Resonant wave motion in a non-homogeneous system

Ellermeier, W.

doi:10.1007/bf00943505

Cited by 5 publications

(6 citation statements)

References 2 publications

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“…Linear equations such as (12) and (13) are standard in problems where material properties vary with position, see Ellermeier (1994). The crux of the matter arises, of course, in solving (12) and (13) analytically for physically interesting area functions s(x).…”

Section: Formulation and Linear Theorymentioning

confidence: 99%

“…Keller (1977) investigated the case of pipes with slowly varying cross-section and produced a nonlinear differential-integral equation to describe the signal in the pipe. While this equation has been widely investigated, see for example Ellermeier (1994) or Chester (1994), it suffers from the basic defect of describing only small variations of a straight tube. It is therefore inherently incapable of dealing with shapes such as the cones and bulbs used in the experiments of Lawrenson et al (1998).…”

Section: Introductionmentioning

confidence: 99%

“…However, Galiev (1999) points to recent work where an oscillating gas bubble produces a shock wave near the bubble centre. Ellermeier (1994) used a technique similar to that of Chester (1994) to examine the effect of material inhomogeneities on the resonant oscillations of a nonlinear elastic slab. While he produced an equation to describe the relation between the amplitude and frequency of resonant oscillations (i.e.…”

Section: Introductionmentioning

confidence: 99%

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Nonlinear resonant oscillations in closed tubes of variable cross-section

Mortell

Seymour

2004

J. Fluid Mech.

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An axisymmetric tube with a variable cross-sectional area, closed at both ends, containing a polytropic gas is oscillated parallel to its axis at or near a resonant frequency. The resonant gas oscillations in an equivalent tube of constant cross-section contain shocks. We show how cone, horn and bulb resonators produce shockless periodic outputs. The output consists of a dominant fundamental mode, where its amplitude and detuning are connected by a cubic equation – the amplitude–frequency relation. For the same gas, a cone resonator exhibits a hardening behaviour, while a bulb resonator may exhibit a hardening or softening behaviour. These theoretical results agree qualitatively with available experimental results and are the basis for resonant macrosonic synthesis (RMS).

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Section: Formulation and Linear Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nonlinear resonant oscillations in closed tubes of variable cross-section

Mortell

Seymour

2004

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).…”

Section: Introductionmentioning

confidence: 99%

“…Prior to the experiments of Lawrenson et al (1998) analytical work had been concerned largely with the effect of small geometric variations from a straight tube in the presence of shocks in the flow (Chester 1994;Mortell & Seymour 1972;Keller 1977;Ockendon et al 1993;Hamilton et al 2001). 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).…”

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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