Doppler Effect Explained
Before talking about the Doppler effect, the meaning of the term effect in the context of physics must be highlighted. Effect is a synonym for physical phenomenon.
Many discoveries in physics have emerged as the result of casually observed effects. For example: the famous “Newton’s apple”.
In the case of the Doppler effect it was no different. It is said that the person who first observed this effect was C. J. Doppler (1802–1853), a German physicist, whose name was attributed to the name of the phenomenon. The approach to the Doppler effect takes place within the study of wave movement.
Wave movement comprises everything that occurs as a consequence of a disturbance that propagates beyond its place of origin and with the condition that it presents inherent effects of diffraction and interference.
Both mechanical disturbances and electromagnetic fields fall under this concept, and are therefore subject to the Doppler effect.
The Doppler effect consists of the variation in the frequency of the wave movement that is perceived by the observer in relative movement.
In this sense, there are certain important factors to consider, namely:
- the source of disturbance or waves.
- the relative motion of the wave source.
- observers and their reference systems.
- the relative movement of the observers.
- the type of wave, conditioned or not by the material medium.
It is necessary to highlight that the description of the wave movement, for example, propagating in the direction of the X axis with an amplitude ξ and defined speed v, without distortion, follows the differential equation in the form:
Let’s first analyze the Doppler effect present in the behavior of the electromagnetic field.
|| Electromagnetic Doppler
An important factor in this context will be the result subsequent to the effect of a moving charge. For this, let us consider the configuration indicated in the figure below, where the particular effects of crossed fields caused by a moving point charge are highlighted.
The magnetic field associated with the moving point charge can be interpreted as a result of the expression:
Assuming that the charge is not accelerated, then the crossed electric field associated with the configuration will be given by a radial expression as follows.
The two expressions form a system in which, isolating the radial verse in the electric field expression and making the substitution in the magnetic field expression, we then have:
The coefficient μ₀ε₀ that appears in this final expression shows that the electric field and the magnetic field are coupled quantities forming the so-called electromagnetic field.
Thus, when applying Maxwell’s equations to crossed electric and magnetic fields, as a way of finding a solution for the electromagnetic wave predicted by him, that is, for example, for the electric field in rectangular coordinates and in the direction of the X axis, we arrive up to:
which is an expression analogous to the equation that describes a wave movement in the direction of the X axis, as mentioned previously. Consequently, the speed is equal to:
This value is a constant in physics and therefore corresponds to the speed of the electromagnetic wave in a vacuum, that is, the speed of light in a vacuum.
A primary aspect is that electromagnetic waves consist of energy in motion, and therefore the speed of the source relative to some material medium is not taken into account in deducing the electromagnetic Doppler effect.
Once the speed of the electromagnetic wave in a vacuum (symbolized by the letter c) is known, we can assume that such a wave is represented by a harmonic function in the form
this in relation to an observer in an inertial reference system, with ω = c k which is the angular frequency, and k = 2 π / λ which is the wave number.
Given that the speed of electromagnetic waves in a vacuum is equal to the phase speed, then the phase of the waves will be invariant in relation to two inertial frames, within the framework of restricted relativity, but their coordinates will be subject to the Lorentz transformation. Thus, for an observer fixed in relation to a second inertial system, we will have the function rewritten in the form:
and therefore, we can make the following equality:
then we can apply the reciprocal Lorentz transformation in order to only compare coefficients.
and, factoring the variables on the left-hand side,
where v is the speed of the observer of the electromagnetic signal in relation to the wave source and along the propagation line. It is then deduced that
or
and finally the expression of the electromagnetic Doppler effect
The electromagnetic Doppler effect explains the shift in the spectrum of starlight when compared to the standard spectrum. This effect is represented by the redshift parameter z
The figure below obtained from a galaxy spectra simulator visually demonstrates the electromagnetic Doppler effect.
|| Doppler in Mechanical Waves
In the case of mechanical waves, the interest is to obtain the relationship between the frequency ν produced by the wave source and the frequency νʹ recorded by the observer.
To obtain such a relationship, we will use the inherent relationship between the frequency of a wave
and also the scheme shown in the figure below.
Based on the figure above, we can write the kinematic equations of motion
that solving for the observation time of the first wave, we have
and
that solving for the observation time of the second wave, we have
Then the sampling time will be
And, the unknown frequency will be given by
When a mechanical wave is surpassed by its own source, an effect supervening on the Doppler occurs, the shock wave is segregated in a propagation cone whose axis is the source’s line of motion.
|| Comment
Both expressions of the Doppler effect can be simplified according to the scope of interest in such a way that we obtain
1️⃣For the electromagnetic Doppler effect:
this will make sense for when v ⪡ c.
2️⃣For the mechanical Doppler effect:
where
is the value of the observer’s velocity relative to the wave source, and ω = 2πν is the angular frequency.
