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SOLUTIONS OF HARMONIC OSCILLATORS BY FOURIER SERIES
Preprint · February 2024
DOI: 10.13140/RG.2.2.21558.88649
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Alberto Carlos Bertuola
Federal Institute of Education, Science and Technology of São Paulo
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Victo S. Filho
H4D Scientific Research Laboratory
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SOLUTIONS OF HARMONIC OSCILLATORS BY FOURIER SERIES 
A. C. Bertuola1 and V. S. Filho2 
1Instituto Federal de Educação Ciência e Tecnologia de São Paulo, Campus São Paulo, Brasil 
2H4D Scientific Research Laboratory, São Paulo, Brazil 
 
Abstract 
In this work, it is used the Fourier series as instrument to obtain the solutions of differential equations of motion of the 
simple harmonic oscillator, of the damped harmonic oscillator and of the forced harmonic oscillator with damping. For damped 
movement it is also shown the efficiency of the complex functions and as they arise from this formalism. It is also presented 
the stationary solution used to describe the motion of the oscillator with damping and subjected to an external force responsible 
for the important term of amplitude which describes the resonance phenomenon. We show that it is possible to obtain the 
general solutions of the differential equation of all regimes de oscillation studied in an undergraduate course in Exact Sciences. 
The mathematical physics analyses of all cases of oscillation are described in detail in this paper. 
Keywords: Oscillator, damping, resonance, Fourier series 
 
1. Introduction 
In a higher level course in Exact Sciences, the 
phenomena of local oscillations are certainly studied at some 
point in the basic learning cycle [1-4]. The problem chosen for 
study is the one-dimensional mass-spring system, which 
means a body of mass m (kg) linked to a spring of spring 
constant k (N/m), as shown in the simplified scheme in Fig. 1. 
The harmonic oscillator has been intensely 
researched in many scientific papers over time [5-9] and there 
are interesting topics of research related to it until recent years 
[10,11], even considering that it is a classical problem and a 
relatively simple physical system. The upper frame of Fig. 1 
shows the diagram of forces acting on the oscillator, without 
the presence of any kind of friction or resistance. 
The weight force is identified and calculated by 
P=mg, considering the value g = 9.8 m/s2 for the acceleration 
of gravity. The normal component of reaction force is N and 
the component of elastic force is obtained by Fk = -kx (Hooke's 
law). Then, the student has his first contact with the simple 
harmonic motion, when Newton's second law is presented in 
the form of a differential equation of this one-dimensional 
motion. 
 
Figure 1: Local oscillation in the unidimensional 
mass-spring problem. 
 
