Resonant acoustic oscillations with damping: small rate theory

Mortell, Michael P.; Symour, B. R.

doi:10.1017/s0022112073002636

Cited by 55 publications

(36 citation statements)

References 19 publications

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“…A long wave running into this section from the pipe is partially reflected and partially transmitted, or somehow dissipated. The simplest model for this process, which was also proposed by Seymour & Mortell (1973), is to assume that the reflected wave is proportional to the incoming one, with a proportionality coefficient ranging between + I and -1. In the same fashion as with the length correction there are theoretical estimates for the value of this factor, but they cannot be trusted for waves of any reasonable amplitude.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear gas oscillations in pipes. Part 1. Theory

Jiménez

1973

J. Fluid Mech.

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The problem of forced acoustic oscillations in a pipe is studied theoretically. The oscillations are produced by a moving piston in one end of the pipe, while a variety of boundary conditions ranging from a completely closed to a completely open mouth at the other end are considered. All these boundary conditions are modelled by two parameters: a length correction and a reflexion coefficient equivalent to the acoustic impedance.The linear theory predicts large amplitudes near resonance and nonlinear effects become crucially important. By expanding the equations of motion in a series in the Mach number, both the amplitude and wave form of the oscillation are predicted there.In both the open-and closed-end cases the need for shock waves in some range of parameters is found. The amplitude of the oscillation is different for the two cases, however, being proportional to the square root of the piston amplitude in the closed-end case and to the cube root for the open end.

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Section: Introductionmentioning

confidence: 99%

Nonlinear gas oscillations in pipes. Part 1. Theory

Jiménez

1973

J. Fluid Mech.

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“…An exception is Ellermeier (1993Ellermeier ( , 1994b who outlined a Duffing-like expansion for a strong inhomogeneity. Also, but to a lesser extent, the effects of rate-dependent properties of the gas and changes in the impedance at the end of the cylinder on preventing shocks were examined by Seymour & Mortell (1973), Mortell & Seymour (1972), Ellermeier (1983) and Sturtevant (1974). These latter introduced a damping mechanism to prevent the shocks, but the order of magnitude of the output was not increased.…”

Section: Introductionmentioning

confidence: 99%

Resonant oscillations of an inhomogeneous gas between concentric spheres

2011

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We investigate the effects of nonlinearity, geometry and stratification on the resonant motion of a gas contained between two concentric spheres. The emphasis is on whether the motion is continuous, and on how the inhomogeneity, geometry or nonlinearity can move the motion to a shocked state. Linear undamped theory yields a standing wave of arbitrary amplitude and an eigenvalue equation in which the higher eigenvalues are not integer multiples of the fundamental; the system is said to be dissonant. Higher modes, generated by the nonlinearity, are not resonant and consequently shocks may not form. When the output is shockless, the amplitude is two orders of magnitude greater than that of the input. When the eigenvalues for a homogeneous gas are not sufficiently dissonant and shocks form, the introduction of a stratification in the gas can restore dissonance and allow a continuous output. Similarly, the introduction of an inhomogeneity can change a continuous motion to a shocked one, as can an increase in the input Mach number, or an increase in the geometrical parameter. Various limits of the eigenvalue equation are considered and previous results for simpler geometries are recovered; e.g. a full sphere, a cone and a straight tube.

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“…The resonance of plane standing waves in a closed tube continues to attract particular attention, [1][2][3][4][5][6][7][8][9] leading to the development of the corresponding part of nonlinear wave theory. Of primary concern in the nonlinear plane wave resonance is, from both theoretical and application points of view, the formation of shock waves, which causes the energy dissipation at a shock front and restrains the growth of wave amplitude.…”

Section: Introductionmentioning

confidence: 99%

“…Of primary concern in the nonlinear plane wave resonance is, from both theoretical and application points of view, the formation of shock waves, which causes the energy dissipation at a shock front and restrains the growth of wave amplitude. In the plane wave resonance with shock waves, therefore, the maximum wave amplitude in the quasisteady state oscillation is limited to O͑ ͱ M͒, [1][2][3][4][5][6][7][8][9] where M is the acoustic Mach number defined at the sound source and usually rather small compared with unity ͑typically M Շ 10 −3 ͒.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear analysis of periodic modulation in resonances of cylindrical and spherical acoustic standing waves

Kurihara

Yano

2006

Physics of Fluids

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The nonlinear resonance of cylindrical acoustic standing waves of an ideal gas contained between two coaxial cylinders is theoretically investigated by the method of multiple scales. The wave motion concerned is excited by a small-amplitude harmonic oscillation of the radius of the outer cylinder, and the formulation of the problem includes the wave phenomenon in a hollow cylinder without the inner one as a limiting case. The spherical standing wave in two concentric spheres is also studied in parallel. The resonance occurs if the driving frequency falls in a narrow band around the linear resonance frequency, and in the weakly nonlinear regime, no shock wave is formed in contrast to the plane wave resonance. A cubic nonlinear equation for complex wave amplitude can then be derived by the method of multiple scales. Using a first integral of the cubic nonlinear equation, we shall demonstrate that the resonant oscillation is accompanied by a periodic modulation of amplitude and phase when the dissipation effect due to viscosity and thermal conductivity is negligible. The period of the modulation varies as the minus two-thirds power of the acoustic Mach number defined at the outer cylinder or sphere and decreases with an increase in the radius ratio of the inner and outer cylinders or spheres. When the dissipation effect is small but not negligible, the modulation is slowly weakened and the resonant oscillation approaches a steady state oscillation, which corresponds to the steady solution examined in earlier works.

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Resonant acoustic oscillations with damping: small rate theory

Cited by 55 publications

References 19 publications

Nonlinear gas oscillations in pipes. Part 1. Theory

Nonlinear gas oscillations in pipes. Part 1. Theory

Resonant oscillations of an inhomogeneous gas between concentric spheres

Nonlinear analysis of periodic modulation in resonances of cylindrical and spherical acoustic standing waves

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