# A review of the classical Doppler effect based on the mathematical
description of the phase function

Óscar Alejos and José María Muñoz

###### Abstract

This paper aims to shed some more light on one of the best known phenomena in
the field of physics, the Doppler effect, in particular, on its classical
version. Although, as mentioned, it is a phenomenon already described more
than 150 years ago, the general case, even considering constant velocities,
continues to be the subject of very recent bibliographical works which either
show it incompletely or start from a not entirely accurate description of it.
Here, general expressions of both frequency shift and aberration are obtained
by geometrical description and, more interestingly, by describing the phase of
the wave in the reference frame in which the propagating medium is at rest.
The coherence between the two points of view permits to contrast the validity
of the proposed results, and allows for a generalization of the resulting
expressions for any reference frame, even those where the propagating medium
is not still.

###### keywords:

Doppler effect , phase function , frequency shift , aberration

††journal: Results in Physics

[first]organization=University of Valladolid,addressline=Department of
Electricity and Electronics, city=Valladolid, postcode=47011, country=Spain

##  1 Introduction

Doppler effect (DE) is a very well-known topic in Physics since its discovery
by Doppler in 1842[1]. Many textbooks [2, 3, 4, 5] include amidst their
contents references to this effect, both in its classical and relativistic
version. Within the classical study, the particular case of sound waves is
discussed, where the wave requires a propagating medium. However, only the
situation when the observer is within the trajectory of the emitter is
considered in most textbooks, i.e., emitter and observer positions are
collinear with emitter’s trajectory. The non-collinear case is indeed the
subject of recent scientific papers [6, 7, 8, 9], where even some attempts to
unify the classical and relativistic DEs are made. However, some presented
results do not consider the nonlinear dependence of phase on time at the
observer’s location, since they compare one period of the emitted signal with
what it is called a period perceived by the observer, i.e., the interval of
time between to identical vibration states of the wave. Some other
bibliographic references[10] present an incomplete treatment of the
noncollinear case.

Accordingly, the present paper attempts to give a clear view of classical DE,
in particular by contrasting the results provided by a purely geometrical
description with those resulting from what can be considered as a more
complete description in terms of the spatial and temporal dependence of the
phase of the propagating wave in the medium.

The work is organized according to the following structure. First, the case of
a stationary observer in the reference frame in which the propagating medium
is at rest is reviewed, the emitter moving at constant velocity in that
reference frame. Initially, the problem is approached by the purely
geometrical construct, justifying the aberration effect when the observer is
not in the path of the emitter. The corresponding expression for the frequency
shift, defined as the instantaneous variation of the phase perceived by the
observer, is also obtained. In the same section, the problem is addressed in a
much more intuitive way, i.e., by constructing the function that defines the
phase at each instant and at each point in space in the observer’s reference
frame. Aberration and frequency shift appear naturally from the derivatives of
the phase function with respect to time and position, checking the consistency
between this and the previous approach. In the following section, the case of
a moving observer in a reference frame where the propagating medium is still
and the emitter at rest is briefly considered. Finally, the more general case
of moving observer and emitter is addressed. This study provides with a
expression of the frequency shift due to the DE in terms of relative positions
and velocities, which can be easily extended to any reference frame,
particularly, those where the propagating medium is not still and wave fronts
drift due to the medium velocity.

##  2 Moving emitter

###  2.1 Geometrical construct

Consider that, at a given time instant $t$, the emitter is oscillating and its
phase is $\varphi$. Without loss of generality, the observer can be located at
the origin of coordinates in its reference frame so that the emitter at that
instant is at the position given by a certain $x$-coordinate and the distance
$h$ between the observer and the line defining the trajectory of the emitter,
as shown in fig. 1. Assume that at that instant the distance between the
emitter and the observer is $c\tau$, where $c$ is the propagation velocity in
the medium, which will be taken to be at rest in the observer’s reference
frame. In fact, $\left(c\tau\right)^{2}=x^{2}+h^{2}$.

Figure 1: Geometry of the problem of a moving emitter. To facilitate the
viewing of the drawing, the distance $vdt$ has been exaggeratedly enlarged.