2 
 
The upper frame in Fig. 1 shows the diagram of 
forces acting on the body for the case of the simple harmonic 
motion. This physical modelling of the simple harmonic 
oscillator is explicitly given by the differential equation 
 𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝝎𝟎
𝟐𝒙 = 𝟎 , (1) 
in which the natural frequency of oscillation is calculated by 
means of the definition 𝜔0 = √
𝑘
𝑚
 . Equation (1) is 
mathematically called ordinary differential equation, of 
second order, linear, homogeneous and with constant 
coefficients. 
In the following a damped motion is introduced, 
which is recognized as more realistic than the case of the 
simple harmonic oscillator. The body in the middle frame of 
Fig. 1 presents the same previous forces in the diagram 
considering an additional viscous friction force, which is 
calculated using the equation 𝐹𝛾 = −𝑏�̇�, where b is the so-
called viscous friction coefficient and �̇� =
𝑑𝑥
𝑑𝑡
 is the adopted 
notation for the first time derivative, representing the 
horizontal component of the velocity. The differential 
equation of this motion is mathematically described by 
 𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝟐𝜸
𝒅𝒙
𝒅𝒕
+ 𝝎𝟎
𝟐𝒙 = 𝟎 , 
(2) 
where 2𝛾 =
𝑏
𝑚
 and 𝛾 is called the damping parameter. 
Finally, the student learns about the damped and 
forced harmonic oscillator, in which arises the important 
phenomenon of resonance. The lower frame in Figure 1 shows 
the addition of the external force F(t) in the diagram of forces. 
The differential equation that rules this movement is given by 
 𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝟐𝜸
𝒅𝒙
𝒅𝒕
+ 𝝎𝟎
𝟐𝒙 = 𝒇(𝒕) , 
(3) 
with 𝑓(𝑡) =
𝐹(𝑡)
𝑚
. 
The differential equations (1), (2) and (3) require the 
search for two solutions, which when combined linearly are 
called general solutions of the differential equations. For 
example, the general solution of the differential equation (1) 
can be obtained by means of various mathematical methods. 
One way to obtain this simple harmonic oscillator solution 
from a physics point of view [1] is to observe the oscillator's 
own movement, recognizing its periodic behaviour. Two 
elementary functions with these properties are candidates for 
solution of (1). The sineand cosine trigonometric solutions are 
conjectured and replacing each one in Eq. (1), it is found that 
the sine function is one that describes the movement of a 
simple harmonic oscillator when it is initially released from 
rest, but displaced from its origin. On the other hand, the 
cosine function describes the movement of the oscillator, 
when it begins its periodic movement from the origin, but with 
a certain value for its initial velocity. The most general initial 
conditions happen when the oscillator is released from a 
different position from the origin, with a certain value for its 
speed. For these initial conditions, the solution is a linear 
combination of sine and cosine functions. 
The main objective of the present work is to obtain 
general solutions for oscillators governed by the differential 
equations (1), (2) and (3) using the Fourier series written in 
the form 
 𝒙(𝒕) = 
𝒂𝟎
𝟐
∑ [𝒂𝒏𝒄𝒐𝒔(𝒏𝝎𝒕)+𝒃𝒏𝒔𝒆𝒏(𝒏𝝎𝒕)]∞
𝒏=𝟏 , (4) 
which is presented in the classic calculation book [2] and the 
most advanced mathematical physics book [3]. This 
alternative method has already been implemented in the cases 
of the harmonic oscillator where the Fourier series only 
involves real functions [12]. Here we intend to address a 
complete study of all of the motions, including those in which 
is needed the complex analysis in the solutions obtained by the 
Fourier series. 
In next section, the general solution for the simple 
harmonic oscillator motion is presented using the series (4), 
highlighting the importance of the initial conditions. In the 
following, it is described the result of the calculations for the 
cases of motion with damping, including the subcritical, 
critical and supercritical regimes. At last, it is also studied the 
forced oscillator motion with damping, in which is considered 
 
3 
 
an external cosine force to materialize the relevant resonance 
phenomenon. 
2. Simple Harmonic Oscillator and Fourier Series 
 Replacing the solution defined by the Fourier series 
(4) and its second derivative in the simple harmonic oscillator 
equation (1), after some algebraic calculations, one obtains 
𝒂𝟎 = 𝟎 and 𝝎 =
𝝎𝟎
𝒏
 . When these values are replaced again in 
the Fourier series (4), the proposed general solution assumes 
the analytical form 
 𝒙𝑺(𝒕) = (∑ 𝒂𝒏
∞
𝒏=𝟏 )𝒄𝒐𝒔(𝝎𝟎𝒕)(∑ 𝒃𝒏
∞
𝒏=𝟏 )𝒔𝒆𝒏(𝝎𝟎𝒕), (5) 
in which 𝑥𝑠(𝑡) is the general solution of the simple harmonic 
oscillator. 
Note in Eq. (5) that the coefficients of the 
trigonometric functions are series in which, a priori, nothing 
can be said about their convergence. Using the most general 
initial conditions at position and velocity (𝒙(𝟎), �̇�(𝟎)) =
(𝒙𝟎, 𝒗𝟎), the coefficients take on the values 
 
