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(2005-08-20)   Rationalized Beaufort Wind Scale
In "force n" weather, the wind speed is proportional to   n^3/2  =  nÖn

The widely-used  Beaufort scale  was devised in 1806, by Sir Francis Beaufort (1774-1857), rear admiral, hydrographer to the Royal Navy.  It was adopted by the British Admiralty in 1838, and has been in international use since 1874.  Originally, the Beaufort Wind Scale did not refer to specific wind speeds, but to the effect of the wind on a full-rigged ship, and the amount of sail which should be carried.  Since "force 12" meant a wind that 'no canvas can withstand', the original scale did not extend beyond that point.

Each Beaufort number still corresponds to a variety of common observations which can be made at sea or inland.  For example, in a "force 0" condition:  'Smoke rises vertically. Sea is like a mirror.'

Since 1946, the Beaufort scale has been defined in terms of the speed of the wind, measured by an anemometer placed 10 meters above the ground.

"Force n" means a wind speed around  V.n^3/2,  where V is a speed of about 1.871 mph  (we're told that the 1946 scale was officially based on a speed of 0.836 m/s, or about 1.87008 mph, which is slightly too low to be consistent with modern tables).  Any speed  V, in mph, between 62Ö26/169 and 146Ö46/529 yields agreement with the rounded "mph" scale below  (and also with the "km/h" scale, which is somewhat less restrictive).

Most tables  erroneously  give 18 mph instead of 17 mph as the upper limit for a moderate breeze; this is inconsistent with the rest of the table, for any value of  V.

To find the Beaufort number corresponding to a given speed, one divides that speed by V, and finds the whole number closest to the cubic root of the square of that ratio.  As a result of this modern definition, the Beaufort scale can be extended beyond the traditional limit of "force 12" for extremely violent winds.

We have not traced the existence of a "standard" value of V; we shall simply note that a value V = 0.8365 m/s (or any value between 0.83626 m/s and 0.8368 m/s) will agree with the above tables in mph or km/h, but that (unexplicably) tables published in knots imply a value of V falling in the incompatible range of 0.8401 m/s to 0.8433 m/s (once the inconsistent value of 16 knots published for the upper limit of a moderate breeze is lowered to 15 knots).

Wheather reports for sailors commonly use the Beaufort scale or quote wind speeds in knots.  Otherwise, the media may prefer different units for wind speeds in different parts of the World:  m/s (Sweden, Denmark), km/h (France, Germany, Canada), mph (United States).

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(2005-08-20)   Saffir / Simpson scale for hurricanes
The customary scales for hurricanes (Beaufort force 12 and "above").

In August 1969, Hurricane "Camille" hit the Mississipi-Alabama coast with what would be "force 23" winds in an extended Beaufort scale:  200 mph to 213 mph.  However, the Beaufort scale is rarely extended  (if ever)  beyond force 12.  Instead, the strength of hurricanes is described with the following scale:

The Saffir / Simpson Hurricane Scale

Cat.	Pressure at center	Wind (km/h)	Surge (m)	Damage
1	above 980 hPa	120 to 153	1.2m to 1.5m	Minimal
2	965 hPa to 980 hPa	154 to 177	1.8m to 2.4m	Moderate
3	945 hPa to 965 hPa	178 to 209	2.7m to 3.7m	Extensive
4	920 hPa to 945 hPa	210 to 249	4.0m to 5.5m	Extreme
5	below 920 hPa	250 +	5.5m +	Catastrophic

In the Atlantic, the record-breaking hurricane season of 2005 included three category-5 hurricanes, named Katrina, Rita and Wilma (in chronological order).  At this writing (Oct. 2005) Wilma is the most intense hurricane ever observed in the Atlantic basin, featuring the lowest sea-level atmospheric pressure ever recorded in the Western Hemisphere outside of tornadoes  (882 hPa).  In the Northwest Pacific Ocean, only 9  typhoons  have surpassed the intensity of Wilma.  (The terms  typhoon  and  hurricane  describe the same phenomenon, but are used in different parts of the Globe.)

The costliest hurricane ever was  hurricane Katrina  (August 23 to 31, 2005) which caused an estimated $200 billion in damages and at least 1281 fatalities  (official count at this writing).  After hitting land as a mere category-1 hurricane north of Miami on August 25, the eye of Katrina made landfall again in Lousiana at 6:10am (CDT) on Monday, August 29, 2005.  as a category-4 hurricane...  By 11 am, the storm surge had breached the levee system protecting  New Orleans  from Lake Pontchartrain.  Most of the city was subsequently flooded.

