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Dipolar modes of solar-like oscillations of red giant stars are analyzed asymptotically. Because of the high mass concentration in the helium core, the oscillations of the stars are composed of internal gravity waves in the core and acoustic waves in the envelope. The two types of oscillations interact with each other through a thin intermediate evanescent region to form an eigenmode of the mixed character. The process of the eigenmode formation is analyzed by assuming that the wavelength of the oscillations is much shorter than the scale height of the equilibrium stellar structure. Special care is paid to the following two points: (1) the effect of the perturbation to the gravitational potential is fully taken into account; (2) the interaction between the gravity waves in the core and the acoustic waves in the envelope can be strong. The condition that every eigenfrequency of the oscillations should satisfy is formulated. Also discussed are the amplitude ratio between the core and the envelope, and the transmission and reflection of the progressive-wave solutions at the intermediate evanescent region. The analysis should be of fundamental use in the interpretation of the observed solar-like oscillations in red giant stars.