Definition and main properties

Harmonic motion is a particularly simple type of periodic motion: the law of motion of a point moving in harmonic motion, that is, the function that describes the position of the point as a function of time, is a simple sinusoid of constant amplitude. (For a quick introduction to trigonometric functions see the page the sine and cosine functions) Consequently, harmonic motion is completely described by just three parameters: the period of the oscillation (or its frequency), the amplitude of the oscillation, and its phase (which indicates the initial position of the oscillation). If the velocity of a point follows the harmonic law, then the acceleration (derivative) and the position (integral) of that point also follow a harmonic law. The sum of several harmonic motions of equal frequency is still a harmonic motion with the same frequency (this fact is extremely useful when used together with the superposition principle)

For a further illustration of these concepts, see also the section interference of harmonic waves.

In the animation above, the red and blue points move in exactly harmonic motion. They are the projections of the black point, which moves in uniform circular motion.

Why study harmonic motion

You will surely have thought that, apart from the mathematical curiosity of being the projection of an extremely simple motion such as that of a point "going round" a circle at constant speed, harmonic motion is not suited to describing the myriad of oscillations that occur in nature (think of the motion of the pen of a seismograph, the oscillation of the structures of a building during an earthquake, the vibration of the string of a musical instrument, etc.). Harmonic motion really seems too simple: why waste time studying it instead of solving more realistic problems?

1. First of all, precisely because it is simple, and so we can understand all its aspects, and derive properties and theorems from it exactly.
2. But above all because harmonic motion has a very deep universality. Thanks to its mathematical properties, it appears (almost in spite of ourselves, we might say) in all branches of physics and, surprisingly, even when the motion or oscillation being studied is not harmonic at all. In particular:
   • Fourier's theorem states that (almost) any oscillation (that is, any periodic function) can be expressed as a sum of harmonic functions
   • (Almost) every oscillating system is well approximated by a harmonic oscillator if the amplitude of the oscillations is small.

We leave the analysis of Fourier's theorem to another section, while we illustrate the last point below by means of an example.

Harmonic approximations

Suppose we have a system in stable equilibrium. It could be

1. a ball at the bottom of a hole;
2. a pendulum hanging from a thread in its vertical position;
3. the string of a musical instrument;
4. any structure that "stands up" (a bridge, a house, etc.);
5. all the waters of planet Earth;
6. a molecule;
7. a crystal.

• If this system is moved away from its equilibrium position in any way, restoring forces will act on it, tending to bring it back to equilibrium.
• In general these forces will be complicated functions of the displacements of the elements of the system.
• These forces must vanish in the equilibrium configuration (this is precisely the definition of "equilibrium");

This is where the harmonic approximation comes in:

• if we imagine that the displacement from the equilibrium condition is small, we can assume (what mathematicians call a first-order expansion) that these restoring forces are proportional to the displacement.
• Under these hypotheses it can be shown that the overall motion of the system can be reduced to a set of as many simple harmonic motions as there are oscillating elements making up the system (see normal modes)

In what follows we analyse harmonic motion more rigorously and, in particular, we will look at the case of the mass-spring system, which can be used as a prototype for all harmonic oscillations.

Kinematics of harmonic motion

Uniform circular motion

There is a geometric situation in which harmonic oscillation appears in an extremely natural way. To describe it, we need to briefly introduce the kinematics of uniform circular motion. Consider a point P that travels around a circle at constant speed. To describe this motion geometrically, it is useful to introduce some quantities with an immediate physical interpretation:

• the period T, defined as the time needed to complete one full turn. It is measured, of course, in seconds;
• the frequency ${\displaystyle \nu }$ or ${\displaystyle f}$, defined as the number of turns completed per unit time. It is measured in hertz, a synonym for turns per second. It is easy to see that period and frequency are inversely proportional, that is,

${\displaystyle T=\frac{1}{\nu }\text{ and }\nu =\frac{1}{T}}$

(1)

• the tangential (or linear) velocity v, defined, as for any "uniform" motion, as the ratio between the distance travelled and the time taken to travel it. If we take one full turn as the distance travelled, we obtain

${\displaystyle v=\frac{2\pi R}{T}=2\pi R\cdot \nu }$

(2)