The emitter moves with velocity $v$ in observer’s reference frame, so that
after a time $dt$ it travels a distance $vdt$. At instant $t+dt$, the phase of
the emitter will be $\varphi+d\varphi$, with $d\varphi=2\pi ft$, $f$ being the
emitter frequency. This new phase will be measured at observer’s position a
time interval $\tau^{\prime}$ later. This means, actually, that the phase
measured at the observer position changes in a quantity $d\varphi$ after a
time $dt^{\prime}$, which can be computed as
$dt^{\prime}=\tau^{\prime}+dt-\tau$. Since, at instant $t+dt$, the emitter is
at position $x+vdt$, then $\left(c\tau^{\prime}\right)^{2}=(x+vdt)^{2}+h^{2}$.
According to this,

$\displaystyle dt^{\prime}$ $\displaystyle=\tau^{\prime}+dt-\tau=$ (1)
$\displaystyle=\frac{\sqrt{\left(x+vdt\right)^{2}+h^{2}}}{c}+dt-\frac{\sqrt{x^{2}+h^{2}}}{c}\approx$
$\displaystyle\approx
dt\left(1+\frac{x\frac{v}{c}}{\sqrt{x^{2}+h^{2}}}\right)\text{.}$

The instant frequency $f^{\prime}$ measured at the observer position is then
given by $f^{\prime}=\frac{1}{2\pi}\frac{d\varphi}{dt^{\prime}}$. By renaming
$\beta=\frac{v}{c}$, and rewritting
$\cos\vartheta=-\frac{x}{\sqrt{x^{2}+h^{2}}}$, then

$f^{\prime}=\frac{f}{1-\beta\cos\vartheta}\text{.}$ (2)

As a consequence of this expression, all observers located along a line whose
direction is given by the angle $\vartheta$ perceive the same frequency
$f^{\prime}$, but, importantly, at different time instants, depending on the
distance from each observer to the emitter along such a line. However, before
dealing with the aberration effect, it is necessary to comment that the above
expression is derived from the one obtained in special relativity, as
indicated in some textbooks. It is sufficient to consider that in this
expression, all times are referred to observer’s frame, which is the same as
that of the reference frame of the emitter in the framework of Galilean
relativity. The missing factor $\frac{1}{\gamma}=\sqrt{1-\beta^{2}}$ in
expression (2) appears naturally if one considers the time dilation associated
with the change of reference frame within special relativity.

Going back to the subject of the aberration, this means that, when the
observer perceives a certain frequency $f^{\prime}$, the emitter has moved a
certain distance, and subtends a different angle with respect to the observer,
i.e., the sound comes from one direction, but the emitter is seen in a
different direction. Both directions can be related to each other.

Figure 2: Aberration. The direction from which the sound comes and the
direction in which the emitter is seen are different.

Consider now fig. 2. When the observer perceives the frequency $f^{\prime}$
coming from the direction given by the angle $\vartheta$, a time $\tau$ has
elapsed since the phase variation of the emitter produced such a sound, due to
the distance $c\tau$ at which the emitter was located from the observer. In
that time $\tau$, the emitter has moved a distance $v\tau$, so that the
distance the emitter is from the observer can be written as a certain factor
$\alpha$ of the distance $c\tau$. Applying the cosine law, it is found that

$\alpha^{2}+\beta^{2}+2\alpha\beta\cos\theta=1\text{.}$ (3)

Additionally, sine law establishes that

$\sin\vartheta=\alpha\sin\theta\text{.}$ (4)

Both equations can be combined to give the desired result linking both angles
$\vartheta$ and $\theta$, as it is demonstrated in A. In this way,

$\displaystyle\sin\vartheta$
$\displaystyle=\sin\theta\left(\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta\right)\text{,}$
(5a) $\displaystyle\cos\vartheta$
$\displaystyle=\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\text{.}$
(5b)

Now, using (5b), expression (2) can be written in its final form, showing the
frequency shift due to DE, i.e,

$f^{\prime}=\frac{f}{1-\beta\left(\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\right)}\text{.}$
(6)

Importantly, expression above reduces to the well-known formulae for collinear
displacement of the emitter with the line joining the emitter and the
observer, that is, for $\theta$ equal to either $0$ (emitter approaching the
observer):

$f^{\prime}=\frac{f}{1-\beta}\text{,}$ (7)

or $180\degree$ (emitter moving away from the observer):

$f^{\prime}=\frac{f}{1+\beta}\text{.}$ (8)

Additionally, DE vanishes when $\vartheta=90\degree$, and the observer sees
the emitter at an certain angle $\theta_{v}\neq 90\degree$ given by the
condition

$\sin\theta_{v}=\frac{1}{\sqrt{1+\beta^{2}}}\text{,}$ (9)

once the observer has been overtaken by the emitter.

