Artin–Mazur zeta function: Difference between revisions

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the whole set doesn't have to be finite, only the ones for each n
 
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{{hatnote|This article is on the type of seismic wave. For the term used in electrocardiography, see [[P wave (electrocardiography)]]. P-wave can also refer to a type
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of electronic  wavefunction in atomic physics; see [[atomic orbital]].}}
{{Earthquakes}}
[[Image:Onde compression impulsion 1d 30 petit.gif|thumb|305px|Plane P-wave]]
[[Image:Ondes compression 2d 20 petit.gif|thumb|305px|Representation of the propagation of a P-wave on a 2D grid (empirical shape)]]
 
'''P-waves''' are a type of [[Linear elasticity#Elastic wave|elastic wave]], called [[seismic waves]] in seismology, that can travel through a [[Continuum mechanics|continuum]]. The continuum is made up of [[gas]]es (as [[sound]] waves), liquids, or solids, including the [[Earth]]. P-waves can be produced by [[earthquakes]] and recorded by [[seismometer|seismographs]]. The name P-wave is often said to stand either for '''primary wave''', as it has the highest velocity and is therefore the first to be recorded; or '''pressure wave''',<ref name="John">{{cite book|last=Milsom|first=J.|title=Field Geophysics|publisher=John Wiley and Sons|year=2003|series=The geological field guide series|volume=25|page=232|isbn=978-0-470-84347-5|url=http://books.google.com/?id=T7CKj8bqVlwC&pg=PA179&dq=%22P-wave%22+pressure+wave+geophysics&cd=3#v=onepage&q=&f=false|accessdate=2010-02-25}}</ref> as it is formed from alternating [[compression (physical)|compression]]s and [[rarefaction]]s.
 
In [[isotropic]] and homogeneous solids, the mode of propagation of a P-wave is always [[Longitudinal wave|longitudinal]]; thus, the particles in the solid have vibrations along or parallel to the travel [[Direction (geometry)|direction]] of the wave energy.
 
==Velocity==
The velocity of P-waves in a [[Homogeneity (physics)|homogeneous]] [[isotropic]] medium is given by
:<math>v_p= \sqrt{ \frac {K+\frac{4}{3}\mu} {\rho}}= \sqrt{ \frac{\lambda+2\mu}{\rho}} </math>
where ''K'' is the [[bulk modulus]] (the modulus of incompressibility), ''<math>\mu</math>'' is the [[shear modulus]] (modulus of rigidity, sometimes denoted as ''G'' and also called the second [[Lamé parameters|Lamé parameter]]), ''<math>\rho</math>'' is the [[density]] of the material through which the wave propagates, and ''<math>\lambda</math>'' is the first [[Lamé parameters|Lamé parameter]].
 
Of these, density shows the least variation, so the velocity is mostly ''controlled'' by ''K'' and ''μ''.
 
The [[elastic moduli]] [[P-wave modulus]], <math>M</math>, is defined so that <math>M = K + 4\mu/3</math> and thereby
:<math>v_p = \sqrt{M/\rho}.\ </math>
 
Typical values for P-wave velocity in earthquakes are in the range 5 to 8&nbsp;km/s.<ref>{{cite web|url=http://hypertextbook.com/facts/2001/PamelaSpiegel.shtml |title=Speed of Sound through the Earth |publisher=Hypertextbook.com |date= |accessdate=2011-12-14}}</ref> The precise speed varies according to the region of the Earth's interior, from less than 6&nbsp;km/s in the Earth's crust to 13&nbsp;km/s through the core.<ref>{{cite web|url=http://earthquake.usgs.gov/learn/topics/seismology/keeping_track.php |title=Seismographs - Keeping Track of Earthquakes |publisher=Earthquake.usgs.gov |date=2009-10-27 |accessdate=2011-12-14}}</ref>
 
==Seismic waves in the Earth==
{{See also|Core-mantle boundary|Mohorovičić discontinuity|Low-velocity zone|Lehmann discontinuity}}
[[File:Speeds of seismic waves.PNG|thumb|300px|Velocity of seismic waves in the Earth versus depth.<ref name=Helffrich>{{cite journal |journal=Nature |year=2002 |volume=412 |issue=2 August |title=The Earth's Mantle |author=GR Helffrich & BJ Wood |page=501; Figure 1 |url=http://www.phys.uu.nl/~sommer/master/Structure%20and%20Evolution/articles%20for%20presentation/9.pdf |publisher=Macmillan Magazines | doi=10.1038/35087500 }}</ref> The negligible ''S''-wave velocity in the outer core occurs because it is liquid, while in the solid inner core the ''S''-wave velocity is non-zero.]]
 
Primary and secondary waves are body waves that travel within the Earth. The motion and behavior of both ''P''-type and ''S''-type in the Earth are monitored to probe the interior structure of the Earth. Discontinuities in velocity as a function of depth are indicative of changes in phase or composition. Differences in arrival times of waves originating in a seismic event like an earthquake as a result of waves taking different paths allow mapping of the Earth's inner structure.<ref name=Negendank>
 
{{cite book |title=New Frontiers in Integrated Solid Earth Sciences
  |author=Justin L Rubinstein, DR Shelly & WL Ellsworth |editor= S. Cloetingh, Jorg Negendank |chapter=Non-volcanic tremor: A window into the roots of fault zones |quote=The analysis of seismic waves provides a direct high-resolution means for studying the internal structure of the Earth...
|page=287 ''ff'' |url=http://books.google.com/books?id=7AIPPoWf3KIC&pg=PA287 |isbn=90-481-2736-X |year=2009 |publisher=Springer}}
 
