The build-up of the heat field in shallow coastal water due to a point source has been investigated using an analytical solution of a time-integral form derived by extending the solutions by Holley(1969) and also presented in Harleman (1971). The uniform water depth is assumed with non-isotropic turbulent dispersion. The alongshore-flow is assumed to be uni-directional, spatially uniform and oscillatory. Due to the presence of the oscillatory alongshore-flow, the heat build-up occurs in an oscillatory manner, and the excess temperature thereby fluctuates in that course and even in the quasi-steady state. A series of calculations reveal that proper choices of the decay coefficient as well as dispersion coefficients are critical to the reliable prediction of the excess temperature field. The dispersion coefficients determine the absolute values of the excess temperature and characterize the shoreline profile, particularly within the tidal excursion distance, while the decay coefficient determines the absolute value of the excess temperature and the convergence rate to that of the quasi-steady state. Within the e-folding time scale $1/k_d$ (where $k_d$ is the heat decay coefficient), heat build-up occurs more than 90% of the quasi-steady state values in a region within a tidal excursion distance (L), while occurs increasingly less the farther we go to the downstream direction (about 80% at 1.25L, and 70% at 1.5L). Calculations with onshore and offshore discharges indicate that thermal spreading in the direction of the shoreline is reduced as the shoreline constraint which controls the lateral mixing is reduced. The importance of collecting long-term records of in situ meteorological conditions and clarifying the definition of the heat loss coefficient is addressed. Interactive use of analytical and numerical modeling is recommended as a desirable way to obtain a reliable estimate of the far-field excess temperature along with extensive field measurements.