An unusual spatiotemporal phenomenon was observed in propagating fronts in the bromate-sulfite system with an initial pH gradient, but not in chlorate-sulfite nor iodate-sulfite systems. Periodic acceleration and deceleration of the front were observed, which appeared as a "jumping" wave. An experimental method was developed to produce the pH gradient systematically and reproducibly in a tube. The effect of the gradient was studied. With the use of infrared and visible imaging the competition between chemically and thermally induced convection was observed. A qualitative model is proposed that includes the effect of pH gradient and the specific mechanism of the bromate oxidation of sulfite.