• the angular velocity (or angular frequency) ${\displaystyle \omega }$, defined as the ratio between the central angle swept (measured in radians) and the time taken to sweep it. If we choose the full angle (${\displaystyle 2\pi }$ radians) as the angle swept, we obtain

${\displaystyle \omega =\frac{2\pi }{T}=2\pi \cdot \nu }$

(3)

Note that, thanks to the choice of measuring angles in radians (this is the physical reason that makes this measure "natural"!), comparing (2) with (3) we obtain:

${\displaystyle v=\omega \cdot R}$

(4)

• Finally, it can be shown that uniform circular motion is an accelerated motion; at first sight this statement seems counterintuitive: in everyday language acceleration is associated with a change in speed (which does not seem to be present in uniform circular motion). However, one must observe that even if the velocity does not change in magnitude, it continuously changes direction (physicists say that the instantaneous velocity is a vector tangent to the trajectory). It can be shown that the acceleration in uniform circular motion:
  • is directed, instant by instant, towards the centre of the circle (for this reason it is called centripetal acceleration)
  • is constant (in magnitude) and its value is given by the formula

${\displaystyle a_c=\frac{v^2}{R}={\omega }^2\cdot R}$

(5)

Harmonic motion as the "projection" of uniform circular motion

We are now ready to present a remarkable example of harmonic motion: we project the point P onto a straight line and call Q its projection. It is immediately evident that the point Q will oscillate between the position ${\displaystyle x=-R}$ and the position ${\displaystyle x=R}$. We will show that Q moves in simple harmonic motion, that is, with a law of motion represented by a sinusoidal function. To show this, we will assume:

• that at the instant t=0 the point P is in a position that corresponds, for the point Q, to the abscissa x=0
• that the point P moves with constant angular velocity ${\displaystyle \omega }$ and has therefore swept, as an immediate consequence of (3), after the time t (starting from the initial instant t=0) an angle α=ωt
• that the kinematic quantities (position, velocity and acceleration) describing the motion of Q are the projections onto the line of the corresponding quantities relating to P (think, for example, of a point of light that, at night, comes towards us in the direction perpendicular to the "visual" line of our eyes; this point will appear to us to be stationary. If the point of light approaches obliquely, we will perceive only the velocity parallel to the visual line and not the whole velocity of the point of light).

With these assumptions it is easy to show that

• the position of Q at the instant t is given by

${\displaystyle x(t)=R\cdot \sin (\omega t)}$

(6)

having applied the definition of sine to the right triangle OPH;

• the velocity of Q at the instant t is given by

${\displaystyle v(t)=v\cdot \cos (\omega t)=\omega R\cdot \cos (\omega t)}$

(7)

having applied the definition of cosine to the right triangle PST and the relation (4) between the velocity v and the angular velocity ${\displaystyle \omega }$;

• the acceleration of Q at the instant t is given by

${\displaystyle a(t)=-{\omega }^{2}R\cdot \sin (\omega t)}$

(8)

having used again the definition of sine and formula (5) for the centripetal acceleration.

The minus sign is due to the fact that, as is easily seen from the figure, the acceleration always has the opposite sign to the x(t) of the point. Comparing (6) and (8), we obtain the remarkable result

${\displaystyle a(t)=-{\omega }^2\cdot x(t)}$

(9)

which can be summarized by saying that in harmonic motion the acceleration a(t) is always proportional to the position x(t), with a negative constant of proportionality. This provides a further characterization of harmonic motion (besides that of having a sinusoidal law of motion), which will be very useful for identifying the "harmonicity" of the motion of other oscillating systems.

One last remark: it is possible to recover the results of equations (6) to (9) without assuming that the point P is, at the instant t=0, in a position that corresponds, for the point Q, to the abscissa x=0. This can be done simply by introducing a phase angle (or simply a phase) ${\displaystyle \varphi }$ that indicates the initial position of the point P. This angle must be added to ${\displaystyle \omega t}$ to give the position of the point P after a time t. All the formulas can be rewritten taking this into account. For example, (6) will become

${\displaystyle x(t)=R\cdot \sin (\omega t+\varphi )}$

(6')

Dynamics of harmonic motion

The previous example can hardly have convinced the reader of the central role of harmonic motion in the oscillations that really occur in nature: it looks like little more than a geometric curiosity. However, the further characterization of harmonic motion that it allowed us to derive will enable us to show, in the next section, that a simple mass-spring system moves in harmonic motion.

