Physical Significance/Interpretation of Angular Wave Number.

But this is not what k represents. In summary, the angular wave number 'k' represents the number of radians of phase that occur over a length of 1 meter in a sinusoidal wave on a string, and has units of radians per meter. It does not represent the number of wavelengths per radian.
  • #1
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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.
 
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  • #2
ApuroopS said:
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.
 

1. What is the physical significance of angular wave number?

The angular wave number, denoted by k, is a fundamental quantity used in the study of waves. It represents the number of radians in one wavelength of a wave, and is closely related to the wave's frequency and wavelength. In other words, it describes how rapidly the wave is oscillating in space or time.

2. How is angular wave number used in wave equations?

In many wave equations, such as the wave equation for electromagnetic waves or the Schrödinger equation for quantum mechanical waves, angular wave number is a key parameter. It is used to define the spatial or temporal variations of the wave, and can help determine the behavior of the wave in different media or potential fields.

3. What is the relationship between angular wave number and wave speed?

Angular wave number is directly proportional to the wave speed, with the constant of proportionality being the wave's frequency. This means that the higher the angular wave number, the faster the wave is moving. This relationship is described by the equation v = ω/k, where v is the wave speed, ω is the angular frequency, and k is the angular wave number.

4. How does angular wave number affect wave interference?

Angular wave number plays a crucial role in determining the interference patterns of waves. When two or more waves with different angular wave numbers intersect, their amplitudes can either reinforce or cancel each other out, resulting in constructive or destructive interference. This phenomenon is important in many applications, such as in optics and acoustics.

5. Can angular wave number be negative?

Yes, angular wave number can be negative. This means that the wave is propagating in the opposite direction of the conventionally defined positive direction. In some wave equations, negative angular wave numbers can also represent waves with imaginary components, which have important implications in quantum mechanics and other fields of physics.

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