###  2.2 Instantaneous phase determination

Since in the framework of Galilean relativity, the wave equation is not
invariant with respect to coordinate transformations, the calculation of the
phase of a wave in a reference frame in which the emitter is in motion with
respect to the observer cannot be performed by simply considering circular
wave fronts in emitter’s reference frame and applying the corresponding
coordinate transformation. The procedure is therefore not so straightforward,
but must be approached from the wavefronts created by the emitter at each of
the points it travels through.

To determine the phase of each point $\left(x,y\right)$ in space, it is then
necessary to first estimate the coordinate point $\left(x_{0},0\right)$ where
the emitter was located when the corresponding wavefront that has reached that
point $\left(x,y\right)$ was emitted. If the emitter is at position
$\left(x_{s},0\right)$ at the current instant, it has been moving for a time
$\tau=\frac{x_{s}-x_{0}}{v}$ since it emitted the wavefront. In that time, the
wavefront has propagated a distance $c\tau$, as shown in the fig. 3. According
to the above, the following pair of equations can be proposed

$\displaystyle x_{s}-x_{0}$ $\displaystyle=v\tau\text{,}$ (10a)
$\displaystyle\left(x-x_{0}\right)^{2}+y^{2}$
$\displaystyle=\left(c\tau\right)^{2}\text{.}$ (10b)

By combining both equations, the second degree equation

$\left(c^{2}-v^{2}\right)\tau^{2}-2\left(x-x_{s}\right)v\tau-\left[\left(x-x_{s}\right)^{2}+y^{2}\right]=0$
(11)

holds, that can be solved for positive $\tau$’s, in such a way that the
product $c\tau$ equals

$c\tau=\frac{1}{1-\beta^{2}}\left[\beta\left(x-x_{s}\right)+\sqrt{\left(x-x_{s}\right)^{2}+\left(1-\beta^{2}\right)y^{2}}\right]\text{.}$
(12)

Figure 3: A train of wavefronts. The phase at a point $\left(x,y\right)$
depends on the position $\left(x_{0},0\right)$ at which the corresponding
wavefront was emitted, and the time elapsed until the front reaches that
point. In that time, the emitter has advanced a certain distance until it
reaches position $\left(x_{s},0\right)$. The angle $\theta$ defines the
direction in which the observer sees the emitter at each instant.

Since the phase at the $\left(x,y\right)$ coordinate point results from a
propagation at velocity $c$ of a wavefront starting at the
$\left(x_{0},0\right)$ coordinate point, the phase difference between the two
points is proportional to the distance $c\tau$ by a factor
$k=\frac{2\pi}{\lambda}$, where $\lambda=\frac{c}{f}$, obviously. Likewise, if
the phase at the point $\left(x_{0},0\right)$ at the instant the wavefront
started was $\varphi_{0}$, at a later instant $\tau$, the phase at that point
will be $\varphi_{0}+2\pi f\tau$. Accordingly, at the subsequent $\tau$
instant the phase at the $\left(x,y\right)$ point can be calculated as
$\varphi_{0}+2\pi f\tau-kc\tau$. Without loss of generality, it can be
considered that the emitter was at the origin of coordinates at instant $t=0$
and its phase was $\varphi_{0}=0$, thus resulting in the expression of the
dependence of the phase with position and time

$\varphi\left(x,y,t\right)=2\pi
ft-k\frac{\beta\left(x-vt\right)+\sqrt{\left(x-vt\right)^{2}+\left(1-\beta^{2}\right)y^{2}}}{1-\beta^{2}}\text{.}$
(13)

Some examples of wavefronts generated by a moving emitter are shown in fig.4
using color maps. The advance of the fronts is shown at different time
instants within an interval equal to the period of oscillation, given by
$\frac{1}{f}$, and for different velocities of the emitter. The perfectly
spherical shape of the wavefronts can be verified in all cases.

