A bore is a transition between different uniform flows of water. If a long wave of elevation travels in shallow water it steepens and forms a bore. The bore is undular if the change in surface elevation of the wave is less than 0.28 of the original depth of water. This paper describes the growth of an undular bore from a long wave which forms a gentle transition between a uniform flow and still water. A physical account of its development is followed by the results of numerical calculations. Finite-difference approximations are used in the partial differential equations of motion. For undular bores, numerical calculations show that (i) the relationship between relative elevation and relative velocity given by long wave theory is approached for an undular bore, (ii) the amplitude of first crest of an undular bore approaches a finite limit approximately at an exponential rate, and (iii) the distance between the first two crests increases without bound, approximately logarithmically.