In this study, the combustion instability characteristics are experimentally investigated in a partially premixed gas turbine model combustor. The combustor is operated with methane and preheated air as the fuel and oxidizer, respectively, at atmospheric pressure. The experiment is carried out at various equivalence ratios and flow rates of fuel and air to investigate the effect on the combustion instability frequency transition. According to the experimental results, the transition of the combustion instability frequency to higher longitudinal mode occurs because of the flow rate variation. To explain the frequency shift phenomenon, the concept of convection time is introduced, which is mostly affected by the flame position and exit velocity of the fuel-air mixture. The flame positions are measured using OH planar laser-induced fluorescence (OH-PLIF), and the flow field information is obtained using particle image velocimetry to calculate the convection time. The measurement results show that the injection velocities of fuel and air are also important factors in determining the combustion instability frequency as well as the equivalence ratio, which is a crucial parameter of the flame position. As a result, it is found that the decrease in convection time owing to a closer distance from the dump plane to the flame and a faster exit velocity of the fuel-air mixture causes the combustion instability frequency mode shift. Additionally, the structural characteristics of the flame are analyzed using high-speed OH-PLIF measurement. The differences in the flame structure between the stable and unstable flames in the 2nd and 3rd longitudinal modes are analyzed. The change in the unburned mixture is mainly observed and the relationship between the dynamic pressure, heat release rate, and length of the unburned region is also analyzed.