Editing P wave (section)

===Velocity===
In [[isotropic]] and homogeneous solids, a P&nbsp;wave travels in a straight line [[Longitudinal wave|longitudinally]]; thus, the particles in the solid vibrate along the axis of propagation (the direction of motion) of the wave energy. The velocity of P&nbsp;waves in that kind of medium is given by
<math display="block">v_\mathrm{p} \; = \; \sqrt{ \frac{\, K + \tfrac{4}{3} \mu \;}{\rho} } \; = \; \sqrt{ \frac{\, \lambda + 2 \mu \;}{\rho} } </math>
where {{mvar|K}} is the [[bulk modulus]] (the modulus of incompressibility), {{mvar|μ}} is the [[shear modulus]] (modulus of rigidity, sometimes denoted as {{mvar|G}} and also called the second [[Lamé parameters|Lamé parameter]]), {{mvar|ρ}} is the [[density]] of the material through which the wave propagates, and {{mvar|λ}} is the first [[Lamé parameters|Lamé parameter]].

In typical situations in the interior of the Earth, the density {{mvar|ρ}} usually varies much less than {{mvar|K}} or {{mvar|μ}}, so the velocity is mostly "controlled" by these two parameters.

The [[elastic moduli]] [[P wave modulus]], <math>M</math>, is defined so that <math display="inline">\, M = K + \tfrac{4}{3} \mu \,</math> and thereby
<math display="block">v_\mathrm{p} = \sqrt{ \frac{\, M \;}{\rho} } </math>

Typical values for P&nbsp;wave velocity in earthquakes are in the range 5 to 8&nbsp;km/s. 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 13.5&nbsp;km/s in the lower mantle, and 11&nbsp;km/s through the inner core.<ref>
{{cite journal
 |last1=Dziewonski |first1=Adam M.
 |last2=Anderson |first2=Don L.
 |year=1981
 |title=Preliminary reference Earth model
 |journal=Physics of the Earth and Planetary Interiors
 |volume=25 |issue=4 |pages=297–356
 |doi=10.1016/0031-9201(81)90046-7
 |bibcode=1981PEPI...25..297D
}}
</ref>

{| class="wikitable sortable collapsible"
|+ Velocity in common rock types<ref>
{{cite web
 |title=Acoustic Logging
 |date=2011-12-12
 |department=Geophysics
 |publisher=U.S. [[United States Environmental Protection Agency|Environmental Protection Agency]]
 |url=http://www.epa.gov/esd/cmb/GeophysicsWebsite/pages/reference/methods/Borehole_Geophysical_Methods/Logging_Techniques_and_Tools/Acoustic_Logging.htm
 |access-date=2015-02-03
}}
</ref>
|-
! scope="col" | Rock Type
! scope="col" | Velocity [m/s]
! scope="col" | Velocity [ft/s]
|-
| Unconsolidated [[sandstone]] || 4,600–5,200 || 15,000–17,000 
|-
| Consolidated sandstone || 5,800 || 19,000
|-
| [[Shale]] || 1,800–4,900 || 6,000–16,000
|-
| [[Limestone]] || 5,800–6,400 || 19,000–21,000
|-
| [[Dolomite (rock)|Dolomite]] || 6,400–7,300 || 21,000–24,000
|-
| [[Anhydrite]] || 6,100 || 20,000
|-
| [[Granite]] || 5,800–6,100 || 19,000–20,000
|-
| [[Gabbro]] || 7,200 || 23,600
|-
|}

Geologist [[Francis Birch (geophysicist)|Francis Birch]] discovered a relationship between the velocity of P&nbsp;waves and the density of the material the waves are traveling in:
<math display="block"> v_\mathrm{p} = a ( \bar{M} ) + b \, \rho </math>
which later became known as [[Birch's law]]. (The symbol {{math|''a''()}} is an empirically tabulated function, and {{mvar|b}} is a constant.)