Mass-spring systems

Consider a mass m connected to a spring as in the figure alongside, where we have introduced a reference frame in which the mass m occupies the position x=0 when the spring is at rest. If the spring is stretched (or compressed) and the mass is at a new position with abscissa x(t), it is subject to a restoring force, called the elastic (or harmonic) force, proportional to the stretching (or compression) of the spring, with a negative constant of proportionality (note that if x(t) is negative, that is, for the compressed spring, the restoring force is positive, and vice versa)

${\displaystyle F_{\mathrm{e}\mathrm{l}}=-k\cdot x(t)}$

(10)

The constant k gives a measure of the "stiffness" of the spring: more precisely, it is the restoring force per unit length.

Applying Newton's second law to the mass-spring system, we obtain

${\displaystyle -k\cdot x(t)=m\cdot a(t)}$

(11)

from which

${\displaystyle a(t)=-\frac{k}{m}\cdot x(t)}$

(12)

The acceleration is therefore proportional to the displacement, with a negative constant of proportionality. But this is precisely the characterization of harmonic motion introduced with equation (9), provided we take

${\displaystyle \omega =\sqrt{\frac{k}{m}}}$

(13)

The mass m will then move according to the law of motion

${\displaystyle x(t)=A\cdot \sin (\omega t+\varphi )}$

(6")

with a time-varying velocity given by

${\displaystyle v(t)=\omega A\cdot \cos (\omega t+\varphi )}$

(7")

where, to free ourselves completely from the origin of harmonic motion as the projection of a circular motion, we have called the amplitude A. Obviously ω too loses its kinematic meaning of angular velocity; it becomes

• in a purely algebraic sense, the quantity through which the period of the oscillation is calculated, by means of the relation

${\displaystyle \omega =\frac{2\pi }{T}}$

(3)

• in a more physical sense, the quantity whose square gives the restoring force per unit length and unit mass, since

${\displaystyle {\omega }^2=\frac{k}{m}=\frac{F}{x\cdot m}}$

Note that, in the case of the mass-spring system, the period of the oscillation becomes

${\displaystyle T=\frac{2\pi }{\omega }=2\pi \sqrt{\frac{m}{k}}}$

(14)

Recalling (1), it is then easy to obtain the natural frequency or resonance frequency f₀ of the oscillator

${\displaystyle f_0=\frac{1}{T}=\frac{1}{2\pi }\sqrt{\frac{k}{m}}}$

(15)

Note that the period and the frequency are independent of the amplitude of the oscillations.

As the values of the constants A and φ vary, (6") gives infinitely many laws of motion: the oscillating system will select only one of them, so as to satisfy given initial conditions in which it is at the instant t=0. Fixing the initial conditions of a dynamical system means specifying:

• the initial position (in the case in question, the value of ${\displaystyle x(0)}$
• the initial velocity (in the case in question, the value of ${\displaystyle v(0)}$

For example

• in the obvious case in which the system is, at the instant ${\displaystyle t=0}$, at rest in equilibrium at the point with coordinate ${\displaystyle x=0}$, imposing on equations (6") and (7") the initial conditions ${\displaystyle x(0)=0\text{ and }v(0)=0}$ means solving the system

${\displaystyle \{\begin{array}{ll}A\cdot \sin \varphi =0& \\ \omega A\cos \varphi =0\end{array}}$

which immediately gives the solution

${\displaystyle A=0}$

${\displaystyle \varphi }$ arbitrary (with A=0 the phase has no meaning)

which corresponds to the absence of motion (the system is in stable equilibrium).

• in the more interesting case in which the system is, at the instant ${\displaystyle t=0}$, at rest at the point with coordinate ${\displaystyle x=A_0}$, imposing on equations (6") and (7") the initial conditions ${\displaystyle x(0)=A_0\text{ and }v(0)=0}$ means solving the system:

${\displaystyle \{\begin{array}{ll}A\cdot \sin \varphi =A_0& \\ \omega A\cos \varphi =0\end{array}}$

which immediately gives the solution

${\displaystyle A=A_0}$

${\displaystyle \varphi =\frac{\pi }{2}}$.

