Frequency and phase in sin:

The equation y(x) = sin(f * x + p). What does f represent and what does p represent?

In the equation y(x) = sin(f * x + p), the variables f and p represent parameters that affect the behavior of the sine function.

f: The variable f represents the frequency of the sine wave. It determines how many oscillations occur over a given interval of x values. The frequency f controls the rate at which the sine function completes a full period, which is 2π radians or 360 degrees. Higher values of f result in more oscillations within the same interval, while lower values lead to fewer oscillations.

p: The variable p represents the phase shift of the sine wave. It indicates the horizontal displacement or translation of the wave along the x-axis. Specifically, it determines the initial position of the wave at x = 0. The phase shift p is measured in radians or degrees and can move the wave to the left or right. Positive values of p shift the wave to the left, while negative values shift it to the right.

By adjusting the values of f and p, you can modify the frequency and position of the sine wave, respectively, in the y(x) equation.

Equations of f frequency; T time period; v velocity; λ wavelength; λ₀ source wavelength; Δλ changed wavelength; T(deg) phase shift in degree; z gravitational or cosmic redshift:

Author ORCID: 0000-0003-1871-7803

Given below are the simplified expressions based on the equation and the relationship between the variables. Where, f is the frequency; T time period; v velocity; λ is the wavelength; λ₀ is the source wavelength; Δλ observed change in wavelength; T(deg) phase shift in degree; z gravitational/cosmic redshift. When,  f = 1/T = v/λ = v/λ₀; T(360) => T(deg) = z * 360; For 1° phase shift, T(deg)  = T/360 = (1/f)/360 = z * 360; z = Δλ/λ₀. 

• f is the frequency.

• We are given f = 1/T = v/λ = v/λ₀.
  

• Therefore, frequency f is equal to the reciprocal of the time period T, the velocity v divided by the wavelength λ, and also v divided by the source wavelength λ₀

• The specific velocities of the waves involved (343 m/s) for acoustic waves or (299,792,458 m/s) for electromagnetic waves.

• T time period.

• We are given a complete time period T in 360°, T(360) => T(deg) = z * 360.

• This equation represents the phase shift in degrees T(deg) being equal to the gravitational or cosmic redshift z multiplied by 360.

• For a complete time period T, the phase shift in degrees T(deg) is equal to z * 360, where z represents the gravitational or cosmic redshift.

• v velocity

• We are given v/λ = f = 1/T = v/λ₀.
  

• From these equations, we can see that v is equal to the product of the frequency f and the wavelength λ, and it is also equal to the product of the frequency f and the source wavelength λ₀.

• The specific velocities for acoustic waves (343 m/s) and electromagnetic waves (299,792,458 m/s) indicate the speed at which the waves propagate.

• λ is the wavelength

• We are given v/λ = f = 1/T =  v/λ₀.

• This equation indicates that the wavelength λ is equal to the velocity v divided by the frequency f, and it is also equal to the source wavelength λ₀.

• The wavelength (λ) is related to the velocity (v) and frequency (f) through the equation λ = v/f. Using the specific velocities provided, the wavelength can be calculated by λ = v/f.

• λ₀ is the source wavelength

• We are given v/λ₀ = v/λ = f = 1/T.

• Therefore, the source wavelength λ₀ is equal to the velocity v divided by the frequency f, and it is also equal to the wavelength λ.

• The source wavelength (λ₀) can be obtained by dividing the velocity (v) by the frequency (f), using the specific velocities given. Thus, λ₀ = v/f.

• Δλ observed change in wavelength

• We are given z = Δλ/λ₀.
  

• This equation represents the gravitational or cosmic redshift z being equal to the change in wavelength Δλ divided by the source wavelength λ₀

• T(deg) phase shift in degree

• We are given T(deg) = T/360 = (1/f)/360 = z * 360.

• This equation states that the phase shift in degrees T(deg) is equal to the time period T divided by 360, which is also equal to the reciprocal of the frequency f divided by 360, and it is equal to the gravitational or cosmic redshift z multiplied by 360. For a complete time period T, the phase shift in degrees T(deg) is equal to z * 360, where z represents the gravitational or cosmic redshift

• z gravitational/cosmic redshift.

• We are given z = Δλ/λ₀.

• This equation indicates that the gravitational or cosmic redshift z is equal to the change in wavelength Δλ divided by the source wavelength λ₀.

These are the simplified expressions based on the given equations and the relationships among the variables.

Reference : Relativistic effects on phaseshift in frequencies invalidate time dilation II 

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