Figure 4: Wavefronts generated by a moving emitter at different instants and
for different velocities of the emitter. Check the sphericity of wavefronts.

Once the function that establishes the phase as a function of position and
time has been determined, the propagation characteristics, i.e., frequency and
wave vector, can be easily derived. To do so, it suffices to consider
$f^{\prime}=\frac{1}{2\pi}\frac{d\varphi}{dt}$ for the former and
$\mathbf{k}^{\prime}=-\nabla\varphi$ for the latter. Starting with the
frequency, it turns out that

$f^{\prime}=f\left\\{1+\frac{\beta}{1-\beta^{2}}\left[\beta+\frac{x-vt}{\sqrt{\left(x-vt\right)^{2}+\left(1-\beta^{2}\right)y^{2}}}\right]\right\\}\text{.}$
(14)

which explicitly shows the time dependence of the instantaneous frequency
perceived by the observer. This expression can be rewritten in terms of the
angle $\theta$ defined in Figs. 2 and 3, yielding

$f^{\prime}=f\left[1+\frac{\beta}{1-\beta^{2}}\left(\beta+\frac{\cos\theta}{\sqrt{1-\beta^{2}\sin^{2}\theta}}\right)\right]\text{.}$
(15)

Proper handling of the above expression (see B) leads to a formulation of the
expression as follows

$f^{\prime}=\frac{f}{1-\beta\left(\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\right)}\text{,}$
(16)

i.e., the result given in (6).

Besides, the wave vector can be obtained as

$\displaystyle\mathbf{k}^{\prime}$
$\displaystyle=\frac{k}{\left(1-\beta^{2}\right)\sqrt{1-\beta^{2}\sin^{2}\theta}}\times$
(17)
$\displaystyle\times\left[\left(\beta\sqrt{1-\beta^{2}\sin^{2}\theta}+\cos\theta\right)\hat{\mathbf{x}}+\left(1-\beta^{2}\right)\sin\theta\hat{\mathbf{y}}\right]\text{,}$

This expression can be straightforwardly rewritten in terms of the angle
$\vartheta$ in fig.2 (see C), resulting

$\mathbf{k}^{\prime}=\frac{k}{1-\beta\cos\vartheta}\left(\cos\vartheta\hat{\mathbf{x}}+\sin\vartheta\hat{\mathbf{y}}\right)\text{,}$
(18)

as it can be expected, since the direction of propagation at each point is
perpendicular to the wavefronts, then pointing in the direction of the segment
joining this point and that were the wavefront was started. This result
accounts for the aberration effect previously mentioned. Additionally, the
observed and the emitted wavelenghts, i.e., $\lambda^{\prime}$ and $\lambda$,
fulfill the expected relationship

$\lambda^{\prime}f^{\prime}=\lambda f=c$ (19)

Figure 5 shows a first example of these results. An emitter moving with
velocity given by $\beta=0.5$ pass by an observer located at a distance of
$5\lambda$ away from emitter’s trajectory. Time in these graphs is taken as
nondimensional by multiplying it by emitter’s frequency $f$, and emitter and
observer are at the closest positions when $t=0$. The dependence of the phase
on time is shown in a), where shorter wavelenghts can be seen at the initial
instants and these wavelengths become larger as the emitter progresses. b)
shows how the instantaneous frequency perceived by the observer decreases in
time. It is important to note the effect of aberration, since the observer
perceives a frequency equal to emitter’s frequency not when emitter and
observer are at the closest positions $\left(\theta=90\degree\right)$, but at
a latter time instant given by $t=\frac{h}{c}=\frac{h}{\lambda f}$, i.e., when
$\vartheta=90\degree$.

Figure 5: DE for a moving emitter, passing by an observer located at a
distance equal to $5\lambda$ to emitter’s trajectory with a velocity given by
$\beta=0.5$: a) temporal evolution of phase vibration at observer’s location,
and b) instantaneous frequency perceived by the observer. Time is expressed in
nondimensional units.

