Wave 2 was the ocean gravity wave generated when wave 1 passed the slope. Wave 1′ was the transmitted pressure-forced wave traveling at the same speed at shallow depths. When wave 1 passed the slope, it split into waves 1′ and 2. The moving air pressure wave induced the incident pressure-forced wave (wave 1), which travels at the same speed as the air pressure wave. Figure 2a illustrates the different snapshots of wave profiles. Figure 2 shows the synthetic test of an air pressure wave traveling at 310 m/s passing over the slope. We extracted the wave profile at the center of the y direction, where y = 556 km. The pressure-forced wave traveled at the speed of the Lamb wave, which is much faster than the ocean gravity waves, which traveled at the long wave speed, \(\sqrt \) when \(t = 0\). At the source, the Lamb wave induced a sea surface disturbance (hereafter referred to as a pressure-forced wave), and ocean gravity waves were simultaneously generated due to the conservation of mass (Kubota et al. 2016), was caused by the air pressure pulse excited by the eruption, a pulse known as a Lamb wave (Lamb 1911 Matoza et al. The fast-traveling tsunami wave, which was a type of meteotsunami (Hibiya and Kajiura 1982 Monserrat et al. 2022), much faster than the conventional tsunami speed of approximately 200 m/s for an average ocean depth of 4 km. Furthermore, the recorded data revealed that the tsunami traveled at a speed of approximately 300–315 m/s (Kubota et al. After the eruption, global tsunami monitoring systems and tide gauges observed tsunami signals earlier than theoretically expected for tsunami waves (Carvajal et al. The Hunga Tonga-Hunga Ha'apai volcano, located in the Kingdom of Tonga in the South Pacific, approximately 65 km north of the Tongan capital island of Tongatapu, erupted violently at UTC 04:14, 15 January 2022 (USGS). We explained the wave separation phenomenon and reproduced the waveforms of different splitting stages observed by the stations near the Japan Trench. Because the ocean gravity wave was slower than the pressure-forced wave near Japan, it was separated from, and traveled behind, the pressure-forced wave. Simultaneously, an ocean gravity wave is generated due to the conservation of water volume. Our simulations show that changes in water depth can amplify or decrease the pressure-forced wave. We found that the pressure-forced waves split and generated ocean gravity waves after passing the Japan Trench based on the S-net data. This was the first case in which ocean bottom monitoring systems widely observed an air pressure-induced tsunami. This tsunami was driven by the pressure-forced wave traveling at the speed of the Lamb wave and, thus, was much faster than conventional tsunamis. The 2022 eruption of the Hunga Tonga-Hunga Ha'apai volcano excited an atmospheric Lamb wave, which induced a fast-traveling tsunami.
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