# Investigation on Ship Hydroelastic Vibrational Responses in Waves
## 2 Ship Hydroelasticity Theory in Regular and Irregular Waves
### Hydrodynamic Theory in Ship Hydroelasticity  
The hydrodynamics in the present paper is based on the potential flow theory [21] which assumes the fluid to be continuous, incompressible and inviscid, with the fluid flow being assumed to be irrotational. In steady state, the fluid velocity potential in reference frame \(\alpha vz\) can be multiplicatively decomposed as [22]:  
\[\Phi(x,y,z,t)=\text{Re}(\phi(x,y,z)e^{\text{i}\omega_{e}t}) \tag{3}\]  
where \(\varphi\) is the space-dependent part of \(\emptyset\), i is the imaginary unit, \(\omega_{e}\) is the wave encounter frequency which is defined as:  
\[\omega_{e}=\omega+Uk_{0}\cos\beta \tag{4}\]  
where \(\omega\) is the natural wave circular frequency, \(k_{0}\) is the wave number.  
The spatial part of \(\varphi\) can be further additively decomposed as:  
\[\varphi(x,y,z)=\zeta_{A}\varphi_{I}(x,y,z)+\zeta_{A}\varphi_{D}(x,y,z)+\sum_{m =1}^{n}v_{m}\varphi_{Rm}(x,y,z) \tag{5}\]  
where \(\zeta_{A}\) denotes the incident wave amplitude, \(\varphi_{I}\) denotes the incident wave potential of unit wave amplitude, \(\varphi_{D}\) denotes the diffraction wave potential of unit wave amplitude, \(\varphi_{Rm}\) denotes the radiation wave potential corresponding to unit motion amplitude of the \(m\)-th degree of freedom of the vessel, \(\
u_{m}\) is the motion amplitude of the \(m\)-th degree of freedom, \(m\) is bigger than 6 with indices \(m<6\) denoting rigid degrees of freedom and \(m>6\) denoting elastic degrees of freedom, respectively.  
The incident wave potential \(\varphi_{I}\) can be obtained by the incident wave field of regular waves. To express those velocity potentials in a more convenient and uniform fashion during the solution procedures, in the following contents we denote \(\varphi_{I}\) as \(\varphi_{0}\), with \(\varphi_{D}\) being denoted as \(\varphi_{n+1}\). \(\varphi_{Rm}\) is replaced by \(\varphi_{r}\) where \(r\) also ranges from 1 to \(n\) which is the same as \(m\). Then we can jointly denote \(\varphi_{I}\), \(\varphi_{Rm}\) and \(\varphi_{D}\) as \(\varphi_{r}\) where \(r\) actually ranges from 0 to \(n+1\). The boundary value problem for the determination of \(\varphi_{Rm}\) and \(\varphi_{D}\) can be expressed as follows [21]:  
\[\left\{\begin{array}{l}\
abla^{2}\varphi_{r}=0(in\;\Omega_{F})\\ -\omega_{e}^{2}\varphi_{r}+g\frac{\partial}{\partial z}\varphi_{r}=0(on\;z=0) \\ \frac{\partial\varphi_{r}}{\partial\mathbf{n}}=\left\{\begin{array}{l}\left( \text{i}\omega_{e}\mathbf{u}_{r}-U\frac{\partial}{\partial x}\mathbf{u}_{r} \right)\cdot\mathbf{n}(r=1,2,\cdots,n)\\ -\frac{\partial\varphi_{0}}{\partial\mathbf{n}}(r=n+1)\\ \end{array}\right.\left(on\;S_{B}\right)\\ \left\{\begin{array}{l}\frac{\partial\varphi_{r}}{\partial z}=0(z=-H)\\ \
abla\varphi_{r}=0(z\rightarrow-\infty)\\ \end{array}\right.\\ \lim_{R\rightarrow\infty}\sqrt{R}(\frac{\partial\varphi_{r}}{\partial R}- \text{i}k\varphi_{r})=0(R\rightarrow\infty)\end{array}\right. \tag{6}\]  
where \(\mathbf{n}\) is the normal vector and is defined as positive when it points from the wet boundary surface into the ship interior, \(\mathbf{u}_{r}\) denotes the \(r\)-th mode of vertical displacement of the hull. The present study adopts the zero speed Green's function [23] based boundary element method to numerically solve \(\varphi_{Rm}\) and \(\varphi_{D}\).