### 1. Introduction

### 2. Technical Background

### 2.1 UNDEX Model

*t*and stand-off distance

*R*where the gas bubble volume acceleration

*V¨*is presented in Eq. (2).

*a*, and vertical upward migration,

*u*, of a gas bubble based on the doubly asymptotic approximation (DAA). The rates of the radius change and vertical displacement are given by Eqs. (3) and (4), respectively. In order to calculate the radius of the gas bubble and its migration, Eqs. (3) and (4) should be integrated with the seven initial conditions at

*t*= 7

_{I}*T*. By integrating Eq. (2), the first condition of the initial radius of the gas bubble and second condition of the initial radial velocity of the gas bubble can be obtained. i.e.,

_{c}*a*(

*t*) =

_{I}*a*=

_{I}*V̇*(

*t*)/4

_{I}*πa*

^{2}(

*t*) and

_{I}*ȧ*(

*t*) =

_{I}*ȧ*=

_{I}*V̇*(

*t*)/4

_{I}*πa*

^{2}(

*t*), respectively.

_{I}*φ*

_{l}_{0}(

*t*), that is shown in Eq. (8) can be obtained. Similarly, Eqs. (4) and (7) produce the fourth condition of

_{I}*φ*

_{g}_{1}(

*t*), as given in Eq. (9). The fifth and sixth conditions are the initial location

_{I}*u*(

*t*)= 0 and the initial velocity

_{I}*u̇*(

*t*) = 0. The last condition corresponds to the fluid potential

_{I}*φ*

_{l}_{1}(

*t*) = 1/2(

_{I}*g*/

*c*)

_{l}*a*

_{I}^{2}.

*φ*_{l}_{0}: fluid velocity potential corresp. to 3^{rd}initial condition*φ*_{l}_{1}: fluid velocity potential corresp. to 4^{th}initial condition*c*: the speed of sound in the fluid_{l}*φ*_{g}_{1}: gas bubble potential*ρ*: gas bubble density_{g}*P*: gas bubble pressure constant (=_{g}*K*(_{c}*V*/_{c}*V*))^{γ}*V*: initial volume of the gas bubble_{c}*V*: volume of gas the bubble*γ*: specific heat ratio of the gas bubble*ζ*: impedance ratio (=*ρ*_{g}*c*/_{g}*ρ*_{l}*c*)_{l}*K*: adiabatic pressure constant_{c}*t*: initial time_{I}*p*: initial pressure at the source point (=_{I}*p*+_{atm}*ρ*_{l}*gd*)_{I}*d*: initial depth of charge mass_{I}*p*: atmospheric pressure_{atm}*g*: gravitational constant0

*P*, is determined using the stand-off distance,

_{R}*R*, of Eq. (13), where

*x**and*

_{s}

*x**are the coordinates of the stand-off and charge points, respectively. The time pressure term,*

_{c}*P*, should be distinguished by the shock phases: the primary shock wave (

_{t}*t*≦7

_{I}*T*) and the gas bubble wave (

_{c}*t*>7

_{I}*T*). The charge constants are far smaller than unity, thus the stand-off distance has a minor effect for the primary shock wave phase, thus the stand-off distance may be often assumed to be constant.

_{c}### 2.2 Definition of PVSS

*f*and

_{i}*S*in Eq. (14) are the natural frequency and the SRS of the relative displacement, respectively. A PVSS,

_{d}*S*, can be calculated by multiplying the relative displacement for each frequency by the own frequency. The PVSS unit is the same as the units of velocity.

_{pv}*X*(

*z*) and

*Y*(

*z*) are

*z*transformations of the input acceleration

*x¨*(

*t*) and the relative displacement response z(

*t*). The coefficients of

*b*

_{0},

*b*

_{1}and

*b*

_{2}are dependent on the types of response, and they are determined using Eqs. (16), (17), and (18), respectively.

### 2.3 PVSS Criteria

*V*

_{0}, accleration

*A*

_{0}, and displacement

*D*

_{0}in the BV043 criteria. Those criteria can be compared directly to the PVSS obtained through the experiments or numerical simulations. The BV043 specifies to use the standard shock accelerations of a half sine wave or a triangular wave, hence a consistent campaign of experiments has been possible.

### 3. UNDEX Shock Responses

### 3.1 UNDEX Conditions

*w*, stand-off distance,

_{c}*R*, and wave incident angle,

*θ*. In the case that the KSF is given, one of the unknows of charge mass, and incident angle, the stand-off distance can be determined.

*corresponding to the heave motion was determined from Eq. (20) based on a reference (ABS, 2021). Because the added mass is faily dependent on the hull form, more rigorous challeges are necessary to determine the added mass.*

_{a}*w*= 544.31 kg of the charge type HBX-1. The wave incident angle of

_{c}*θ*= 90o was assumed to generate the worst UNDEX pressure field. With the charge mass, wave incident angle, and KSF, the stand-off distance of 43.1m was decided. The UNDEX model of Geers and Hunter (2002) was used to generate the primary and gas bubble pressure fields. The charge propeties are summarized in Table 2.