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1n then the nth eigenmode satisfies the resonance condition; all other modes are out of resonance. The repetition of the calculation below for a single nonresonant mode indicates that its contribution to the resulting pressure is at least Q times smaller (Q is the quality factor of the resonator) than the contribution of the resonant mode. Since we only consider the fundamental or second resonance, all nonresonant modes are of higher order. This additionally reduces the contributing pressures duo to increased damping. As a result, we expect the total contribution of the nonresonant modes to be on the order of 1/Q as well. Precise calculations made for a test case (the wall friction is the only absorption mechanism) confirm this conclusion.
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Regarding the first equality in Eq. (29), this is a trivial consequence of the perturbation theory. The second equality is more questionable. It was proved to be an asymptotically precise result if the different quality factors have considerably different magnitudes. In the following, exactly this situation will be studied. However, if the magnitudes are the same, Eq. (29) is only an approximation whose error might be on the order of 100%.
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