Figure 6 shows more examples of these results. In the case a) the observer is
at a distance equal to $5\lambda$ away from emitter’s trajectory, who is
moving at different velocities. Since the distance holds for all cases, the
instant at which the frequency shift vanishes remains constant by the
condition $ft=\frac{h}{\lambda}=5$, i. e., the emitter is farther from the
observer as the velocity increases, according to (9). Besides, frequency
variation becomes sharper as emitter’s trajectory approaches observer’s
position, as shown in b).

Figure 6: DE for a moving emitter, passing by an observer. Instantaneous
frequency perceived by an observer: a) located at a distance equal to
$5\lambda$ to emitter’s trajectory, when the emitter moves with different
velocities, and b) located at different distance from emitter’s trajectory,
when the emitter moves with velocity $\beta=0.5$. (time expressed in
nondimensional units).

##  3 Moving observer

###  3.1 Geometrical construct

Consider now the case of a moving observer. The emitter is at rest at the
reference frame where the propagating medium is still. With no loss of
generality, the emitter has been located at the origin of coordinates. At a
given time instant $t$, the observer perceives a phase $\varphi$. This means
that the phase of the emitter is $\varphi+kc\tau$ at that instant, according
to the drawing in fig.7. After a time interval of $dt$, the observer has moved
a distance $vdt$ and perceives a different phase $\varphi+d\varphi^{\prime}$.
The phase of the emitter at that instant must be $\varphi+kc\tau+2\pi fdt$.
Taking into account the new distance $c\tau^{\prime}$ between the observer and
the emitter, the equation $\varphi+d\varphi^{\prime}=\varphi+kc\tau+2\pi fdt-
kc\tau^{\prime}$ holds, so that

$\displaystyle d\varphi^{\prime}$ $\displaystyle=2\pi fdt-
kc\left(\tau^{\prime}-\tau\right)=$ (20) $\displaystyle=2\pi
f\left(dt-\frac{\sqrt{\left(x+vdt\right)^{2}+h^{2}}}{c}+\frac{\sqrt{x^{2}+h^{2}}}{c}\right)\approx$
$\displaystyle\approx 2\pi
fdt\left(1-\frac{v}{c}\frac{x}{\sqrt{x^{2}+h^{2}}}\right)\text{.}$

Figure 7: Geometry of the problem of a moving observer. Again, the distance
$vdt$ has been exaggeratedly enlarged to facilitate the viewing of the
drawing.

The frequency at observer’s position is obtained as

$f^{\prime}=\frac{d\varphi^{\prime}}{dt}=f\left(1-\beta^{\prime}\cos\theta\right)\text{,}$
(21)

where $\beta^{\prime}$ is defined by considering the velocity
$v=\beta^{\prime}c$ of the observer. This is indeed the DE expected expression
for moving observers. In this case, no aberration must be considered, since
wave vectors are directed radially from the emitter, being then perpendicular
to the wave fronts, as it will be stated in the next subsection.

###  3.2 Instantaneous phase approach

Within emitter’s reference frame the medium is still, so that (13) is
rewritten as

$\varphi\left(x,y,t\right)=2\pi ft-k\sqrt{x^{2}+y^{2}}\text{.}$ (22)

leading to concentric wave fronts centered at emitter’s location, as shown in
fig.8. The frequency $f^{\prime}$ perceived by the observer is obtained from
the derivative of the previous expression with time. The use of the convective
derivative $\frac{d}{dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla$ is
now necessary to yield again
$f^{\prime}=f\left(1-\beta^{\prime}\cos\theta\right)$, as in the previous
subsection. As for the wave vector, the use of the $\nabla$ operator results
in
$\mathbf{k}=-\nabla\varphi=k\left(\cos\theta\hat{\mathbf{x}}+\sin\theta\hat{\mathbf{y}}\right)$.

Figure 8: DE for a moving observer, passing by an observer. The observer moves
with velocity $\mathbf{v}$ and perceives the wave fronts moving in the
direction of the wave vector $\mathbf{k}$.