{
∑ 𝒂𝒏 = 𝒙𝟎
∞
𝒏=𝟏
∑ 𝒃𝒏 =
𝒗𝟎
𝝎𝟎
∞
𝒏=𝟏
 . 
(4) 
The equalities in the system (6) demonstrate the 
convergences of the numerical series using only physical 
considerations by means of the initial conditions, whose finite 
values of convergence correspond to the initial position and 
the initial velocity of the system. 
Substituting the equations of the system (6) into Eq. 
(5), we obtain 
 𝒙𝑺(𝒕) = 𝒙𝟎𝒄𝒐𝒔(𝝎𝟎𝒕)
𝒗𝟎
𝝎𝟎
𝒔𝒆𝒏(𝝎𝟎𝒕), (5) 
which is the general solution to the differential equation of the 
simple harmonic oscillator. 
To end this section, it was previously stated that this 
movement is idealized, that is, it is not easy to perform such 
an experiment in a didactic laboratory. This can be clarified in 
a more experimental view when the movement of the simple 
harmonic oscillator is simulated, reducing as much as possible 
the influence of friction and damping in the experiments. 
Furthermore, it is necessary that the experimental data are 
collected in a short time interval, when the maximum and 
minimum amplitude values remain almost constant. 
3. Damped Harmonic Oscillator and Fourier Series 
 The damped harmonic oscillator develops a 
movement whose amplitude decreases in time. The general 
solution must satisfy the limit 
𝐥𝐢𝐦
𝒕→∞
𝒙(𝒕) = 𝟎 , (6) 
which is the physical condition to be obeyed by the amplitude 
of oscillation. Differentiating twice the series in Eq. (4) and 
substituting the results of these calculations into the 
differential equation (2), we effectively obtain 𝑎0 = 0 and a 
system of equations that is transformed to matrix notation as 
 [
𝝎𝟎
𝟐 − 𝒏𝟐𝝎𝟐 +𝟐𝜸𝒏𝝎
−𝟐𝜸𝒏𝝎 𝝎𝟎
𝟐 − 𝒏𝟐𝝎𝟐
] [
𝒂𝒏
𝒃𝒏
] = [
𝟎
𝟎
] . (9) 
 The matrix equation (9) indicates that if one of the 
coefficients (𝑎𝑛 , 𝑏𝑛) is null, the other will also be null. The 
condition for the existence of a non-trivial solution is to 
guarantee that the determinant of the first matrix is null, that 
is, 
 (𝝎𝟎
𝟐 − 𝒏𝟐𝝎𝟐)
𝟐
+ 𝟒𝜸𝟐𝒏𝟐𝝎𝟐 = 𝟎 . (10) 
To isolate the term 𝑛𝜔 in Eq. (10), it is compulsory 
to use the definition of imaginary number (𝑖 = √−1), so that 
one will have to work in the domain of the complex numbers. 
Carrying out the necessary calculations in Eq. (10), it is 
modified to the result 
 (𝒏𝝎 ± 𝜸𝒊)𝟐 = 𝝎𝟎
𝟐 − 𝜸𝟐 , (11) 
which will be used to classify the types of possible damping: 
critical, subcritical and supercritical. 
 
