05 July 2023

Derivation: How to calculate redshift using values of phase shift in frequency of wave equation?

Derived by Soumendra Nath Thakur. (ORCID: 0000-0003-1871-7803)

Summary

To calculate redshift using the values of phase shift in frequencies, follow the derivations. Here's a step-by-step explanation for each case:


Gravitational or Cosmic Redshift:

Start with the formula for 1° phase shift: Z = (λ₀/360) / λ₀, where λ₀ is the wavelength at the source.

Rearrange the equation to express λ₀ in terms of frequency: λ₀ = 1/f₀, where f₀ is the frequency at the source.

Substitute the expression for λ₀ into the redshift formula: Z = ((1/f₀)/360) / (1/f₀).

Simplify the equation to obtain the final formula for gravitational or cosmic redshift:

  • Z = (1/360f₀) / (1/f₀) = 1/360.

Therefore, for gravitational or cosmic waves, the redshift (Z) can be calculated as 1/360 for each 1° phase shift.


Doppler Redshift:

Begin with the formula for 1° phase shift: Z = (λ(rest)/360 - λ(rest)) / λ(rest), where λ(rest) is the wavelength at rest.

Similarly to the previous derivation, express λ(rest) in terms of frequency: λ(rest) = 1/f(rest), where f(rest) is the frequency at rest.

Substitute the expression for λ(rest) into the redshift formula: Z = ((1/f(rest))/360 - (1/f(rest))) / (1/f(rest)).

Simplify the equation to obtain the final formula for Doppler redshift: 

  • Z = (1/360f(rest) - 1/f(rest)) / (1/f(rest)).

Therefore, for Doppler redshift, the redshift (Z) can be calculated based on the given phase shift and the frequencies (or wavelengths) at rest.

In both cases, you'll need to know the frequency (or wavelength) at the source/rest and apply the appropriate formula to calculate the redshift. Additionally, keep in mind the specific velocities of the waves involved, whether it's the speed of sound (343 m/s) for acoustic waves or the speed of light (299,792,458 m/s) for electromagnetic waves.

Description:

The value of a redshift is denoted by the letter Z, corresponding to the fractional change in wavelength, positive for redshifts, negative for blueshifts, and by the wavelength ratio 1 + z, which is >1 for redshifts, <1 for blueshifts. And so, red-shift (z,>1) is the displacement of spectral lines towards longer wavelengths (Δλ+λ₀) > λ₀ i.e. the red end of the electromagnetic spectrum.

Where, velocity of the wave v = fλ, where acoustics waves speed 343 m/s and electromagnetic waves speed 299792458 m/s. 
                                   
For gravitational or cosmic waves, wavelength at the source is λ₀ and observed change in wavelength is Δλ.

The time interval T(deg) for 1° of phase is inversely proportional to the frequency (f). We get a wave corresponding to the time shift.

1° phase shift = T/360.

Since, T = 1/f,

1. Derivation of formula for gravitational and cosmic waves:

For 1° phase shift, T(1°) = T/360 = (1/f)/360 = ΔT. 

Since, λ = T = 1/f.

λ₀ = T₀ = 1/f₀. Where, T₀ is the period and f₀ frequency at the source.

Δλ = T₀/360 = (1/f₀)/360 for gravitational or cosmic waves.

Or, Δλ = λ₀/360 = (1/f₀)/360 

Since, Z = Δλ/λ₀ 

For 1° phase shift, Z = {(1/f₀)/360}/(1/f₀) or, (λ₀/360)/λ₀. formula for gravitational and cosmic redshift... (1).

2. Derivation of formula for Dopplar redshift:

For 1° phase shift, T(1°) = T/360 = (1/f)/360 = ΔT. 

Since, λ = T = 1/f.

λ(rest) = T(rest) = 1/f(rest). Where, T(rest) is the period and f(rest) frequency at the source.

λ(obs) = T(rest)/360 = {1/f(rest)}/360 for Doppler redshift.

Or, λ(obs) = {1/f(rest)}/360 = λ(rest)/360.

And since, Z = λ(obs)-λ(rest)}/λ(rest)

For 1° phase shift, Z = {1/f(rest)}/360 -1/f(rest)} / 1/f(rest) or λ(rest)/360 - λ(rest)}/λ(rest) formula for Dopplar redshift ... (2).

Therefore, depending upon acoustic or, electromagnetic wave, we can calculate respective values of f₀ or f(rest), to obtain respective values of redshift (Z), using vales of respective phase shift in frequencies f₀ or f(rest) for gravitational and cosmic redshifts, or Doppler shift respectively, applying respective velocities of the waves, whether acoustics wave or electromagnetis wave.

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