##  4 Moving emitter and observer

This case will be addressed directly from the point of view of the phase
function. As in the previous sections, the velocities will be taken with
respect to the reference frame in which the propagating medium is at rest, so
that, without loss of generality, the emitter will be considered to move with
velocity $\hat{\mathbf{v}}=v\hat{\mathbf{x}}$, with $v=\beta c$, and the
motion of the observer is given by the vector
$\mathbf{v}=v_{x}\hat{\mathbf{x}}+v_{y}\hat{\mathbf{y}}$. The use of the
convective derivative yields the expression for the frequency shift as

$f^{\prime}=f\left[1-\frac{\beta\left(\beta_{x}-\beta\right)+\frac{\left(\beta_{x}-\beta\right)\cos\theta+\beta_{y}\left(1-\beta^{2}\right)\sin\theta}{\sqrt{1-\beta^{2}\sin^{2}\theta}}}{1-\beta^{2}}\right]\text{,}$
(23)

where $\beta_{x}=\frac{v_{x}}{c}$ and $\beta_{y}=\frac{v_{y}}{c}$, and
$\theta$ being the angle between emitter’s trajectory and the relative
instantaneous position between the observer and the emitter, as shown in
fig.9. Interestingly, despite being a rather complex expression, it predicts
that the DE vanishes when both the observer and emitter move parallel with the
same velocity, i.e., $\beta=\beta_{x}$ and $\beta_{y}=0$. More importantly,
this expression depends indeed on relative positions, through the
$\theta$-angle, and relative velocities, that between the emitter and the
reference frame in which the propagating medium is at rest, and that between
the observer and the emitter. This means, in particular, that the result is
valid for any other reference frame. Hence, by defining
$\left(\beta^{\prime}_{x},\beta^{\prime}_{y}\right)$ as the components of the
relative velocity of the observer with respect to the emitter, previous
equation can be rewritten as

$f^{\prime}=f\left[1-\frac{\beta\beta^{\prime}_{x}\sqrt{1-\beta^{2}\sin^{2}\theta}+\beta^{\prime}_{x}\cos\theta+\beta^{\prime}_{y}\left(1-\beta^{2}\right)\sin\theta}{\left(1-\beta^{2}\right)\sqrt{1-\beta^{2}\sin^{2}\theta}}\right]\text{,}$
(24)

where $\beta$ will still be used to refer to the relative velocity between the
emitter and the reference frame in which the propagating medium is at rest.

Figure 9: DE for the case when both emitter and observer are moving with
respect to the reference frame where the propagating medium . The velocity of
the emitter is given by vector $v\hat{mathbf{x}}$, while the observer moves
with velocity $v_{x}\hat{\mathbf{x}}+v_{y}\hat{\mathbf{y}}$. The angle
$\theta$ defining the relative instantaneous position between observer and
emitter is also displayed.

As an example, fig.10 shows the case where the propagating medium moves with
respect to the emitter. The medium velocity is defined as
$-v\hat{\mathbf{x}}$, and three different velocities are chosen, corresponding
to $\beta=-0.25,-0.5,\text{and}-0.75$. These are indeed the cases shown in
fig.4 but seen from the emitter’s reference frame. Due to the moving medium, a
drift of the wave fronts occurs. This drift promotes the gathering of wave
fronts to the right of the emitter and its spreading to the left or,
equivalently, different propagating velocities of the wave depending on
direction. As it has been mentioned above, despite these different propagating
velocities, a stationary observer with respect to the emitter will not notice
any frequency shift, since the wavelengths in the direction of the observation
will enlarge or shorten accordingly.

Figure 10: Wavefronts resulting from drift associated with a moving medium in
the emitter’s reference frame. The sphericity of the wavefronts hold, but
their centers shift with respect to the emitter’s position. The negative
values of $\beta$ mean that within this reference frame, the medium moves with
velocity $-v\hat{\mathbf{x}}$.

##  5 Acknowledgments

The authors would like to acknowledge the support of GIR Materiales Magnéticos
of the University of Valladolid and the fruitful discussions with its
components that originated and shaped the contents of this work.