Hurricane Names

Normally, the names of Hurricanes comes from a preapproved yearly list of 21 names with initals A through W (skipping Q and U) which is reused every 6 years, except that names of major hurricanes are retired and replaced...  The 2005 season had so many major storms that the last ones had to be named after letters from the Greek alphabet  (Alpha, Beta, Gamma, Delta, Epsilon, Zeta).

The following names have been retired, as of 2009:   Agnes (1972), Alicia (1983), Allen (1980), Allison (2001), Andrew (1992), Anita (1977), Audrey (1957), Betsy (1965), Beulah (1967), Bob (1991), Camille (1969), Carla (1961), Carmen (1974), Carol (1954), Celia (1970), Cesar (1996), Charley (2004), Cleo (1964), Connie (1955), David (1979), Dean (2007), Dennis (2005), Diana (1990), Diane (1955), Donna (1960), Dora (1964), Edna (1968), Elena (1985), Eloise (1975), Fabian (2003), Felix (2007), Fifi (1974), Flora (1963), Floyd (1999), Fran (1996), Frances (2004), Frederic (1979), Georges (1998), Gilbert (1988), Gloria (1985), Gustav (2008), Hattie (1961), Hazel (1954), Hilda (1964), Hortense (1996), Hugo (1989), Inez (1966), Ike (2008), Ione (1955), Iris (2001), Isabel (2003), Isidore (2002), Ivan (2004), Janet (1955), Jeanne (2004), Joan (1988), Juan (2003), Katrina (2005), Keith (2000), Klaus (1990), Lenny (1999), Lili (2002), Luis (1995), Marilyn (1995), Michelle (2001), Mitch (1998), Noel (2007), Opal (1995), Paloma (2008), Rita (2005), Roxanne (1995), Stan (2005) and Wilma (2005).

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(2005-08-20)   Fujita scale for tornadoes
Local twisters are primarily measured against a 6-rung scale  (F0 to F5).

Within tornadoes, the wind can reach speeds in excess of 280 mph  (450 km/h).  If the Beaufort scale was applicable, this would mean force 28 or 29.  Instead, all tornadoes are ranked using the following scale, from weakest to strongest:

The Fujita Tornado Scale

Fn	Effects	Wind speed  (km/h)
F0	Twisted antennas, broken branches	60 to 110
F1	Uprooted trees, vehicles turned over	120 to 170
F2	Lifted rooves, small projectiles	180 to 250
F3	Walls tipped over, large projectiles	260 to 330
F4	Houses destroyed, some trees lifted	340 to 410
F5	Large structures lifted, incredible damages	420 to 510

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(2006-12-02)   Measuring in Decibels  (dB)
A general-purpose logarithmic scale for physical  power.

In a given medium, a signal carries a certain power  (or a power flux)  proportional to the square of an associated "amplitude"  (which may be variously defined).

The "amplitude" of an oscillating linear system is preferably defined as the  RMS of an  intensive  quantity  (like voltage).  The square of that amplitude divided by the RMS of the corresponding power is called "impedance".  If we divide by a power flux instead, what we obtain is known as a "characteristic impedance" (see below the example of sound, where the amplitude is a pressure).  Conversely, since the characteristic impedance of the vacuum is traditionally expressed in ohms  (it's 376.73...W)  the "amplitude" of the electromagnetic field should be expressed in V/m, which identifies the electric field E.

The  relative magnitude  of two signals may be expressed equivalently as a logarithmic function of the ratios of their powers (P) or as the same logarithmic function of the squares of their amplitudes (A).  If decibels (dB) are used, the  relative  magnitude of the signal (compared to some other signal of rererence) is defined by either of the following expressions, which involve  decimal  logarithms.

Relative magnitude (or  level)  in dB   =   10  log( P/P₀ )   =   20  log( A/A₀ )

When the amplitude doubles, the power becomes 4 times as high and the level is raised by roughly 6 dB.  If the amplitude is multiplied by 10, the power is 100 times higher and the level is raised exactly 20 dB.

From relative ratios to absolute measurements :

Decibels are most useful to express ratios of related signals (for example the signals at the input and the output of an electronic amplifier).  However, specifying a conventional "reference" signal readily establishes an "absolute" decibel scale.  Each choice of a particular reference establishes a different "absolute" scale. 

