General features of resonance

Resonance is the phenomenon by which an oscillating system can absorb energy from an external source particularly efficiently only at one (or more) very precise frequencies.

Let us examine its features through an example:

• We know from experience that a small push on a swing makes it oscillate at a very precise frequency, called the natural frequency, which depends on the length of the swing but not on the details of the force applied during the push (see natural frequencies).
• If, instead of pushing only once, we repeat the push periodically in time, we can study how the swing responds as the frequency of our pushes changes.
• We will find that, if the force applied from outside has a frequency close to the natural frequency of the swing, the swing tends to oscillate more and more widely.
• For frequencies of the external force much higher or much lower than the natural frequency, instead, our pushing is almost always "out of time": sometimes it goes along with the motion and sometimes it brakes it, and on average it transfers very little energy to the swing.
• We conclude, therefore, that the natural frequency of oscillation is also the frequency at which an external force transfers energy to the oscillating system most efficiently.

Every physical system characterised by natural frequencies of oscillation can resonate with an external source.

Single oscillator

Even though studying the resonance of real systems can be very complicated, it is simple and useful to visualise some general features of the phenomenon in the case of a single oscillator. The details of the calculation are left to the specific sections (see the links at the bottom of the page), while here we only show and comment on the following two graphs.

• We study a single ideal (harmonic) oscillator of natural frequency ${\displaystyle {\omega }_0}$. In the graphs, therefore, the point with abscissa ${\displaystyle \frac{\omega }{{\omega }_0}=1}$ corresponds to the resonance point.
• The graphs show several lines, each corresponding to a different value of the parameter ${\displaystyle \gamma }$, which represents the friction forces in the system: ${\displaystyle \gamma =0}$ means there is no friction; increasingly large ${\displaystyle \gamma }$ corresponds to increasing friction forces.
• NOTE: the graphs show the frequency response of the system, not the details of its motion. The motion may in fact go through a transient phase before settling into an oscillation at a given frequency. Here, however, we are not interested in the transient, so the graphs refer to the amplitude and phase of the pure oscillatory motion that results after some time.
• NOTE: the graphs describe general features of the phenomenon, while the numerical details may depend on the type of quantities involved. For example, the phase shift takes different values depending on whether the displacement or the velocity (or the voltage and the current, in electrical systems) of the resulting motion is measured, but the qualitative features are identical in both cases. The phase-shift graph below refers to the velocity..

Amplitude of the oscillatory motion as the frequency of the external force varies	Phase shift of the oscillatory motion as the frequency of the external force varies
The amplitude, starting from its static value (at zero frequency the external force is a constant), grows as the frequency approaches the resonance value. At resonance a maximum value is reached, which is larger the smaller the friction in the system. The amplitude decreases as the frequency moves beyond the resonance value. At very high frequencies the amplitude always tends to zero.	The phase shift decreases as the frequency approaches the resonance value. For frequencies below resonance the velocity leads (positive phase shift) the force. At resonance the phase shift is always zero, regardless of the friction in the system: the velocity is in phase with the force, that is, the force always pushes in the direction of motion. This is precisely why the energy transfer is greatest at resonance. The phase shift becomes negative as the frequency moves beyond the resonance value. This means that the velocity is again out of phase with the force, but now lags behind it. Without friction a limiting case is reached: the phase shift is always 90° in magnitude

variable mass	variable spring constant
variable friction
	These animations show the effect on the amplitude response of a single oscillator of changing its mass, its spring constant and its friction coefficient. An increase in mass lowers the resonance frequency, while an increase in the spring constant raises it. Friction, instead, damps the response mainly at the resonance frequency (which also drops slightly as friction increases). The graphs are on a log-log scale, and show on the abscissa the frequency f of the driving force relative to the natural frequency f0 of the system with ${\displaystyle m=1}$, ${\displaystyle k=1}$ and ${\displaystyle r=0.1}$. The ordinate is proportional to the maximum amplitude ${\displaystyle X_0}$ of the displacement. ${\displaystyle F_0}$ is the maximum of the applied external force

Systems of many oscillators

An equally general property of oscillating systems is the following: a system made of many oscillators coupled to one another still has resonances, which, however, generally do not coincide with those of its components.

