Physical Significance/Interpretation of Angular Wave Number.

[ApuroopS]

My doubt is on the angular wave number 'k'

in case of a sinusodal wave on a string,
the angular wave number

k = (2*pi)/lambda

lambda being the wave length of the wave.
What does 'k' signify in physical terms ?

for e.g.

general wave number = 1/lambda

again, lambda being the wavelength of the wave.

the physical significance for this is basically the number of wavelengths per unit distance


so along those lines, does the angular wave number stand for number of wavelengths per radian ??
Or something else ?

any help in clarifying this will be appreciated.
Cheers.

btw, my apologies in advance for the bad presentation, but can't actually texify from my cellphone.

 

[Redbelly98]

My doubt is on the angular wave number 'k'

in case of a sinusodal wave on a string,
the angular wave number

k = (2*pi)/lambda

lambda being the wave length of the wave.
What does 'k' signify in physical terms ?
.
.
.
so along those lines, does the angular wave number stand for number of wavelengths per radian ??
Or something else ?

It is something else.

First, keep in mind that angular quantities can be measured in terms of either radians or cycles, where 1 cycle is 2π radians. And, even though radians and cycles are typically considered dimensionless (unitless) quantities, it can be useful to include radians in cycles in tracking the units.

So, the factor of "2π" in the equation for k can really be thought of as "2π radians per cycle" or "2π radians / 1 cycle. Also, think of the units of λ as being in meters per cycle, rather than simply meters. (If we are using meters as our basic unit of length -- but we could easily use feet or furlongs as well.)

k = (2π radians)/(1 cycle) * (1/λ)

Since λ has units of meters per cycle, the above gives units of "radians per meter" for k. In other words, it is the number of radians of phase that occur over a length of 1 meter.

As an example, suppose a wave has a wavelength of 2 m -- or really 2 m/cycle. A cycle of the wave is then 2 meters long, which also corresponds to 2π radians, or in other words it's π radians in 1 meter. So k is π radians per meter. We typically don't include the radians in the units, so we just say it's π m^-1.

...does the angular wave number stand for number of wavelengths per radian?

The number of wavelengths per radian is always 1/(2π), for any wave.