The quality of certain processes over time can be characterized by a sequence of functional profiles. Effective monitoring of such processes requires timely detection of changes in the underlying signal of noisy profiles, since deviations from a desired in-control function can indicate an out-of-control process. Challenges of this problem include maintaining the computational efficiency needed to consider profiles in real time and the adaptability to detect functional changes of any form. This paper proposes a novel Bayesian methodology to obtain the analytic posterior distribution of the change point in the sequence and to use this to assess the state of the process at each time point. Unlike other Bayesian approaches that monitor changes in the parameters of observed profiles, the method uses wavelets to avoid restrictive parametric assumptions on the form of the functions. Additionally, the method adaptively incorporates the sparsity of the wavelet representation in the selection of a prior distribution for the coefficients. The method's effectiveness is demonstrated through a simulation comparison with another wavelet-based profile monitoring method.