How to Calculate Wave Number and Its Importance in Physics
Wavenumber is a crucial topic in physics. Wavenumber is the reciprocal of the wavelength of a wave. It is specially used to calculate the number of cycles per unit distance. Wavenumber has a magnitude with no direction.
A Wave number definition states that “A wave's spatial frequency, measured in cycles per unit distance or radians per unit distance”. The Wavenumber is the magnitude of the wave vector in multidimensional systems. Wavenumber is a scalar quantity.
Wavenumber definition in spectroscopy and chemistry fields is defined as the number of wavelengths per unit distance.
angular wavenumber is a quantity in theoretical physics that is defined as the number of radians per unit distance.
Wavenumber Formula in Spectroscopy
Wavenumber \[\bar{v}\] is a term used in spectroscopy to describe a frequency that has been separated by the speed of light in a vacuum. The formula of Wave number in spectroscopy and chemistry fields is given as,
\[\bar{v}=\frac{1}{\lambda }=\frac{\omega }{2\pi c}=\frac{v}{c}\]
Where \[\bar{v}\] is the spectroscopy wavenumber.
λ is the wavelength sometimes called a spectroscopic wavenumber which is equal to the spatial frequency.
\[\omega =2\pi v\] is the angular frequency.
By using Planck’s relation spectroscopic wavenumber can be converted into energy per photon as follows,
\[E=hc\bar{v}\]
Where E is the energy per photon,
h is the reduced Planck’s constant = 6.62607004 x 10-34m2kg/s
c is the speed of light
\[\bar{v}\] is the Spectroscopic Wavenumber
Spectroscopic wavenumber can be converted to the wavelength of light as follows,
λ=\[\frac{1}{n\bar{v}}\]
Where λ is the wavelength,
n is the refractive index of the medium,
\[\bar{v}\] is the Spectroscopic Wavenumber
The SI unit of measurement of Spectroscopic Wavenumber is reciprocal of a meter that is m-1. The CGS unit of measurement of Spectroscopic Wave number is reciprocal of a centimetre that is cm-1.
Wavenumber Formula for Wave Equations
The formula for angular wave number in theoretical physics is given by
K = \[\frac{2\pi }{\lambda }=\frac{2\pi v}{v_p}=\frac{\omega }{v_p}\]
Where k is the angular wave number,
λ is the wavelength,
\[\omega =2\pi v\] is the angular frequency,
Vp is the phase velocity
For the special case of an electromagnetic wave propagating at the speed of light in a vacuum, the Wave equation k is given by
k = \[2\pi \frac{E}{hc}\]
Where k is the angular wave number,
E is the energy of the wave,
h is the reduced Planck’s constant = 6.62607004 × 10-34 m2 kg / s
c is the speed of light
For the special case of a matter-wave in the non-relativistic approximation that is in the case of a free particle (a particle that has no potential energy), the Wave equation k is given by
k = \[\frac{2\pi }{\lambda }=2\pi \frac{p}{h}=2\pi \frac{\sqrt{2mE}}{h}\]
Where k is the angular wave number,
λ is the wavelength,
p is the momentum of the particle,
h is the reduced Planck’s constant = 6.62607004 × 10-34 m2 kg / s
m is the mass of the particle
E is the kinetic energy of the particle
Applications of Wavenumber
A wavenumber is used to calculate the spatial frequency.
Other than spatial frequency, wavenumber can be used to define other quantities such as optics and wave scatterings in physics.
Wavenumbers and wave vectors are crucial in X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics.
The canonical momentum of quantum mechanical waves is the wavenumber multiplied by the reduced Planck's constant.
Group velocity can be defined by using a wavenumber.
Problems on Wave Number
In this section, we will learn how to find wavenumber by using the different formulas which we discussed till now. Since there are different types of formulas to calculate the wavenumber depending on the various associated terms. Let us solve some problems to understand the concepts of wavenumber more clearly.
Calculate the Angular Wavenumber if the Wavelength of the LightWave is 600 Nanometers. Also, Calculate the Spectroscopic Wavenumber.
Ans: Here we are given only the wavelength of the light wave, so we will use the angular wavenumber which is directly related to the wavelength.
The angular wavenumber formula is given by
k = \[\frac{2\pi }{\lambda }\]
Here is the wavelength of the light wave is given as 600 nanometers which is equal to 600 x 10-9m.
Now substituting the values we will get the angular wavenumber as follows:
k = \[\frac{2\pi }{600\times 10^{-9}}\]
k = 10.48 x 106m-1
To calculate the spectroscopic wavenumber use the formula which describes the relation between spectroscopic wavenumber and wavelength of the light wave.
The spectroscopic wavenumber is given as follows:
\[\bar{v}\]= \[\frac{1}{\lambda }\]
Here λ is the wavelength of the light wave is given as 600 nanometers which is equal to 600 × 10-9m. Now substituting the values we will get the spectroscopic wavenumber as follows:
Now substituting the values we will get the spectroscopic wavenumber as follows:
\[\bar{v}\]= \[\frac{1}{600\times 10^{-9}}\]
\[\bar{v}\]= 1.67 x 106m-1
Conclusion
In atomic, molecular, and nuclear spectroscopy, a wavenumber is defined as the true frequency divided by the speed of light, and which is equal to the number of waves in a unit distance. In optics and wave scattering physics, such as X-ray diffraction, neutron diffraction, electron diffraction, and elementary particle physics, wavenumbers and wave vectors are frequently used.
FAQs on Wave Number: Definition, Formula & Applications
1. What exactly is a wave number in simple terms?
In Physics, the wave number tells you how many full waves or cycles fit into a specific unit of distance. Imagine a wavy line; the wave number is a measure of how compressed or spread out those waves are. A higher wave number means more waves are packed into the same space, indicating a shorter wavelength.
2. How can I calculate the wave number if I know the wavelength?
You can easily calculate the wave number by taking the reciprocal of the wavelength. The formula is:
Wave Number (ν̄) = 1 / Wavelength (λ).
For example, if a wave has a wavelength of 2 metres, its wave number would be 1/2 or 0.5 waves per metre.
3. What is the difference between wave number and angular wave number?
While related, they measure slightly different things.
- The wave number (ν̄) counts the number of complete cycles per unit distance (units: m⁻¹).
- The angular wave number (k) measures the number of radians per unit distance (units: rad/m). It's more common in wave equations and is calculated as k = 2π/λ.
4. What are the correct SI units and symbols for wave number?
The standard SI unit for wave number is reciprocal metres, written as m⁻¹. In chemistry and spectroscopy, it is also very common to see units of reciprocal centimetres (cm⁻¹). The symbol for spectroscopic wave number is ν̄ (nu-bar), while the symbol for angular wave number is 'k'.
5. Why is the concept of wave number so important in spectroscopy?
Wave number is crucial in spectroscopy because it is directly proportional to a wave's energy and frequency. This makes it much easier to analyse how atoms and molecules absorb or emit light. Scientists can look at the wave numbers in a spectrum to directly identify substances and understand their atomic or molecular energy levels, which is more convenient than constantly converting from wavelength.
6. How does wave number help connect a wave's spatial features with its phase?
The angular wave number (k) acts as a bridge between a wave's spatial period (its wavelength) and its phase. The term 'kx' in the general wave equation, y = A sin(ωt - kx), determines how the wave's phase changes as you move along its direction of travel (x). A larger 'k' means the phase changes more rapidly with distance, corresponding to a shorter wavelength and more tightly packed waves.
