Angular wavelength - units

[FrankM]

I know the units for "angular frequency" are radians per second, but I cannot find any site that defines the units for "angular wavelength".

 

There is a difference in the meaning of a numeric value that represents a regular wavelength and that of an "angular wavelength", but I cannot find how this is noted.

[FrankM]

I found one source that gave a definition for "angular wavelength", but I find quite a few sites that are using that term. Most sites do not note the units of measure although I found two that put (degrees) in parentheses, but this doesn't fit the one definition.

 

http://support.livemath.com/expressionistdocs/FM7-encycloped.pdf

 

Go to page 49 and the entry for lambda-bar. The units were not mentioned.

[Jay-qu]

I have never heard of it.. what does it have applications in?

 

That link just says its the wavelength divided by 2pi..

[FrankM]

There aren't a great many web sites that use the term "angular wavelength", but most are dealing with optical, electromagnetic and astronomy subjects, with some quantum mechanics thrown in.

 

In some cases it almost seems that the term was defined to fit a particular scenario, there not being any other suitable definition for a condition.

 

The one site mentioned that contains the symbol "lambda-bar" and the definition for "angular wavelength" intimates that the symbol has been in use for that term. Even so, what units would be used to define wavelength/2Pi?

 

If I treated a wavelength as a vector I could create a definition for "angular wavelength". Given two wavelengths of equal length that represent the legs of a right triangle, the hypotenuse would be the vector result, an "angular wavelength".

 

"Angular frequency" is a well known term but it really means "radian velocity" or "radian frequency" depending upon whether it is used to denote some mechanical rotation or the characteristics of a wave.

 

If I treated a frequency as a vector I could create a pure definition for "angular frequency". Given two "radian frequencies" of equal length that represent the legs of a right triangle, the hypotenuse would be the vector result, an "angular frequency".

 

In both cases you end up with a vector result with a cartesian component. What units would be used to affix to the numeric results?

[CraigD]

I know the units for "angular frequency" are radians per second, but I cannot find any site that defines the units for "angular wavelength".

 

There is a difference in the meaning of a numeric value that represents a regular wavelength and that of an "angular wavelength", but I cannot find how this is noted.

I think FrankM’s difficult is due to “angular frequency” being a misnomer. The wikipedia article “angular frequency” states that is a synonym of “angular speed”, a scalar (directionless) measure of a change in quantity – angle – divided by a change in time. Thus valid units for angular speed are radians/second, degrees/hour, revolutions/minute, etc.

 

Frequency, on the other hand, is the measure of the count of occurrences of an event – high and low air pressure, consecutive crests of a moving wave relative to a fixed or moving reference point at a fixed or moving reference point, firings of a cannon, final assembly of an automobile, etc. – divided by change in time. Thus valid units of frequency are cycles/second (hz), shots per minute, cars per day, etc.

 

Note that only units of frequency with some reference to a distance, or length, in the form of the presence of the phrase “fixed or moving reference point”, have meaningful corresponding wavelength units. These units are always units of length – meters, etc.

 

Because of the relationship between a angular “cycle” – 1 cycle = 2[math]\pi[/math] radians = 360°, etc – angular velocity is therefore a simple multiple of “ordinary” frequency. Angular wavelength is therefore a simple multiple of ordinary wavelength, the inverse of angular frequency. Though of questionable meaningfulness, regardless of the units of angle used, angular wavelength will be in units of length.

[FrankM]

I found another term for "angular frequency", "circular frequency".

 

http://www.diracdelta.co.uk/science/source/a/n/angular%20frequency/source.html

 

The site does not provide a definition for "angular wavelength".

 

....

Because of the relationship between a angular “cycle” – 1 cycle = 2[math]\pi[/math] radians = 360°, etc – angular velocity is therefore a simple multiple of “ordinary” frequency. Angular wavelength is therefore a simple multiple of ordinary wavelength, the inverse of angular frequency. Though of questionable meaningfulness, regardless of the units of angle used, angular wavelength will be in units of length.

 

The website that cited the use of lambda-bar for "angular wavelength" states that it is divided by 2[math]\pi[/math], thus angular wavelength should be a simple division of an ordinary wavelength. If you divide a wavelength by 2[math]\pi[/math] radians you cannot ignore the radian divisor in the descriptive units.

 

It seems scientists are using "angular wavelength" but I have found only one source that gives a definition, and no sources that note the units.

[sanctus]

If you go to http://scienceworld.wolfram.com/physics/h-Bar.html

you see that h and h-bar have the same units, it's because to simplify calculations you just divide by 2pi, without caring about this units. Anyway usually you use h-bar when you want to calculate cross section where you integrate over the solid angle and therefore the not written quantity in radians cancels itself in the integration.

I think with lambda-bar it is just the same.

 

But I'm not 100% sure

[Qfwfq]

I know the units for "angular frequency" are radians per second, but I cannot find any site that defines the units for "angular wavelength".

It would be the reciprocal of wavenumber, which is a far more widely used definition.

 

There is a difference in the meaning of a numeric value that represents a regular wavelength and that of an "angular wavelength", but I cannot find how this is noted.

The difference in meaning is simply that in one case you are talking periods and in the other radians. An angle in radians is a dimensioness quantity (length/length) and so it doesn't change the unit of measure. Although an engineer may find periods handier, being a "more tangible" thing than an angle of one radian, from many points of view the radian-based correspondents are far better and make formal calculations less messy.

[FrankM]

It would be the reciprocal of wavenumber, which is a far more widely used definition.

 

The difference in meaning is simply that in one case you are talking periods and in the other radians. An angle in radians is a dimensioness quantity (length/length) and so it doesn't change the unit of measure. Although an engineer may find periods handier, being a "more tangible" thing than an angle of one radian, from many points of view the radian-based correspondents are far better and make formal calculations less messy.

 

The "wolfram" site indicates that wavenumber has two definitions, which creates confusion unless it is specifically defined before usage or everyone in a particular scientific discipline agrees to one particular definition.

 

I do agree that "radian-based correspondents" would provide terms with a more precise meaning and have unambiguous calculations. "Radian frequency" is a better descriptive term than "angular frequency" and reflects the current assigned units. Then why not "radian wavelength", it being the simple division of wavelength by 2[math]\pi[/math]?

 

The above units can be applied to "radian geometry", another term that is sprinkled throughout a few websites.

[Qfwfq]

The first definition on Wolfram is certainly rare, I wasn't aware of it before looking at that page. By "far more widely used" I meant the "real" one! :)

 

Concerning terminology, angular velocity comes from applications, such as mechanics, where there is an actual rotation. For periodic phenomena in general it isn't necessarily a geometric matter. I wouldn't worry unduly about terms when the merely come from analogy.

 

To the engineer, 'radian' is just one possible instance of angular. To the physicist, it is the canonical instance per antonomasia. To the mathematician, angular and radian are the same thing and degrees or whatever are unnecessary hogwash. :)