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amplitude angle angle of incidence aperture applicable approximately arbitrary axis calculated coefficient comparison condensation condition constant corresponding cylindrical denote density depends determined diameter differential equation dimensions direction distance disturbance effect energy equal experiment expression finite flame fluid fork frequency function given Green's theorem harmonic heat Helmholtz infinite integral intensity investigation kinetic energy length limit loop medium motion mouth node normal velocity observed obtained octave open end parallel period phase Phil pipe piston pitch plane waves plate Pogg positive potential pressure problem propagation quantity quarter period radius rarefaction ratio reflected wave refraction represented resonator result satisfied shewing simple source solution source of sound sphere spherical spherical harmonics suppose surface symmetrical theorem theory tion tone tube vanish velocity of propagation velocity of sound velocity-potential vibration viscosity wave-length zero