This situation corresponds to an oscillation of the system between the points with coordinates x=± A₀, where the motion reverses: the velocity vanishes and changes sign.

The pendulum

A simple pendulum is a mass m hanging from a thread of length l. When the pendulum is displaced by a small angle from the vertical, the component of its weight along the path pulls it back towards the equilibrium position with a force that, for small displacements x, is approximately proportional to the displacement:

${\displaystyle F\approx -\frac{mg}{l}\cdot x}$.

This is an example of the harmonic approximation: comparing with (10) we see that the pendulum behaves like a mass–spring system with ${\displaystyle k=mg/l}$, and therefore

${\displaystyle T=2\pi \sqrt{\frac{l}{g}}}$

(16)

The period depends neither on the mass nor, for small oscillations, on the amplitude: this is the isochronism observed by Galileo, which made the pendulum suitable for regulating clocks. For large oscillations the harmonic approximation no longer holds and the period increases with the amplitude (see the pendulum on the page on non-linear effects).

Mechanical energy in harmonic motion

The mass-spring system has two types of energy:

• kinetic energy, which is the energy that the mass has because of its velocity, and is equal to

${\displaystyle E_{\mathrm{c}\mathrm{i}\mathrm{n}}(t)=\frac{1}{2}m{v}^2(t)}$

• potential energy, which is the energy due to the elastic deformation of the spring, and is equal to

${\displaystyle E_{\mathrm{p}\mathrm{o}\mathrm{t}}(t)=\frac{1}{2}k{x}^2(t)}$.

During the oscillatory motion

• these two energies are alternately transformed into each other and, as we will show, their sum remains constant;
• the potential energy is maximum when the spring is in the position of maximum deformation, that is, for ${\displaystyle x(t)=\pm A}$;
• the kinetic energy is maximum when the mass has the maximum velocity, that is, when it passes through the point ${\displaystyle x(t)=0}$.

To calculate the total energy of the oscillating mechanical system (that is, its mechanical energy) at a given instant t, we must add its kinetic and potential energy:

${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{c}}(t)=E_{\mathrm{c}\mathrm{i}\mathrm{n}}(t)+E_{\mathrm{p}\mathrm{o}\mathrm{t}}(t)}$

(23)

we obtain

${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{c}}(t)=\frac{1}{2}m{v}^2(t)+\frac{1}{2}k{x}^2(t)}$

(24)

Substituting into (24) the equations (6") and (7") already derived for harmonic motion

${\displaystyle x(t)=A\cdot \sin (\omega t+\varphi )}$

${\displaystyle v(t)=\omega A\cdot \cos (\omega t+\varphi )}$

we obtain

${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{c}}(t)=\frac{1}{2}m{\omega }^2{A}^2\cdot {\cos }^2(\omega t+\varphi )+\frac{1}{2}k{A}^2\cdot {\sin }^2(\omega t+\varphi )}$

(25)

from which, since by (13)

${\displaystyle m{\omega }^2=k}$,

collecting ${\displaystyle \frac{1}{2}k{A}^2}$ we obtain

${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{c}}(t)=\frac{1}{2}k{A}^2\cdot ({\cos }^2(\omega t+\varphi )+{\sin }^2(\omega t+\varphi ))}$

(26)

which, by the well-known fundamental relation of trigonometry, finally becomes

${\displaystyle E_{\mathrm{m}\mathrm{e}\mathrm{c}\mathrm{c}}=\frac{1}{2}k{A}^2}$

(27)

Note that, while the kinetic and potential energies separately change in time, the mechanical energy no longer depends on time. It is therefore a constant of the motion.

In a real system, however, friction slowly turns the mechanical energy into heat: the amplitude decreases over time and we speak of damped oscillations. To keep the oscillation going, energy must be supplied from outside (see resonance).

Check your understanding

In-depth study and links

References to harmonic motion are found throughout the text, and in particular in the following sections:

• coupled oscillators
• harmonic waves
• Fourier's theorem
• resonance
• normal modes
• standing waves
• musical instruments
• perception of sound