4 
 
Another way to verify the need to use imaginary 
number is to perform the matrix product in the matrix equation 
(9), obtaining a homogeneous linear system of two equations. 
Then, after some algebraic manipulations, we obtain the 
equality 
 (
𝒃𝒏
𝒂𝒏
)
𝟐
+ 𝟏 = 𝟎 , (12) 
which is satisfied for the values 𝑏𝑛 = ±𝑖𝑎𝑛. If 𝑎𝑛 = 𝑧𝑛 with 
𝑧𝑛 ∈ 𝐶, then 𝑏𝑛 = ±𝑖𝑧𝑛. If 𝑎𝑛 = 𝑧�̅�, where 𝑧�̅� is the complex 
conjugate of 𝑧𝑛, then 𝑏𝑛 = ±𝑖𝑧�̅�. When each of these results 
is inserted into the Fourier series given in Eq. (4), we obtain 
the system of equations 
 {
𝒙(𝒕) = ∑ 𝒛𝒏𝒆𝒊𝒏𝝎𝒕∞
𝒏=𝟏 
𝒙(𝒕) = ∑ �̅�𝒏𝒆−𝒊𝒏𝝎𝒕∞
𝒏=𝟏
 , (13) 
which were previously selected among all possible 
combinations between the values of 𝑎𝑛 and 𝑏𝑛. 
In next subsections, the three damping cases will be 
analysed using both equations (11) and (13), satisfying the 
temporal damping limit given in Eq. (8). 
3.1 Damping Critical Regime 
In this regime, both behaviours of oscillation and 
damping have significant influence in the movement of the 
oscillator, which mathematically is represented by the equality 
𝜔0 = 𝛾. This condition allows us to obtain from Eq. (11) the 
simpler results 𝑛𝜔 = +𝛾𝑖 or 𝑛𝜔 = −𝛾𝑖, which are substituted 
into the two equations of the system (13). One of them is 
selected by means of the limit test (8) and the result is the 
solution 
𝒙𝒄(𝒕) = (∑ 𝒛𝒏
∞
𝒏=𝟏 ) 𝒆−𝜸𝒕 . (7) 
The vector space of solutions to the damped 
harmonic oscillator differential equation has two dimensions 
and requires two linearly independent functions to fully 
describe it. However, using the Fourier series formalism, it 
was only possible to present a single function (14). It is devoid 
of the information that must be present in the other function 
that completes the basis vectors. This other function can be 
conjectured according to the general solution 
 𝒙𝒄(𝒕) = (∑ 𝒛𝒏
∞
𝒏=𝟏 )𝒆−𝜸𝒕 − (∑ 𝒘𝒏
∞
𝒏=𝟏 )
𝝏
𝝏𝜸
𝒆−𝜸𝒕. (15) 
 The functions (𝑒−𝛾𝑡,
𝑑
𝑑𝛾
𝑒−𝛾𝑡) are linearly 
independent and form a complete basis of the solution space 
of the differential equation (2). Writing the terms of the series 
presented in Eq. (15) in their respective exponential forms 
𝑧𝑛 = |𝑧𝑛|𝑒𝛿𝑖 e 𝑤𝑛 = |𝑤𝑛|𝑒𝜌𝑖 and substituting into Eq. (15), 
we obtain 
𝒙𝒄(𝒕) = 𝒆𝜹𝒊(∑ |𝒛𝒏|∞
𝒏=𝟏 )𝒆−𝜸𝒕− 𝒆𝝆𝒊(∑ |𝒘𝒏|∞
𝒏=𝟏 )
𝝏
𝝏𝜸
𝒆−𝜸𝒕 , (16) 
in which 𝒙𝒄(𝒕) is the general solution for the critical regime. 
Assigning the values (𝛿, 𝜌) = (0,0) to guarantee 
the existence of a general solution defined in the set of real 
numbers, Eq. (16) undergoes the modification shown in 
equation 
 𝒙𝒄(𝒕) = (∑ |𝒛𝒏|∞
𝒏=𝟏 )𝒆−𝜸𝒕 − (∑ |𝒘𝒏|∞
𝒏=𝟏 )
𝝏
𝝏𝜸
𝒆−𝜸𝒕. (17) 
Using the initial conditions (𝑥(0), �̇�(0)) = (𝑥0, 𝑣0), 
the series in Eq. (17) converges respectively to the values 
 {
∑ |𝒛𝒏| = 𝒙𝟎
∞
𝒏=𝟏 
∑ |𝒘𝒏| = 𝒗𝟎 + 𝜸𝒙𝟎
∞
𝒏=𝟏
 . (18) 
Substituting the respective real values of 
convergence of the series given in the Eq. (18) into Eq. (17), 
we obtain, in the critical damping regime, the definitive form 
for the general solution 
 𝒙𝒄(𝒕) = 𝒆−𝜸𝒕[𝒙𝟎 + (𝒗𝟎 + 𝜸𝒙𝟎)𝒕] , (19) 
which satisfies the limit (8) and the differential equation (2), 
in the critical condition 𝜔0 = 𝛾. 
The final result displayed in Eq. (19) is the well-
known general solution. The result here obtained is the same 
one presented in Ref. [13], obtained by means of another 
mathematical method. 
 
