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acoustic energy acoustic pressure acoustic waves amplitude apply approximation attenuation coefficient axis Bessel functions boundary conditions component compute consider constant corresponding cross-sectional area density depends derivative determine dipole dissipation distance distribution effects element emission emitted entropy equal equilibrium example expansion expressed far-field finite fluctuations fluid velocity force frequency gases given gives group velocity heat conductivity Helmholtz Helmholtz resonator Hence ideal fluid impedance incident wave infinite integral intensity limit linear magnitude mass mode monochromatic waves motion nondimensional obtain one-dimensional oscillating phase velocity piston plane wave pressure and velocity problem propagation quantity radiation radius ratio region represents resonance result right-hand side rigid Section shown in Figure shows Similarly solution sound waves speed of sound sphere spherical stress tensor Substitution surface temperature thermodynamic transmission coefficient transmitted transverse modes tube unit values vanishes variations vector velocity potential viscosity wall wave equation wavelength written yields zero