Correction: Oscillation frequencies of long-period seismic events at Kusatsu-Shirane volcano, Japan, related to the volume of water vapour in a hydrothermal crack

There are some theoretical and typographical errors in the paper ‘Oscillation frequencies of long-period seismic events at Kusatsu-Shirane volcano, Japan, related to the volume of water vapour in a hydrothermal crack’ (M. Nakano et al. 2025). M. Nakano et al. (2025) used the equation for the sound speed of a misty gas derived by S.W. Kieffer (1982). S. Temkin & R.A. Dobbins (1966), on the other hand, derived a more theoretically justified equation, and we found that there is a slight difference between the two equations. We should have used the equation of S. Temkin & R.A. Dobbins (1966) for calculating the sound speed of a misty gas to estimate crack model parameters. In this correction, we first explain the difference between the two equations and then provide improved crack model parameters estimated using the equation of S. Temkin & R.A. Dobbins (1966). S.W. Kieffer (1982) derived the sound speed of a gas-particle mixture from the equation of state for an ideal gas under the assumption that the mixture behaves like an ideal gas (Appendix A) and expressed the sound speed of a misty gas (a) as: where cp and cv are the specific heat of gas at constant pressure and constant volume, respectively, and |$c_{\rm{p}}^{\prime}$| is the specific heat of particle. R and |${\rm{\Theta }}$| are the specific gas constant and the absolute temperature, respectively. η is the mass ratio of particles to gas, and is defined as: where ρg and ρp are the densities of the gas and particle, respectively, and x is the gas-volume fraction. Here, η is also expressed in terms of the gas-weight fraction (n) as: Substituting eq. (3) into eq. (1) gives equation (22) in M. Nakano et al. (2025): In contrast, S. Temkin & R.A. Dobbins (1966) derived the equation for the sound speed of a misty gas by solving the equation of state for an ideal gas together with the equations for the conservation of mass, momentum and energy for both the gas and particles (equation 32 in S. Temkin & R.A. Dobbins 1966): Here, |${{a}_0}$| is the sound speed in the pure gas, |${{C}_{\rm{m}}}$| is the mass ratio of particles to gas in a unit volume of the undisturbed mixture, i is the imaginary unit, |$\omega $| is the angular frequency, |$\gamma $| is the ratio of specific heat of the gas at constant pressure to the specific heat of the gas at constant volume and |${{\tau }_{\rm{d}}}$| and |${{\tau }_{\rm{t}}}$| are the dynamic and thermal relaxation times of the particles, respectively. If the effects of intrinsic losses are neglected, that is |${{\tau }_{\rm{d}}} = {{\tau }_{\rm{t}}} = 0$| in eq. (5), we obtain the following equation for the sound speed of a misty gas: Here, |${{C}_{\rm{m}}}$| is defined as: where |${{n}_0}$| and |$m^{\prime}$| are the mean number of particles per unit volume and the mass of one particle, respectively. Eq. (7) can be rewritten as: where |${{v}_{\rm{p}}}$| is the volume of one particle, respectively. Here, the total volume of particles in the mixture (⁠|${{V}_{\rm{p}}}$|⁠) is expressed as: where |${{V}_0}$| is the total volume of the mixture. Then, eq. (8) can be written as: Using the following relation: eq. (10) can be rewritten as: Consequently, η defined in eq. (2) is different from |${{C}_{\rm{m}}}$| defined in eq. (12). The latter can be expressed as: By substituting eq. (13) into eq. (6), the sound speed of a misty gas should therefore be rewritten as: Because x depends on n, the value of a in eq. (14) is a function of n, as mentioned by M. Nakano et al. (2025). Fig. 1 shows the sound speed as a function of the gas-weight fraction calculated using eqs (4) (S.W. Kieffer 1982) and (14) (S. Temkin & R.A. Dobbins 1966). The two equations give slightly different values, particularly for small n. Fig. 2 shows the differences between the crack model parameters estimated using eqs. (4) and (14). It is apparent from Fig. 2 that there are slight differences between the estimated values of n and the mass of H2O in the crack (⁠|${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$|⁠) but almost no differences between those of the crack length (L), width (W), volume (V) and a. These results demonstrate that eq. (4) well approximates eq. (14), but the use of eq. (14) is theoretically justified. Sound speed versus the gas-weight fraction for a misty gas with pure gas volume fractions larger than 0.25. Black and red lines were calculated using the equations given by S.W. Kieffer (1982) (eq. 4) and S. Temkin & R.A. Dobbins (1966) (eq. 14), respectively. Temporal changes of crack parameters from January 1989 to January 1993: (a) crack length (L), (b) crack width (W), (c) sound speed (a), (d) gas-weight fraction (n), (e) crack volume (V) and (f) mass of H2O in the crack (⁠|${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$|⁠). Black and red circles represent the crack parameters estimated using the equations for the sound speed given by S.W. Kieffer (1982) (eq. 4) and S. Temkin & R.A. Dobbins (1966) (eq. 14), respectively. Shaded areas in (f) indicate the periods when the estimated frequency values were almost constant. Using eq. (14), we reproduced the relationships between |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| against the oscillation period (T), the quality factor (Q) and the seismic moment (M0) (Fig. 3); the relationships between |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| and the volume of water vapour (⁠|${{V}_{{{{\rm{H}}}_2}{\rm{O}}}}$|⁠) against T (Fig. 