We threw over part of our ballast, and mounted up till the cold and the rarefaction of the air became very troublesome.
"Wonderful Balloon Ascents" by Fulgence Marion
He tries the rarefaction of air, but finds the effect insensible.
"Faraday As A Discoverer" by John Tyndall
The rarefaction of the atmosphere produced that painful oppression known by the name of PUNA.
"In Search of the Castaways" by Jules Verne
This last symptom of the effects of high rarefaction, is, to an Englishman, at least it was to us, always a great relief.
"Forest and Frontiers" by Roualeyn Gordon-Cumming
Rarefaction or exhaustion produced by chimneys.
"A Catechism of the Steam Engine" by John Bourne
According to the wave theory, a condensation and rarefaction are necessary to constitute a sound wave.
"Scientific American Supplement, No. 595, May 28, 1887" by Various
Rarefaction of air facilitates discharge, why, 1375.
"Experimental Researches in Electricity, Volume 1" by Michael Faraday
This rarefaction of the spongy bone is the earliest change seen with the X-rays.
"Manual of Surgery" by Alexis Thomson and Alexander Miles
Towards morning, the painful rarefaction of the air diminished.
"Plotting in Pirate Seas" by Francis Rolt-Wheeler
On the other hand, it was very cold, and I was conscious of that peculiar nausea which goes with rarefaction of the air.
"Danger! and Other Stories" by Arthur Conan Doyle
The bodily eye, for example, cannot see the condensations and rarefactions of the waves of sound.
"Fragments of science, V. 1-2" by John Tyndall
The men suffered greatly from the rarefaction of the air.
"Great Men and Famous Women. Vol. 6 of 8" by Various
The rarefaction of the air began to be severely felt.
"A Winter Amid the Ice" by Jules Verne
Also, rarefaction by the air-pump does not injure air in the least degree.
"Experiments and Observations on Different Kinds of Air" by Joseph Priestley
But the disturbance we call rarefaction moves in air with the same velocity as a condensation.
"The Telephone" by A. E. Dolbear
Professor Coffin is a believer in the generally-received theory of rarefaction, as the cause of all winds.
"The Philosophy of the Weather" by Thomas Belden Butler
He spoke in honeymoon whispers; but the rarefaction of the air was such that every word was audible.
"How to be Happy Though Married" by Edward John Hardy
My ears also ached exceedingly in consequence of the rarefaction of the atmosphere.
"From the Oak to the Olive" by Julia Ward Howe
The air may be rendered dielectrically weak also by rarefaction.
"The inventions, researches and writings of Nikola Tesla" by Thomas Commerford Martin
With these pumps a higher degree of rarefaction can be obtained.
"Physics" by Willis Eugene Tower
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The difference between the two limiters is instead clear when we analyse the density right before the rarefaction fan.
Hydrodynamic simulations with the Godunov SPH
The bottom-right panel shows a zoom-in of pressure behaviour around the onset of the rarefaction fan.
Hydrodynamic simulations with the Godunov SPH
According to Corollary 3.5, one of the incoming waves must be a pure rarefaction, while the other can contain compressive regions which are changed by the interaction at the contact, or there could be no opposite incoming wave.
Shock-free Solutions of the Compressible Euler Equations
To be speciﬁc, consider a backward rarefaction (initially compactly supported in (0, ∞)) interacting with a 3-contact (increasing entropy jump), mr > ml, at x = 0, with no incoming forward wave.
Shock-free Solutions of the Compressible Euler Equations
If there are no incoming backward waves while the incoming forward waves are rarefactive on some piece of the right-most 3-contact, then the reﬂected backward waves are compressive.
Shock-free Solutions of the Compressible Euler Equations
If this characteristic ended at an interior contact, a backward rarefaction would emerge, and later reﬂect another forward wave, a contradiction.
Shock-free Solutions of the Compressible Euler Equations
Rarefactions with two monotonic contacts By way of example, we explicitly construct an eventually noninteracting solution consisting of rarefactions with two increasing entropy jumps (3-contacts).
Shock-free Solutions of the Compressible Euler Equations
By applying Corollary 3.5 at both jumps, it follows that all waves in the solution are rarefactions, so that ˙s ≤ 0 and ˙r ≥ 0 everywhere.
Shock-free Solutions of the Compressible Euler Equations
It remains to show that the backward rarefactions focus in the half-plane t < 0.
Shock-free Solutions of the Compressible Euler Equations
It suﬃces to show that rarefactive data produces global shock-free solutions.
Shock-free Solutions of the Compressible Euler Equations
We then resolve the states across the entropy jumps and propagate the solution outwards as simple (rarefaction) waves.
Shock-free Solutions of the Compressible Euler Equations
According to Corollary 3.5, the waves that are reﬂected back into the domain are always rarefactive, so we obtain global bounds on the derivatives of Riemann invariants.
Shock-free Solutions of the Compressible Euler Equations
Finally, having solved the IBVP inside the domain Ω, we again apply (3.4) to obtain the Riemann invariants outside the domain, which are rarefactions by construction.
Shock-free Solutions of the Compressible Euler Equations
It is clear that these solutions are not eventually noninteracting, as there is always reﬂected rarefaction.
Shock-free Solutions of the Compressible Euler Equations
We analyze a general rarefactive solution, as follows.
Shock-free Solutions of the Compressible Euler Equations
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