Structural acoustics of perforated panels
In this work, radiation and transmission of sound through flexible perforated panels set in infinite rigid baffles are investigated. The treatment is largely analytical using Fourier transforms and contour integrations. Numerical calculations are only used occasionally. The work is largely divided into three parts: the first part involves radiation and transmission studies using the one-way coupled formulation, the second part investigates the same problems using the two-way coupled (or the fully-coupled) formulation and the third part involves derivations of closed form expressions for the modal coupling coefficient using contour integration. In the first part, the panel with perforations is placed in a baffle that is perforated or unperforated. Having an unperforated or a differently perforated baffle presents challenges. It causes a certain coupling of wavenumbers leading to an integral equation. In the literature so far, the baffle has been taken to be similarly perforated, thus, simplifying the situation. The perforations are arrays of circular holes and are mathematically modeled using a perforation ratio. An existing model for a circular hole that transmits sound is used and the collective array is modeled using a perforate impedance. Since, there is an escape of fluid through the perforations as the panel vibrates (radiating or transmitting sound) an averaged fluid particle velocity over the panel surface is derived using fluid continuity and momentum equations. This averaged fluid velocity is then used along with impedances to compute the pressures and sound powers. In addition, the presence of the holes shifts the resonance frequencies and modifies the modeshapes. This shift is accounted for using the Receptance method. The entire derivation is done in the wavenumber domain (spatial Fourier transform). And at the end, numerical calculations are done. For the radiation and transmission problems, the results are presented in terms of the radiation efficiency and the transmission loss, respectively. It is observed that the perforations reduce the in vacuo natural frequencies of the panel. For the radiation problem, analytical expressions for the radiated power and radiation efficiency are derived in an integral form and numerical results are obtained for different perforation parameters such as perforation ratio, hole diameter and number of holes. It is observed that a reduction in the perforate impedance leads to a decrease in the radiated power and also in the radiation efficiency. The effects of resistive and reactive hole impedances on the sound radiation are also discussed. For the transmission problem, it is found that the perforate impedance acts in parallel to the panel impedance and for a real-world scenario, where the perforate impedance is less than the panel impedance, a reduction in the transmission loss (TL) can be achieved with perforations on the panel. For small holes at lower frequencies the resistive impedance dominates over the reactive impedance. This results in a higher TL at lower frequencies for a micro-perforated panel as compared to that for a panel of same perforation ratio but with larger holes. In the second part, the same two problems of radiation and transmission of sound through perforated panels set in rigid baffles are studied using the two-way coupled or fully coupled formulation. In addition to the details presented for the one-way cases above, here two equations are derived where the average fluid particle velocity and the panel velocity depend on each other. Thus, a coupled problem needs to be solved. Due to the inclusion of the fluid loading, a modal coupling coefficient arises in the formulation. This coupling coefficient is indicative of the degree of coupling between the in vacuo panel modes caused by the acoustic fluid. In several of the earlier studies on unperforated panels, in the literature, largely the self modal coupling has been investigated. Only a few studies have presented studies on the cross modal coupling. These studies were restricted to the low frequencies. The formulation is reduced to a single coupled equation and the system of equations (including the modal coupling coefficient) are solved numerically. Again, the results are presented in terms of the radiation efficiency and the transmission loss. The natural frequencies are identified from the peaks in the mean panel quadratic velocity spectrum and compared with results from the literature. It is observed that the radiation efficiency decreases with the increase in the perforation ratio, irrespective of the surrounding acoustic medium. For a given perforation ratio, the water-loaded panel radiation efficiency is found to be less than that for a panel immersed in air. It is also observed that for a light fluid like air, a one-way coupled formulation is adequate. Further, a fully coupled model for the transmission problem is also developed. It is observed that the TL of a perforated panel acquires negative values at low frequencies. This apparent anomaly is resolved by taking into account the additional power component that flows from the baffle region onto the panel at low frequencies. In the last part of the thesis, approximate expressions in closed form are obtained for the modal coupling coefficient using the contour integration. Analytical expressions valid for any given fluid loading conditions are derived for the modal interactions between the corner modes, single and double edge modes and the acoustically fast modes. This is further used to evaluate the natural frequencies and the radiation efficiency of the perforated panel. The results agree very well with those obtained earlier in the thesis using the numerical integration. Also, plots of the resistive and reactive parts of the modal coupling coefficient are presented and discussed.