Fig. 3

Frequency ratio of the fundamental resonant frequency between the theoretical values and numerical simulation results. The theoretical frequencies are calculated from Eq. (1). The simulations were performed under two conditions (Additional file 2: Figs. S2–3): a a frustum pipe is connected to a flat plane, and b a frustum pipe is situated at the center of the crater floor of Aso volcano. These figures show the cases, where the closed radius of the frustum (\(a_\text {c}\)) is 58 m. The sound velocities inside the vent were assumed to be 400 m/s, 600 m/s, 800 m/s, and \(c_\text {air}\) (1-D profiles of the velocities are shown in Additional file 2: Fig. S2–2). At the high velocity inside the vent, the computed fundamental frequency is lower than the theoretical value (a, Additional file 2: Fig. S2–4d). The computed fundamental frequency with topography becomes slightly lower than that with a flat plane (b). The pipe lengths, indicated by the white-filled circles, cannot produce any overtone peaks of the amplitude spectra. c Amplitude spectra obtained from the simulations when \(c = 600\) m/s and \(a_\text {c} = 58\) m (with the volcanic topography). The black line shows the spectrum when the pipe length is 150 m, and the thin gray line shows the spectrum when the pipe length is 25 m. The spectral structure of the 150-m pipe is very similar to the observed structure (Fig. 2a). However, a very short pipe cannot make a clear two-peaked spectral structure such as that observed