4); and the simulated relationships between |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| against T, Q and M0 (Fig. 5). We note that Figs 3, 4 and 5 correspond to Figs 12, 13 and 14 in M. Nakano et al. (2025), respectively. We obtained the following T−|${{m}_{{{{\rm{H}}}_{\rm{2}}}{\rm{O}}}}$| and T−|${{V}_{{{{\rm{H}}}_{\rm{2}}}{\rm{O}}}}$| relationships, which are shown by the fitted lines in Fig 4: Relationships of the mass of H2O in the crack (⁠|${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$|⁠) against T, Q, and M0 in 1989 (magenta triangles), 1990 (blue squares), 1991 (green diamonds) and 1992–1993 (red circles) obtained using the equation for the sound speed in S. Temkin & R.A. Dobbins (1966). Relationships between |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| and |${{V}_{{{{\rm{H}}}_2}{\rm{O}}}}$| against T during 1989–1993 obtained using the equation for the sound speed in S. Temkin & R.A. Dobbins (1966). Solid and dotted lines indicate the power-law fits and their error bounds, respectively. Simulated relationships between |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| against T, Q and M0 calculated using the equation for the sound speed in S. Temkin & R.A. Dobbins (1966). We used values of |${{m}_{{{{\rm{H}}}_2}{\rm{O}}}}$| between 2 × 104 and 8 × 106 kg and constant values of Δd (4.9 × 10−6), W (136.4 m) and d/L (5.9 × 10−4). The value of n ranged from 0.01 to 0.99. The estimated values of the crack parameters, which are also listed in Table S1, are almost identical to those reported in M. Nakano et al. (2025). We found the following typographical errors in M. Nakano et al. (2025). In the third paragraph of Section 3.1, the spelling of ‘Hibert’ was incorrect. The correct sentence is, ‘A bandpass filter was first applied to both observed and synthetic displacement waveforms, and their envelope waveforms were calculated using the Hilbert transform.’ In the fifth paragraph of Section 3.2, the particle radius was incorrectly stated as ‘1 μm’. The correct sentence should read ‘We assumed that |$Q_{\rm{i}}^{ - 1} \ll \ Q_{\rm{r}}^{ - 1}$| for misty gas containing small particles with radii smaller than 10 μm in accord with Taguchi et al. (2018).’ In the eighth paragraph of Section 3.2, the description incorrectly implied that the crack volume V was estimated using a, L, W and d. The correct statement should read ‘Using L, W and d, we estimated the crack volume (V) as:’ In the first paragraph of Section 5.2, the equation number was incorrectly stated as ‘eq. (30)’. The correct sentence should read ‘Our estimates showed that |${{m}_{{{{\rm{H}}}_{\rm{2}}}{\rm{O}}}}$| depended on T (Fig. 13a and eq. 33).’ In the first paragraph of Section 5.3, ‘the 2016 eruption at White Island, New Zealand’ was incorrect. It should read ‘the 2013 eruption at White Island, New Zealand’. In Table 1, the symbol for ‘Oscillation mode’ was misstated as ‘M’. The correct symbol is ‘m’. These theoretical and typographical errors do not affect the discussion and conclusions in M. Nakano et al. (2025). We used seismic data for LP events at the Kusatsu-Shirane volcano provided by the Earthquake Research Institute of The University of Tokyo (http://www.eri.u-tokyo.ac.jp/en/). We provide the derivation of the sound speed of a gas-particle mixture based on S.W. Kieffer (1982). They assumed that the gas-particle mixture consists of sufficiently small particles such that thermal equilibrium is maintained between phases due to rapid heat transfer and collisions between particles are neglected. The latent heat was also neglected. Under these assumptions, a gas-particle mixture can be treated as a pseudogas, allowing us to express the sound speed of the mixture using the equation of state for an ideal gas: where P and |${{\rho }_{\rm{g}}}$| are the pressure and density of the pure gas, respectively, R is the specific gas constant and |${\rm{\Theta }}$| is the absolute temperature. The density of the pseudogas (⁠|${{\rho }_{\rm{f}}}$|⁠) is given as: where |${{\rho }_{\rm{p}}}$| and |${{\rho }_{\rm{g}}}$| are the densities of the particles and the gas, respectively, and n is the gas-weight fraction, expressed as: where |$\eta $| is the mass ratio of the particles to gas. Using eqs (A.2) and (A.3) and assuming |${{\rho }_{\rm{p}}} \gg {{\rho }_{\rm{g}}}$|⁠, we obtain: Eq. (A.4) can be rewritten as: By substituting eq. (A.5) into eq. (A.1), the equation of state for a pseudogas can be approximated by: where |${{R}_{{\rm{mix}}}}$| is the specific gas constant of the pseudogas, defined as: Based on S.W. Kieffer (1982), we assume that the pseudogas is homentropic. Accordingly, the total entropy change of the pseudogas (dS), which is equal to the sum of entropy changes of the particles (⁠|$d{{S}_{\rm{p}}}$|⁠) and of the gas (⁠|$d{{S}_{\rm{g}}}$|⁠), must be zero. Thus: where |${{c}_{\rm{p}}}$| and |$c_{\rm{p}}^{\prime}$| are the specific heats of the gas at constant pressure and of the particles, respectively. By integrating eq. (A.8), we obtain: where: Here, |${{\gamma }_{{\rm{mix}}}}$| is the heat capacity ratio of the pseudogas. Using Mayer’s relation, i.e. |$R = {{c}_{\rm{p}}} - {{c}_{\rm{v}}}$|⁠, eq. (A.10) can be rewritten as: where |${{c}_{\rm{v}}}$| is the specific heat of the gas at constant volume. Then, the sound speed of the pseudogas is given as: Thus, eq. (1) is obtained by substituting eqs (A.7) and (A.11) into eq. (A.12). Here, eqs (A.7), (A.11) and (A.12) are respectively identical to equations (30), (29) and (7) in H. Kumagai & B.A. Chouet (2000) and equations (9), (8) and (4) in M.M. Morrissey & B.A. Chouet (2001).