In this work, the acoustic Helmholtz equation is derived, and its analytical solution and the characteristic equation of the uniform guide mode in single mode fibers are obtained by the method of separation of variables. The normalized frequency of the acoustic mode is defined. By combining the argument approximation of the Bessel function are analyzed the eigenvalue range of the acoustic mode, the cut-off frequency, far from the cut-off state of the acoustic mode induced by backward stimulated Brillouin scattering, the dispersion and the multi-peak Brillouin gain spectrum. The research results indicate that the longitudinal acoustic fundamental mode L₀₁ cannot be cut-off and is mainly confined in the fiber core, which is coupled with the optical fundamental mode LP₀₁ to form the main peak of the Brillouin gain spectrum. The other higher-order acoustic modes all have low cut-off frequencies and are distributed more in the fiber cladding than mode L₀₁ which couples with the optical fundamental mode LP₀₁ to form the subpeaks of the Brillouin gain spectrum. The transverse normalized phase constant and effective refractive index of the acoustic mode increase with normalized frequency increasing. Only longitudinal acoustic modes L_0n contribute to backward Brillouin gain spectrum in single mode fiber. When the GeO₂ concentration is less than 4% and core radius is 4.5 μm, the single mode characteristics of the fiber remain unchanged, but the maximum number of acoustic L_0n modes is 4. With the increase of GeO₂ concentration in the fiber core, the Brillouin gain spectrum is red-shifted and the number of acoustic modes increases, the Brillouin gain peak value of L₀₁ mode gradually increases, and the contributions of higher-order modes decrease. The single-mode fiber with a core’s germanium doped concentration of 3.65% and core radius of 4.3 μm has 4 L_0n modes and 16 L_mn (m>0) modes at a wavelength of 1.55 μm, with one main peak and two subpeaks in the Brillouin gain spectrum appearing due to the acousto-optic coupling of the acoustic L₀₁, L₀₃, and L₀₄ modes with the optical LP₀₁ mode. The single-mode fiber with a core’s germanium doped concentration of 15% and core radius of 1.3 μm has 3 L_0n modes and 7 L_mn (m>0) modes, with the Brillouin gain spectrum having 3 main peaks due to the acousto-optic coupling of the L₀₁, L₀₂, and L₀₃ modes with the LP₀₁ mode. These conclusions are well consistent with the reported experimental phenomena and provide theoretical references for studying and utilizing the SBS acoustic waveguide in optical fibers.