5 
 
3.2 Subcritical Damping Regime 
To study the case of damping in the subcritical 
regime, in which the mathematical condition is 𝝎𝟎 > 𝜸, a 
new parameter 𝛀𝟏 is defined in the set of real numbers, that 
is, 𝛀𝟏
𝟐 = 𝝎𝟎
𝟐 − 𝜸𝟐. When it is substituted into Eq. (11), the 
possible results for 𝒏𝝎 are shown in Table 1. 
Table 1: Possible values for 𝑛𝜔 in the subcritical damping regime. 
𝑛𝜔 
Ω1 + 𝛾𝑖 Ω1 − 𝛾𝑖 
−Ω1 + 𝛾𝑖 −Ω1 − 𝛾𝑖 
 
Among the four values of Table 1, only the two 
values 𝑛𝜔 = ±Ω1 + 𝛾𝑖 are ideal to replace into the system of 
equations (13), precisely to satisfy the limit condition (8). The 
result is the sum of the two selected functions, which is written 
in the form 
 𝒙𝒔𝒃(𝒕) = 𝒆−𝜸𝒕[𝒆(𝛀𝟏𝒕+𝜹)𝒊 + 𝒆−(𝛀𝟏𝒕+𝜹)𝒊](∑ |𝒛𝒏|∞
𝒏=𝟏 ), (20) 
in which 𝒙𝒔𝒃(𝒕) is the general solution for the subcritical 
regime. The coefficients 𝑧𝑛 = |𝑧𝑛|𝑒𝛿𝑖 and 𝑧�̅� = |𝑧𝑛|𝑒−𝛿𝑖 were 
intentionally chosen, forming a pair constituted by a complex 
number and its conjugate which were used in Eq. (20). 
Identifying the sum of complex exponentials with the cosine 
function and taking into account to the initial conditions 
(𝑥(0), �̇�(0)) = (𝑥0, 𝑣0), Eq. (20) assumes its definitive form 
 𝒙𝒔𝒃(𝒕) = 𝒆−𝜸𝒕 [𝒙𝟎𝒄𝒐𝒔(𝛀𝟏𝒕) + (
𝒗𝟎+𝜸𝒙𝟎
𝛀𝟏
) 𝒔𝒆𝒏(𝛀𝟏𝒕)] , (21) 
which satisfies the limit (8). It can also be used to obtain the 
general solution of the simple harmonic oscillator, by means 
of the condition lim
𝛾→0
𝑥𝑠𝑏(𝑡) = 𝑥𝑆(𝑡). It is also possible to 
obtain Eq. (19) using the condition lim
Ω1→0
𝑥𝑠𝑏(𝑡) = 𝑥𝑐(𝑡), 
according result published in Ref. [14]. 
3.3 Supercritical Damping Regime 
This damping regime is the one that happens when it 
is satisfied the condition 𝝎𝟎 < 𝜸. A new parameter 𝛀𝟐 is 
defined in the set of real numbers as 𝛀𝟐
𝟐 = 𝜸𝟐 − 𝝎𝟎
𝟐. When it 
is substituted into Eq. (11), four pure imaginary numbers are 
obtained and presented in Table 2. 
Table 2: Possible values for 𝑛𝜔 in the supercritical damping 
regime. 
𝑛𝜔 
(Ω2 + 𝛾)𝑖 −(Ω2 + 𝛾)𝑖 
(Ω2 − 𝛾)𝑖 −(Ω2 − 𝛾)𝑖 
The two values that are in the first column of Table 2 
are those that satisfy the limit (8). Each of them is inserted into 
the first equation of the system (13). The temporary result is 
the equation 
 𝒙𝒔𝒑(𝒕) = 𝒆−𝜸𝒕[(∑ |𝒛𝒏|∞
𝒏=𝟏 )𝒆𝛀𝟐𝒕 + (∑ |𝜺𝒏|∞
𝒏=𝟏 )𝒆−𝛀𝟐𝒕] , (22) 
in which 𝒙𝒔𝒑(𝒕) is the general solution for the supercritical 
regime. 
 For the function to take on values in the set of real 
numbers, the coefficients that accompany the real 
exponentials in Eq. (22) have to take on real values. 
Exponentials of real exponents can be related to trigonometric 
functions with complex arguments, as presented in the system 
 { 
𝒆𝛀𝟐𝒕 = 𝐜𝐨𝐬𝐡(𝛀𝟐𝒕) + 𝐬𝐢𝐧𝐡(𝛀𝟐𝒕) 
𝒆−𝛀𝟐𝒕 = 𝐜𝐨𝐬𝐡(𝛀𝟐𝒕) − 𝐬𝐢𝐧𝐡(𝛀𝟐𝒕) 
. (23) 
Substituting the equations of the system (23) and the 
initial conditions (𝑥(0), �̇�(0)) = (𝑥0, 𝑣0) into Eq. (22), the 
result is the general solution 
 𝒙𝒔𝒑(𝒕) = 𝒆−𝜸𝒕 [𝒙𝟎 𝐜𝐨𝐬𝐡(𝛀𝟐𝒕) (
𝒗𝟎+𝜸𝒙𝟎
𝛀𝟐
) 𝐬𝐢𝐧𝐡(𝛀𝟐𝒕)]. (24) 
A curious observation is the fact that Eq. (24) is 
obtained by means of the condition lim
Ω1→iΩ2
𝑥𝑠𝑏(𝑡) = 𝑥𝑠𝑝(𝑡), 
according to Ref. [14]. This implies that Eq. (21) has all the 
information necessary to recover as the general solution of the 
simple harmonic oscillator as the solutions of the critical and 
supercritical regimes from the damping oscillator. 
4. Forced Oscillator with Damping 
The movement of the damped oscillator under the 
action of a time-dependent external force can be significantly 
 
