Distance at Given Time Periods FIGURE 3: Changing Form of an Innovation Wave, Acceptance Over e at Given Distances
KARIEL vASELENAK: WAVES OF SPATIAL DIFFUSION D=2 PASTURE GRAZING TUBERCULOSIS GHETTO SUBSIDY CONTROL EXPANS ION FIGURE 4: Number of Acceptances for Three Phenomena Over Time at Various Distances use the model given by Morrill for this purpose. The general wave formula, as iven by morrill is e t! where as= the amplitude or vertical displacement, Ao= the initial amplitude b= the coefficient of frictio d= distance and When this equation is applied to the study of spatial diffusion, the amplitude at would represent the number of adopters at time t, which is a function of Ao, the number of adopters at time to, distance, time, and a coefficient of friction. morrill hints at the existence of th s coefficient of friction by suggesting that early
JOURNAL OF REGIONAL SCIENCE, VOL 13, NO 2, 1973 930-1940 GHETTO PASTURE GRAZING TUBERCULO CONTROL EXPANSION SUBSIDY FIGURE 5: Number of Acceptances for Three Phenomena by Distance for Various Time Periods adopters may be more enthusiastic tellers about an innovation than later ones; in other words, that over time the diffusion wave loses energy through friction and its amplitude therefore decreases. The relationship between energy and ampli tude is defined in the equation representing the total energy of a particle execut- ing harmonic motion E=kaz where a the amplitude and k a constant(Resnick and Halliday [3, p. 354]) Clearly, we can solve for a and then substitute energy into the wave equation. It becomes possible thereby to interchange the concepts of energy with the number of adopters. The implication for diffusion theory is that the total energy of innovation waves can in this manner then be plotted against time or distance. By introducing the concept of energy into the study of spatial diffusion, greater omprehensiveness can be gained for explaining the diverse topics investigated by regional scientists
KARIEL VASELENAK: WAVES OF SPATIAL DIFFUSION If one accepts the above explanation of including a zero point, then distance and time are symmetrical and substitutable in waves of diffusion. In addition to obtaining a closer fit to the data, a more realistic view of the diffusion process is studying waves of spatial diffusion possible to incorporate the concept of energy when REFERENCES 1] Morrell, R. L " Waves of Spatial Diffusion, "Journal of Regional Science, 8(1968), 1-18 The Shape of Diffusion in Space and Time, "Economic Geography, 46(1970), 259- 3] Resnick, R and D. Halliday. Physics. New York: Wiley, 1966