</ref><ref name=Fowler>
 
{{cite book |title=The solid earth: an introduction to global geophysics |author= CMR Fowler |page=100 |chapter=§4.1 Waves through the Earth |quote=Seismology is the study of the passage of elastic waves through the Earth. It is arguably the most powerful method available for studying the structure of the interior of the Earth, especially the crust and mantle. |url=http://books.google.com/books?id=PifkAotvTroC&pg=PA100 |isbn=0-521-58409-4 |publisher=Cambridge University Press |year=2005 |edition=2nd}}
 
</ref>
 
===P-wave shadow zone===
[[Image:Earthquake wave shadow zone.svg|thumb|P-wave shadow zone (from [[USGS]])]]
Almost all the information available on the structure of the Earth's deep interior is derived from observations of the travel times, [[Reflection (physics)|reflection]]s, [[refraction]]s and phase transitions of seismic body waves, or [[normal modes]]. Body waves travel through the fluid layers of the [[Earth#Interior|Earth's interior]], but P-waves are refracted slightly when they pass through the transition between the semisolid [[mantle (geology)|mantle]] and the liquid [[Earth#The core|outer core]]. As a result, there is a P-wave "[[shadow zone]]" between 103° and 142°<ref>Lowrie, William. ''The Fundamentals of Geophysics''. Cambridge University Press, 1997, p. 149.</ref> from the earthquake's focus, where the initial P-waves are not registered on seismometers. In contrast, S-waves do not travel through liquids, rather, they are [[Attenuation|attenuated]].
 
===As an earthquake warning===
 
Earthquake advance warning is possible by detecting the non-destructive primary waves that travel more quickly through the Earth's crust than do the destructive [[S-waves|secondary]] and [[Rayleigh wave]]s, in the same way that lightning flashes reach our eyes before we hear the thunder during a storm.  The amount of advance warning depends on the delay between the arrival of the P-wave and other destructive waves, generally on the order of seconds up to about 60–90 seconds for deep, distant, large quakes such as Tokyo would have received before the [[2011 Tohoku earthquake and tsunami]]. The effectiveness of advance warning depends on accurate detection of the P-waves and rejection of [[ground vibrations]] caused by local activity (such as trucks or construction) as otherwise [[false-positive]] warnings will result. [[Earthquake Early Warning]] systems can be automated to allow for immediate safety actions such as issuing alerts, stopping elevators at the nearest floors or switching gas utilities off.
 
==See also==
* [[Earthquake Early Warning (Japan)]]
* [[Lamb waves]]
* [[Longitudinal wave]]
* [[Love wave]]
* [[Seismic wave]]
* [[S-Wave]]
* [[Surface wave]]
 
==References==
<references/>
* {{cite web|url=http://earthquake.usgs.gov/learning/glossary.php|work=U.S. Geological Survey"|title=Photo Glossary of Earthquakes|accessdate=March 8, 2009}}
 
==External links==
*[https://www.youtube.com/watch?v=2rYjlVPU9U4 Animation of a P-Wave]
*[http://www.eas.purdue.edu/~braile/edumod/waves/WaveDemo.htm Purdue's catalog of animated illustrations of seismic waves]
*[http://bingweb.binghamton.edu/%7Ejbarker/animations.html Animations illustrating simple wave propagation concepts by Jeffrey S. Barker ]
*[http://www.seismicwarning.com/web/technology/waveseparation.php Detection of P-waves and Rejection of Environmental Noise for Accurate Earthquake Early Warning]
 
{{DEFAULTSORT:P-Wave}}
[[Category:Waves]]
[[Category:Fluid dynamics]]
[[Category:Seismology and earthquake terminology]]
 
[[da:P-bølger]]

Latest revision as of 07:46, 21 December 2014

When faced with an impending exam, students need to learn to create a schedule for themselves. No one has to suffer in silence; there is plenty of help for anxiety including professional help and self help. If you have less than this, even though you may have a positive attitude, eventually these results will lower your confidence because of what is stored in the amygdala. Practice leaving the house when you're not really going anywhere. Paranormal terrorism can cause unnatural stress and anxiety for victims.

It is an ordeal at times and if you are prone to worrying unnecessarily you must lighten up your lifestyle. If you're wondering if alcohol can cause panic attacks, the answer is YES. He might also misinterpret involuntary facial tension as an attitude problem. Pennsylvania State University's Learning Center - Test taking and test anxiety. With enough time and effort, you will be able to find a good psychiatrist who you can talk to and also get prescription medication from.

The psychological impact our environment and past experiences has can have a profound effect on how we consciously perceive the world around us and ourselves. If you suffer with anxiety and panic attacks, the prospect or the reality of living on your own may seem a rather less than attractive proposition than it would otherwise. The anxiety that prevents you from living life to the fullest needs to be cured so you can face challenges head on and move forward with your life. If this has been happening to you lately, you are likely having panic attacks. Experts agree that proper education about asthma is your first step into protecting yourself from further aggravating the said disease.

In addition to using up all of that unwanted adrenaline, you will reap the benefits of overall health that comes with regular exercise. Luckily, anxiety is a treatable disorder and there are many ways to help your teen. You should definitely learn the way long you'll have to take the drug. Have you ever been concerned about your own risk if there is a family history of such disorders. One of the best means to slow down is to have entertainment.

The exercise doesn’t necessarily have to be rigorous either. He also worries about an upcoming math test, and considers getting "sick" so he can miss it. Likewise anxiety disorders can often lead to depression. Neuroplasticity is all about the brain reorganizing its neuron-to-neuron connections in response to new circumstances and environments. If they are unable to avoid it then they painfully endured it.

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