6 
 
simplified if the study is restricted to consider an external 
temporal force per unit mass of the harmonic type 
𝑓(𝑡) = 𝑓0cos(𝜔𝑡), where 𝜔 is an external frequency and is 
considered to have a well-defined value. The differential 
equation (3) in this simple model of external force is given by 
 
𝒅𝟐𝒙
𝒅𝒕𝟐 + 𝟐𝜸
𝒅𝒙
𝒅𝒕
+ 𝝎𝟎
𝟐𝒙 = 𝒇𝟎𝐜𝐨𝐬(𝝎𝒕) . (25) 
One of the solutions of this equation is known as 
associated homogeneous differential equation, already 
previously obtained to describe the cases of damping. The 
other is the so-called particular solution, which is of greater 
interest because it contains the information that describes the 
resonance phenomenon. It is possible to know a little more 
about a system by varying the value of 𝜔 and verifying when 
and how the phenomenon of resonance manifests itself. 
The particular solution is determined using Fourier's 
series (4) and its derivatives, which are substituted into Eq. 
(25) and, after algebraic manipulations, some important 
results are found. The first one to mention is that 𝑎0 = 0; the 
second is that 𝑎𝑛 = 𝑏𝑛 = 0 for 𝑛 > 1 and finally, the 
inhomogeneous linear system 
 { 
(𝝎𝟎
𝟐 − 𝝎𝟐)𝒂𝟏 + 𝟐𝜸𝝎𝒃𝟏 = 𝒇𝟎 
−𝟐𝜸𝝎𝒂𝟏 + (𝝎𝟎
𝟐 − 𝝎𝟐)𝒃𝟏 = 𝟎 
 . (26) 
 From the linear system (26), the coefficients a₁ and 
b₁ are obtained and the solution (4) is transformed into the 
particular solution 
𝒙𝒑(𝒕) = [
(𝝎𝟎
𝟐−𝝎𝟐)𝒇𝟎
(𝝎𝟎
𝟐−𝝎𝟐)
𝟐
+𝟒𝜸𝟐𝝎𝟐
] 𝒄𝒐𝒔(𝝎𝒕) + [
𝟐𝜸𝝎𝒇𝟎
(𝝎𝟎
𝟐−𝝎𝟐)
𝟐
+𝟒𝜸𝟐𝝎𝟐
] 𝒔𝒆𝒏(𝝎𝒕). (27) 
 The resonance region can be determined by 
calculating the extreme values of the particular solution (27) 
in the variable 𝜔. But this task can be less difficult if there is 
an additional modification of (27) to the convenient analytic 
expression 
 𝒙𝒑(𝒕) = 
𝒇𝟎
√(𝝎𝟎
𝟐−𝝎𝟐)
𝟐
+𝟒𝜸𝟐𝝎𝟐
𝐜𝐨𝐬(𝝎𝒕 + 𝝋), (28) 
in which was defined in the calculations the value 
 𝝋 = ± 𝐭𝐚𝐧−𝟏 (
𝟐𝜸𝝎
|𝝎𝟎
𝟐−𝝎𝟐|
) . (29) 
The analytical expression given in Eq. (28) is surely 
more adequate to explore the phenomenon of resonance [13] 
by analysing the amplitude term in the particular solution of 
the forced motion with damping. 
5. Final Remarks 
 By choosing the Fourier series as Ansatz for the 
differential equations of the simple harmonic oscillator, of 
damped harmonic oscillator and of forced harmonic oscillator, 
the general solutions of the main oscillation regimes were 
obtained, covering all the cases studied in undergraduate 
courses in Exact Sciences. The convergence of each numerical 
series obtained in Fourier method is successfully presented by 
means of the initial conditions. The method used is in practice 
a way to obtain the respectivegeneral solutions for each type 
of movement of oscillation. During the calculations, at certain 
times, some mathematical knowledge is required and 
performed, such as trigonometric and hyperbolic functions, 
complex numbers and series, just to cite a few. At other times, 
a certain effort had to be done to obtain the solutions. Besides, 
it is needed the development of a physical intuition to 
understand the phenomena associated to the movements. 
These procedures will certainly be very useful for students 
who want to reproduce the results, especially in order to learn 
so many interesting concepts used in this Fourier series 
formalism. 
References 
[1] FEYNMAN R. P., LETGHTON, R. B. and SANDS, M. 1963 The 
Feynman Lectures on Physics. Reading: Addison-Wesley Publishing 
Company, v. 1 
 