Example:

• It is enough to take a simple mass-spring system and connect it to an identical one, by means of a spring equal to the other two. The details of the calculations for this system can be found in the section coupled oscillators. Here it is enough to note that, while the two component systems both have the same resonance frequency ω₀, the compound system has two distinct frequencies: one identical to that of the components, ω₁=ω₀, and the other higher: ${\displaystyle {\omega }_2=\sqrt{3}{\omega }_0}$.

Consequently, it is not enough simply to break a system down into oscillating subsystems to identify its resonances easily; a particular analysis called modal analysis, which is the study of normal modes, must be carried out. For details of this procedure, and many examples, see the relevant sections.

The following graphs illustrate the typical response of a system of several oscillators. In each graph the elementary resonances (of the normal modes) of the system are shown in black. The red line is their sum, and is the overall frequency response of the system.

Amplitude as the frequency of the external force varies	Phase shift as the frequency of the external force varies
The resonance peaks can all be different from one another, that is, have different resonance frequencies, but also different amplitudes and widths.	The overall phase shift need not be zero at each individual resonance. In this case it keeps decreasing as the frequency increases but, of course, there are more complex cases.

From the position and width of the peaks, the physical characteristics of each of the normal oscillators of the system can be deduced (to see how, consult the page on impedance). However, these parameters do not necessarily correspond to any of the systems that physically make up the system (see the example given in the section systems of many oscillators and on the page coupled oscillators)

Check your understanding

Examples of resonance phenomena

Natural phenomena

Resonance is a fundamental and universal phenomenon. It characterises all systems that have the two properties needed for oscillation (corresponding to inertia and elasticity in mechanical systems), and it can be observed in a great many natural phenomena.

• Resonance and tides
• The acoustic illusion of the seashell
• Resonant emission and absorption of light by atoms
• Planets and orbital resonances
• Resonances in buildings and structures: bridges, skyscrapers and seismic dampers, engines

Common physical systems and devices

Resonance can also occur in man-made devices and machines. Here, however, we must distinguish two situations:

1. the phenomenon is wanted, because it allows a signal to be amplified or selected. This is the case of radio, the laser, filters. Resonance makes it possible to tune the device, that is, to optimise its response in one or more very precise frequency bands.
2. the phenomenon is unwanted, because the system in resonance is subjected to stresses that can compromise its integrity or operation. This is the case of buildings and structures (houses, bridges, etc.), whose materials risk deteriorating or breaking if subjected to excessive stresses. The design must allow for possible resonant stresses and act to detune the structure, damping its response at the critical frequencies.

In both cases, knowing the phenomenon allows us to control it, either favouring or inhibiting resonant behaviour depending on the type of structure. Below, the examples are listed according to the type of quantities involved. Each category contains examples of both "wanted" and "unwanted" resonance.

Acoustic

• Helmholtz resonators
• The body of musical instruments
• Resonance and architectural acoustics

Electrical and electronic

• Band-pass filters
• Oscillating circuits and radio
• The microwave oven

Optical

• Fabry-Pérot cavity (etalon)

Magnetic

• Nuclear magnetic resonance

Less common devices

• The FEL (free-electron laser)
• The undulator
• The klystron

Further reading and links

Normal modes and resonance are not phenomena peculiar to elastic systems; they are common to all oscillating systems, whether electrical, acoustic, mechanical, macroscopic (that is, described by classical physics) or quantum (that is, described by quantum physics).

• Questions and answers on resonance
• For a quantitative treatment of resonance, see the pages illustrating the concept of impedance, in particular impedance of the harmonic oscillator, impedance of a vibrating string, electrical impedance, acoustic impedance
• For the calculation of the resonance frequencies of various mechanical systems of particular interest for musical instruments, see normal modes of a string, normal modes of a circular membrane, normal modes of a rectangular membrane
• Related questions and answers:
  • Is it possible to shatter a glass with only the force of the human voice?