## 2 ARNC Using an ESG
### Anti-Sound Radiation Using the Cowl Panel Excited by the ESG  
The dimension and boundary condition of the cowl panel can affect the sound radiation property and capability of the ESG-cowl pair. The actual cowl panel of the test car used in this study had a complex shape, but it was roughly similar to a rectangular plate. Most edge parts were welded to the car body, and this indicates that the cowl panel can be assumed as a thin rectangular plate (length \(\times\) width = \(a\times b\)) with a completely clamped boundary condition, C-C-C-C. When an ESG excites the cowl panel with a force at the location \((\eta,\ \zeta)\), as illustrated in Figure 1, the ESG-cowl pair can radiate anti-sound to the driver's ear to suppress road noise. The governing equation of undamped vibration at an arbitrary position \((y,z)\) on the cowl panel at time \(t\) can be expressed as [17]:  
\[ \begin{array}{l}D\Big{(}\frac{\partial^{4}}{\partial t^{2}}+2\frac{\partial ^{4}}{\partial y^{2}\partial z^{2}}+\frac{\partial^{4}}{\partial z^{4}}\Big{)} w(y,z,t)+\rho h\frac{\partial^{2}w(y,z,t)}{\partial t^{2}}\\ =f(y,z,t),\end{array}  \tag{1}\]  
where \(D\) is the flexural rigidity of the cowl panel, which is given by \(Eh^{3}/12\big{(}1-v^{2}\big{)}\). The symbols \(E\), \(h\), \(v\), \(\rho\), and \(w(y,z,t)\) are [PERSON]'s modulus, the thickness, Poisson's ratio, the density, and the displacement of the panel, respectively. The external force exerted on the panel is defined by \(f(y,z,t)\). Considering the cowl panel is excited by a harmonic force with the frequency \(\omega\), then \(w(y,z,t)\) and \(f(y,z,t)\) can be represented as \(W(y,z)e^{i\omega t}\) and \(F(y,z)e^{i\omega t}\), respectively. The concentrated force \(F(y,z)\) generated by the ESG at the location \((\eta,\ \zeta)\) of the cowl panel can be expressed by \(F\delta(y-\xi)\delta(z-\eta)\), where \(F\) is the force amplitude and \(\delta(\cdot)\) is theDirac delta function. Finally, the displacement amplitude \(W(y,z)\) of the cowl panel can be written as [17]:  
\[ \begin{array}{ll}&W(y,z)\\ &=\sum\limits_{m}^{\infty}\sum\limits_{n}^{\infty}\frac{\text{${\text{F}Y_{m}( \eta)Z_{n}(\zeta)}$}}{D(I_{1}I_{2}+2I_{3}I_{4}+I_{5}I_{6})-\rho h\omega^{2}I_{ 2}I_{6}}Y_{m}(y)Z_{n}(z),\end{array}  \tag{2}\]  
where \(Y_{m}(\cdot)\) and \(Z_{n}(\cdot)\) are the eigenfunctions of the mode shapes of the cowl panel to the \(y\) and \(z\) directions, respectively, and \(m\) and \(n\) are positive integers that denote mode numbers. Additionally, \(I_{1}-I_{6}\) are the values related to the eigenfunctions (see reference [17] for further details). At the ear location \(\mathbf{r}_{ear}=\left[ \begin{array}{cc}x_{ear}&y_{ear}&z_{ear}\end{array}  \right]^{\text{T}}\), the sound pressure \(p(\mathbf{r}_{ear},t)\), which is radiated from the cowl panel as shown in Figure 1, can be defined by the Rayleigh's integral and given by [18]:  
\[ \begin{array}{ll}p(\mathbf{r}_{ear},t)&=\frac{i\omega\rho_{0}}{2\pi}e^{i \omega t}\int\limits_{S}^{t}\frac{V(y,z)e^{-i\omega tR}}{R}dS\\ &=\frac{i\omega\rho_{0}}{2\pi}e^{i\omega t}\int_{0}^{\rho}\int_{0}^{a}\frac{i \omega W(y,z)e^{-i\omega R}}{R}\;dydz,\end{array}  \tag{3}\]  
where \(\rho_{0}\) is the density of the air, and \(S\) and \(k\) are the area and the wavenumber of the cowl panel, respectively. The symbol \(R=|\mathbf{r}_{ear}-\mathbf{r}|\), where \(\mathbf{r}=\left[ \begin{array}{cc}0&y&z\end{array} \right]^{\text{T}}\), and \(V(y,z)\) is the velocity amplitude, which is equal to the amplitude of \(j\omega W(y,z)\). In practice, the actual sound pressure at the driver's ear position may differ greatly from the estimated one through Equations (2) and (3) under ideal conditions. This is because the cowl panel in a car has a complex shape and many connections with surrounding structures, and the radiated sound from the cowl panel is affected by the acoustic characteristics in the cabin. Thus, the radiated sound by the ESG-cowl pair measured at the driver's ear can be estimated by using Equations (2) and (3).