[2] NUSSENZVEIG, H. M. 2002 Curso de Física Básica. 4. ed. São 
Paulo: Editora Edgard Blücher Ltda, v. 2 
 [3] PISKUNOV, N. 1969 Differential and Integral Calculus. 
Moscow: Mir Publishers 
 
[4] ARFKEN, G. B. and WEBER, H. J. 1995 Mathematical Methods 
for Physicists. 4. ed. San Diego: Academic Press 
 
[5] MUNGAN, C. E. 2009 Chemical potential of one-dimensional 
simple harmonic oscillators. Eur. J. Phys., 30(5) 1131 
[6] CHANG, L. N., MINIC D., OKAMURA, N. and TAKEUCHI, 
T. 2002 Exact solution of the harmonic oscillator in arbitrary 
dimensions with minimal length uncertainty relations. Phy. Rev. D 
65(12) 125027 
 
[7] PATIL, S. H. 2006 Harmonic oscillator with a δ-function 
potential, Eur. J. Phys., 27(4) 899 
 
 
7 
 
[8] POTTORF, S., PUDZER, A., CHOU, M. Y. and HASBUN, J. E. 
1999 The simple harmonic oscillator ground state using a variational 
Monte Carlo method, Eur. J. Phys., 20(3) 205 
[9] HIRA, K. 2013 Derivation of the harmonic oscillator propagator 
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[10] FAN, X-Y. et al., 2022 Studying Heisenberg-like Uncertainty 
Relation with Weak Values in One-dimensional Harmonic 
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[11] GHOSE, I. and SEM P. 2021 The variational method applied to 
the harmonic oscillator in the presence of a delta function potential. 
Eur. J. Phys. 42 045406 
[12] CASTRO, A. S. 2014 Harmonic oscillator: An analysis via 
Fourier series. Rev. Bras. Ensino Fís. 36(2) 2701 
 [13] SYMON, K. R. 1996 Mecânica. Rio de Janeiro: Editora 
Campus Ltda 
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damped harmonic ascillator revisited. Rev. Bras. Ensino Fis. 27(3) 
327. 
 
 
 
 
 
 
 
 
 
 
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