The most popular such scale  (especially among electrical engineers)  is the  decibel-milliwatt  (dBm)  for which the zero level (0 dBm) is a signal whose total (harmonic) power is one milliwatt  (1 mW).

L   =   10  log ( P / 1 mW )  dBm

Power (P)	0.1 mW	1 mW	10 mW	100 mW	1 W	10 W
Level (L)	-10 dBm	0 dBm	10 dBm	20 dBm	30 dBm	40 dBm
	-40 dBW	-30 dBW	-20 dBW	-10 dBW	0 dBW	10 dBW

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(2010-01-03)   Measuring Sound in Decibels  (dB)
Sound Intensity Level  (SIL)  and  Sound Pressure Level  (SPL)

As sound propagates, it carries a certain power per unit area of a small surface perpendicular to the direction of propagation.  This physical quantity, called  sound intensity,  is measured in  watt per square meter  (W/m² ).

When expressed in decibels, that acoustic power  (per unit of receiving area) is called  sound intensity level.  The reference level  (0 dB)  is, by convention, a sound whose intensity is  10^-12  W/m².  The  level  (L)  of a sound whose intensity is  I  (expressed in  W/m² )  is, therefore:

L   =   [ 10  log ( I ) + 120 ]  dB (SIL)

According to the general scheme outlined above,  the  amplitude  of a soundwave is most commonly defined as its  acoustic pressure  p  (which is equal to the the RMS of the rapid local variations in air pressure).

A sound intensity of  10^-12 W/m²  corresponds to an acoustic pressure  p_o  which depends on temperature and pressure.  The above is  rigorously  equivalent to:

L   =   [ 20  log ( p / p_o ) ]  dB (SIL)

However, in daily practice, a  different  sound reference is often used which is defined by an acoustic pressure of  exactly  20 mPa, regardless of ambient conditions  This gives rise to a slightly different scale, called  "sound pressure level"  and identified by the acronym SPL  (which is, unfortunately, often omitted).

L	=	20  log ( p / 20 mPa )	dB (SPL)
	»	[ 20  log ( p )  +  94 ]	dB (SPL)

The SPL approximation is commonly used by practioners who are satisfied with the mere measurement of acoustic pressure.  The SPL scale is usually  assumed  to coincide numerically with the (correct) SIL scale for dry air at room temperature under normal pressure...  Let's  check  that:

The characteristic acoustic impedance corresponding to a sound having an intensity   I = 10^-12 W/m²   and an acoustic pressure   p = 20 mPa   is equal to:

Z   =   p² / I   =   400 Pa.s / m

For dry air under normal pressure, this would correspond to a toasty temperature of about  40°C.  Conversely, at room temperature  (20°C)  Z  would be around  413.2 Pa.s/m  which yields  p_o = 20.33 mPa.  This gives:

L   =   [ 20  log ( p )  +  93.84 ]  dB (SIL)       (air,  1 atm,  20°C)

So, the two formulas would match perfectly around  40°C  and would be less than  0.2 dB  off at room temperature.  Good enough.

The loudest possible sound is 191 dB.  Isn't it?

This popular piece of trivia is to be taken with a grain of salt, since some of the natural assumptions normally describing sound make little or no physical sense when the saturation limit is approached.  Never mind, here goes nothing...

If a sound is a perfect sinewave, the acoustic pressure which appears in the SPL formula is about  70%  (i.e., 1/Ö2)  of the maximum deviation from ambient pressure.  Disallowing negative pressures, the latter quantity cannot exceed the ambient pressure  (which we assume to be the normal atmospheric pressure of  101325 Pa).  So, the acoustic pressure (RMS) cannot exceed 71647.6 Pa.

The  saturation level  for a sinewave would thus be about  191 dB.  Formally, a  square wave  could be  3 dB louder  (194 dB).  However, neither answer is satisfactory, because most assumptions about sound collapse well below such pathological levels.  In particular, large pressure disturbances are dissipative  (they heat up the air itself)  and cannot be described as waves in a  linear  system  (power flux need not be proportional to the square of acoustic pressure).

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(2006-12-11)   Apparent and Absolute Magnitude of Stars
The absolute magnitude of a star is its apparent magnitude 10 pc away.

Ptolemy rated all stars wisible with the naked-eye, from brightest (first magnitude) to faintest (sixth magnitude).  It turns out that a star of the first magnitude in this ancient system is about 100 times as bright as a star of the sixth magnitude.  Thus, in 1854, the British astronomer N.R. Pogson proposed to turn that ptolemaic rating system into a strict logarithmtic scale, where a difference of 5 magnitudes would separate two stars whose brighnesses are in a ratio of 100 to 1.

