Ocean bottom pressure data observed at 113 sensors at S-net (National Research Institute for Earth Science and Disaster Resilience, 2019) were downloaded (Figure 1). The observed data are filtered in the periods between 100 and 3600 s using the bandpass filter in Seismic Analysis Cord (SAC) developed by Goldstein et al, (2003) (Supplement S1).

The governing equations to be solved by the numerical simulation is briefly explained. Linear approximation of Euler`s momentum equation is:

$$\frac{\partial u}{\partial t}=-\frac{1}{\rho }\frac{\partial p}{\partial x}$$

$$\frac{\partial v}{\partial t}=-\frac{1}{\rho }\frac{\partial Pp}{\partial y}$$

1

$$\frac{\partial w}{\partial t}=-g-\frac{1}{\rho }\frac{\partial p}{\partial z}$$

where *u*, *v*, and *w* are the velocity in x, y, z-direction, respectively, *p* is the pressure, *g* is gravity acceleration, and \(\rho\) is the water density. By using longwave, or shallow water, approximation, third equation in (1) becomes

$$\frac{\partial w}{\partial t}=-g-\frac{1}{\rho }\frac{\partial p}{\partial z}=0$$

2

Then, pressure *p* is \(p=-\rho gz+c\) where c is a function of x, y, and t. At the ocean surface where *z* is a wave height (*h*), pressure (*p*) becomes atmospheric pressure (*p**0*) which is an input for the air-sea coupled wave in this study. Pressure, *p*, then becomes \(p=\rho g\left(h-z\right)+{p}_{0}\). The momentum equations (1) then become

$$\frac{\partial u}{\partial t}=-g\frac{\partial h}{\partial x}-\frac{1}{\rho }\frac{\partial {p}_{0}}{\partial x}$$

$$\frac{\partial v}{\partial t}=-g\frac{\partial h}{\partial y}-\frac{1}{\rho }\frac{\partial {p}_{0}}{\partial y}$$

3

Next, continuity equation of the linear long wave is

$$\frac{\partial h}{\partial t}=-\frac{\partial du}{\partial x}-\frac{\partial dv}{\partial y}$$

4

where *d* is the ocean depth. Momentum equations (3) for air-sea coupled waves and continuity equation (4) were numerically solved using a staggered grid system with an input of the atmospheric pressure gradient at each time step. The grid sizes of the numerical computation were set at 1.5 km in both x and y directions.

The atmospheric pressures observed at approximately 3000 points in Japan (Weathernews, https://jp.weathernews.com/news/38708/, 2022) showed that the pressure pulse (a peek amplitude of approximately 2hPa, and the duration of 20-15 mins) passed through Japan from southeast to northwest with a strike of -44°. The shape of the pressure pulse was assumed to be half the wavelength of sine wave. The half-wavelength was set to 300 km, which corresponded to a duration of 16 mins. The peek amplitude was set to 2hPa. First, numerical computation in one dimension with a constant ocean depth of 5500 m was carried out to obtain the steady state of air-sea coupled initial wave. The pressure pulse and the steady state initial wave are entered into the two-dimensional computational domain from the low boundary along the x-axis (Figure 2). The pressure pulse is continuously propagated in y-direction with a constant velocity of 312 m/s. The bathymetry was rotated 44° clockwise (Figure 1 and 2) to match a strike of the pressure wave from Tonga.