## Theory
### Calculation of Radiated Sound Power  
When a structure vibrates in the air, its structural modes are orthogonal and also decoupled, which means that the modes do not affect each other. However, if the structure vibrates under water, then it will form an integrated system with the external water. Therefore, structural vibration is no longer independent, and its modes will be coupled with the water. The coupling characteristics between these modes are specifically reflected in the modal impedance [23; 26]. According to the theory of modal superposition, the sound pressure on the surface of a cylindrical shell can also be described in the form of modal superposition as in Equation (1). Thus, sound pressure \(p\) can be expressed as  
\[p=\sum\limits_{n,m}^{\infty}P_{nm}\phi_{nm} \tag{4}\]  
where \(P_{nm}\) is the modal participant coefficient of the sound pressure. \(P_{nm}\) can be expressed as the product of the displacement modal coefficient and the modal impedance, like \(\sum\limits_{q}^{\infty}W_{nq}Z_{nmq}\). The component \(W_{nq}\) is the modal participant coefficients of vibration displacement, \(Z_{nmq}\) represents the contribution of mode (\(n\),\(q\))'s vibration to mode (\(n\),\(m\))'s sound pressure, called modal radiation impedance. Particularly, if \(q=m\), then \(Z_{mmm}\) will be called modal self-radiation impedance, and if \(q\
eq m\), then \(Z_{nmq}\) will be called modal mutual radiation impedance. According to the definition of radiated sound power \(\Pi\), it can be expressed as the surface integral of the normal sound intensity, which is related to the sound pressure and the normal vibration velocity. The formula can be represented as  
\[\Pi=\iint\limits_{s}\frac{1}{2}\text{Re}(p(j\omega w)^{*})ds \tag{5}\]  
where \(p\) is the sound pressure distribution function on the surface of cylindrical shells, \(j\) is the imaginary unit, \(w\) is the normal displacement distribution function on the surface of cylindrical shells, Re represents the real part of a complex, \(*\) represents the complex conjugate for scalars and complex conjugate transpose for vectors, and \(\omega\) is the angular frequency. When only the (\(n\),\(m\)) order mode participates in the vibration of a cylindrical shell, as well as the participant coefficient of this mode is \(W_{mm}\), the displacement distribution will be \(W_{mm}\phi_{nm}\) and the sound pressure on the surface in this situation can be described as \(W_{nm}Z_{mmm}\phi_{nm}\). Then, according to Equation (5), the radiated sound power \(\Pi_{nm}\) under the circumstance where the participant coefficient is \(W_{nm}\) can be represented as  
\[\Pi_{nm}=\iint\limits_{s}\frac{1}{2}\text{Re}(W_{nm}Z_{mmm}\phi_{nm}\cdot(j \omega W_{mm}\phi_{nm})^{*})ds=\frac{1}{2}\text{Re}\big{(}-j\omega\varepsilon_ {nm}|W_{mm}|^{2}Z_{mmm}\big{)}=|W_{nm}|^{2}\hat{\Pi}_{nm} \tag{6}\]  
where \(\hat{\Pi}_{nm}=\frac{1}{2}\text{Re}(-j\omega\varepsilon_{nm}Z_{mmm})\), it represents the radiated sound power under the circumstance where the participant coefficient is \(1\). Equation (6) indicates that the radiated sound power under the circumstance where the participant coefficient is \(W_{nm}\) is \(|W_{mm}|^{2}\) times that under the circumstance where the participant coefficient is \(1\). When all modes participate in the vibration, on the basis of the theory of modal superposition, the displacement distribution becomes \(\sum\limits_{g,t}^{\infty}W_{gt}\phi_{gt}\). As mentioned above, the sound pressure on the surface in this situation can be described as \(\sum\limits_{n,m}^{\infty}\sum\limits_{q}^{\infty}W_{nq}Z_{nmq}\phi_{nm}\).