So specified, the modern system of  stellar magnitudes  extends to faint objects (beyond magnitude 6) as well as very bright ones (the brightest stars, the planets, the Moon, the Sun) which are assigned a magnitude less than 1, or even a negative magnitude...  The Sun has a magnitude of  -26.7.  With a magnitude of -1.6, Sirius is the brightest object outside the solar system.  The faintest stars detected so far by the largest telescopes have a magnitude of 23 or so...

As brightness decreases by a factor of  100 ^1/5, magnitude increases by one unit.  This factor is known as  Pogson's ratio, in honor of N.R. Pogson (1829-1891).

100 ^0.2   = 10 ^0.4   =   2.51188643150958...

This simply means that one star magnitude is exactly equal to  4 decibels (4 dB).  However, star magnitudes are very rarely (if ever) expressed in decibels.  Historically, the relation is reversed:  The idea for expressing powers in decibels came from the stellar magnitude system !

There are 20 stars of the first magnitude (magnitude less than 1.5) 60 stars of the second magnitude (magnitude between 1.5 and 2.5) about 180 stars of the third magnitude (between 2.5 and 3.5) etc.  This tripling pattern holds for relatively bright stars but tends to be less explosive thereafter (it looks more like a mere doubling for stars around magnitude 20).

Most physicists would probably prefer to base star magnitudes on their bolometric output powers  (in which all electromagnetic frequencies carry equal weight).  This is rarely done, if ever, except for the Sun itself.

Ideally, the visual magnitude of a star should be based on the power it emits in the visible spectrum, using the same standard photopic response of the human retina on which the definition of the lumen is based  (although the dark-adapted scotopic response might be more relevant to direct telescopic observations by humans).

In actual practice, however, various standard filters are used instead which allow an automated determination of a star's magnitude in various portions of the electromagnetic spectrum.  As the emission spectrum of a star is, in the main, very similar to that of a  blackbody, precise comparisons of such different flavor of magnitudes allow the determination of a star's surface temperature (T).

Regardless of what spectrum-specific "flavor" of star magnitude is used, the  absolute  magnitude of a star is defined as what its apparent magnitude would be if it was observed at a distance of 10 pc  (10 parsecs is about  32.6 light-years).  To determine the absolute magnitude of a star, its distance must first be estimated  (using parallax or other methods)  so that the apparent magnitude can be adjusted, knowing that the observed power flux varies as the inverse square of the distance.

Conversely, the absolute magnitude of some stars may be known from other considerations (e.g., the absolute magnitude of a so-called Cepheid variable star is a function of the period of its oscillation in brightness).  This allows some distances to be estimated from apparent magnitudes, without the need for parallax measurements  (which are certainly not practical for intergalactical distances).

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(2005-11-26)   The Richter Scale of Earthquake Magnitudes
The seismic energy radiated is the basis of a rationalized Richter scale.

The original Richter Scale was devised in 1935 at the California Institute of Technology by Beno Gutenberg and Dr. Charles F. Richter.  More modern versions of that scale have been devised which are adequate to measure the largest earthquakes while being roughly compatible with the traditonal 1935 definition for small earthquakes.

The  1935 Richter Scale  of Richter and Gutenberg  (now called local magnitude) was defined as a  logarithmic  scale;  strictly based on readings from a particular type of instrument then used at CalTech  (the Wood-Anderson torsion seismometer).  Magnitude 0  was arbitrarily assigned to an earthquake that would cause a maximum combined horizontal displacement of 1 micron  (1 micrometer)  on such an instrument at  100 km from the epicenter.  (This reference level is so low that negative magnitudes are very rarely quoted.)  If that amplitude increases by a factor of 10, the  local magnitude  increases by one unit.

The problem with this viewpoint is that the amplitude originally considered by Richter is not a simple function of the energy released, except for the smallest earthquakes.  There are nonlinearities and the duration of the earthquake is also an important factor, especially for very large quakes which may last several minutes...

• Mercalli Intensity Scale: The effects measured at a particular location.
• Wood-Anderson seismographs at Caltech.
• Charles F. Richter & Beno Gutenberg: log E = 11.8 + 1.5 R
• Seismic Moment, Hiro Kanamori: M is about 20000 E.

Richter Magnitude