## Appendix A

By combining equations (3) and (4), it follows

$\left(\frac{\sin\vartheta}{\sin\theta}\right)^{2}+\beta^{2}+2\beta\frac{\sin\vartheta}{\sin\theta}\cos\theta=1\text{,}$
(25)

which can be rewritten as a second degree equation in $\sin\vartheta$ as

$\sin^{2}\vartheta+2\beta\sin\theta\cos\theta\sin\vartheta-\left(1-\beta^{2}\right)\sin^{2}\theta=0\text{,}$
(26)

one of whose solutions is the relationship shown in (5a), i.e.,

$\sin\vartheta=\sin\theta\left(\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta\right)\text{,}$
(27)

This expression can be handled to obtain that corresponding to
$\cos\vartheta$. Indeed,

$\displaystyle\cos^{2}\vartheta$ $\displaystyle=1-\sin^{2}\vartheta=$ (28)
$\displaystyle=\beta^{2}\sin^{4}\theta+2\beta\cos\theta\sin^{2}\theta\sqrt{1-\beta^{2}\sin^{2}\theta}+$
$\displaystyle\qquad+\cos^{2}\theta\left(1-\beta^{2}\sin^{2}\theta\right)=$
$\displaystyle=\left(\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\right)^{2}\text{,}$

and, finally,

$\cos\vartheta=\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\text{.}$
(29)

as it appeared in (5b)

## Appendix B

Expression in (15) can be rewritten as follows

$\displaystyle 1+$
$\displaystyle\frac{\beta}{1-\beta^{2}}\left(\beta+\frac{\cos\theta}{\sqrt{1-\beta^{2}\sin^{2}\theta}}\right)=$
(30)
$\displaystyle=\frac{\sqrt{1-\beta^{2}\sin^{2}\theta}+\beta\cos\theta}{\left(1-\beta^{2}\right)\sqrt{1-\beta^{2}\sin^{2}\theta}}\text{.}$

After multiplying the numerator and the denominator by the factor
$\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta$, the result

$\displaystyle\frac{1-\beta^{2}\sin^{2}\theta-\beta^{2}\cos^{2}\theta}{\left(1-\beta^{2}\right)\sqrt{1-\beta^{2}\sin^{2}\theta}\left(\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta\right)}=$
(31)
$\displaystyle=\frac{1}{1-\beta\left(\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\right)}\text{,}$

holds, which is the factor appearing in (6).

## Appendix C

To check the relationship between equations (17) and (18), it is first
necessary to determine the relationship between their moduli. Thus,

$\displaystyle k^{\prime}$
$\displaystyle=k\frac{\sqrt{\left(\beta\sqrt{1-\beta^{2}\sin^{2}\theta}+\cos\theta\right)^{2}+\left(1-\beta^{2}\right)^{2}\sin^{2}\theta}}{\left(1-\beta^{2}\right)\sqrt{(1-\beta^{2}\sin^{2}\theta}}=$
(32)
$\displaystyle=k\frac{\sqrt{1-\beta^{2}\sin^{2}\theta+\beta^{2}\cos^{2}\theta+2\beta\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}}}{\left(1-\beta^{2}\right)\sqrt{(1-\beta^{2}\sin^{2}\theta}}=$
$\displaystyle=k\frac{\sqrt{1-\beta^{2}\sin^{2}\theta}+\beta\cos\theta}{\left(1-\beta^{2}\right)\sqrt{1-\beta^{2}\sin^{2}\theta}}\text{,}$

which, by virtue of the results in A and B, transforms into

$k^{\prime}=\frac{k}{1-\beta\left(\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}\right)}=\frac{k}{1-\beta\cos\vartheta}\text{.}$
(33)

As for the direction of this vector, the tangent of the corresponding angle
can be worked out as the quotient $\frac{k^{\prime}_{y}}{k^{\prime}_{x}}$,
i.e.,

$\frac{k^{\prime}_{y}}{k^{\prime}_{x}}=\frac{\left(1-\beta^{2}\right)\sin\theta}{\beta\sqrt{1-\beta^{2}\sin^{2}\theta}+\cos\theta}\text{.}$
(34)

By multiplying the numerator and the denominator by the factor
$\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta$ and simplifying the result,
it follows that

$\frac{k^{\prime}_{y}}{k^{\prime}_{x}}=\frac{\sin\theta\left(\sqrt{1-\beta^{2}\sin^{2}\theta}-\beta\cos\theta\right)}{\beta\sin^{2}\theta+\cos\theta\sqrt{1-\beta^{2}\sin^{2}\theta}}=\frac{\sin\vartheta}{\cos\vartheta}=\tan\vartheta\text{,}$
(35)

which demonstrates that the wave vector $\mathbf{k}^{\prime}$ is oriented
along the direction given by the angle $\vartheta$.

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