The resonance frequency is the frequency of a sustained excitation for which a system oscillates with maximum amplitude. It is very close to the system’s natural frequency f₀, a property that depends only on the system itself, as long as damping remains small.
Discover FizziQ
How to measure it in class
FizziQ has a spectrum analyzer (FFT) and a frequency synthesizer. The first reveals the natural frequency of an object excited briefly; the second makes it possible to sweep frequencies in order to plot a true resonance curve.
Steps:
- Choose a simple resonator: glass bottle, test tube, graduated cylinder
- Open FizziQ’s spectrum analyzer and bring it close to the opening
- Blow at an angle across the neck to excite the cavity, and record the frequency of the main peak: this is f₀
- Vary the water level in the bottle, and therefore the air volume, and record f₀ each time
- Plot f₀ as a function of the volume V and check the 1/√V decrease
- Quantitative variant: use the synthesizer to emit a slow frequency sweep, measure the reproduced level with the sound level meter, and plot the amplitude as a function of f to reveal the resonance peak
Scientific activities on this topic
By measuring the resonance frequency of certain cavities, you can in particular calculate the speed of sound very precisely:
- Speed of sound with a test tube: blow into a test tube and derive the speed of sound from the natural frequency
- Helmholtz resonance: the sound of a bottle: measure the frequency emitted when uncorking a bottle and relate it to the volume
- Dominant frequency in a tube: excite a pipe and identify its resonance modes
- Mass-spring oscillator: measuring the natural period: experimentally verify the law T₀ = 2π√(m/k)
- Air columns and pitch: build an instrument from tubes of different lengths
- Spectrum of sung vowels: identify the formants, resonances of the vocal tract that distinguish an “a” from an “i”
- Standing Faraday waves: make standing patterns appear by exciting a liquid at its natural frequency
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The mechanism: an accumulation of energy
Why does the amplitude become so large? Because at resonance, the excitation always “pushes” at the right moment, in the direction of the motion. At each cycle, it supplies a little energy that the system accumulates. The classic image is the swing: by pushing at the natural rhythm of the oscillation, small pushes are enough to obtain large amplitudes. If you push out of time, you brake.
The amplitude does not grow indefinitely, however: it stabilizes when the power dissipated by friction equals the power supplied by the excitation.
The role of damping
Damping governs the sharpness of the resonance peak, which is quantified by the quality factor Q. The weaker the damping, the larger Q, the taller and narrower the peak: the system responds strongly only in a very tight frequency band. Conversely, a strongly damped system has a low, wide peak, and responds weakly over a broad range.
Orders of magnitude: a tuning fork has a Q of a few thousand: it rings for a long time and at a very pure frequency. A car shock absorber, designed not to oscillate, has a Q below 1. It is precisely by increasing damping that a structure is protected from resonance.
An often forgotten detail: when damping increases, the amplitude resonance frequency drops slightly below the natural frequency. In high school this difference is neglected, as it stays below one percent for weakly damped systems.
Natural modes: there are several
An extended system (string, air column, plate) does not have one natural frequency but a series of natural frequencies, called modes. For a guitar string or a tube open at both ends, these modes are f₁, 2f₁, 3f₁…: the harmonic series, which gives instruments their timbre. For a tube closed at one end, only the odd harmonics exist: f₁, 3f₁, 5f₁… This is what distinguishes the sound of a clarinet (closed pipe) from that of a flute (open pipe), at comparable length.
The Helmholtz resonator is an exception: it behaves like an oscillator with a single degree of freedom (the air in the neck plays the role of the mass, the air in the volume that of the spring) and therefore essentially has only one natural frequency.
The myth of the Tacoma bridge
It is the example cited everywhere, and it is wrong. The collapse of the Tacoma Narrows Bridge, on November 7, 1940, was not due to forced resonance: the wind, steady that day at approximately 65 km/h, contained no periodic component tuned to a natural frequency of the deck. The phenomenon at play is aeroelastic flutter: the motion of the deck itself modifies the airflow, which in return amplifies this motion. It is an instability by feedback, in which energy is drawn continuously from the flow, and not a response to an external periodic excitation. The distinction is essential: in a resonance, the excitation exists independently of the system; in flutter, it is generated by the motion itself.
The correct example of destructive mechanical resonance remains breaking step on a bridge: troops marching in step impose a very real periodic excitation, hence the military order to break step. The Millennium Bridge in London, in 2000, experienced a related but more subtle phenomenon, with a coupling between the lateral oscillation of the deck and the gait of the pedestrians.
Where resonance is encountered
The notion goes far beyond mechanics. An RLC electrical circuit has a resonance frequency used to select a radio station. MRI exploits the nuclear magnetic resonance of protons. The microwave oven, on the other hand, does not work by resonance of the water molecule: that is a misconception. Its 2.45 GHz frequency is very far from the resonances of liquid water; the heating comes from the agitation of the dipoles and from dielectric losses.
Six science experiments with the tone generator (FizziQ blog) shows how to sweep frequencies to reveal a resonance peak.
Formula
Natural frequency of a mass-spring oscillator:
f₀ = (1 / 2π) × √(k / m)
Natural frequency of a simple pendulum of length L (small oscillations):
f₀ = (1 / 2π) × √(g / L)
Helmholtz resonator (cavity of volume V, neck of cross-section S and effective length L_eff):
f₀ = (v / 2π) × √(S / (V × L_eff))
Modes of a tube open at both ends (all harmonics):
f_n = n × v / (2L), with n = 1, 2, 3…
Modes of a tube closed at one end (odd harmonics only):
f_n = n × v / (4 × (L + 0.6 R)), with n = 1, 3, 5…
The term 0.6 R is the end correction: the air located just outside the opening takes part in the vibration, so that the tube “sounds” as if it were a little longer than it actually is.
Quality factor, which measures the sharpness of the peak:
Q = f₀ / Δf
where:
- f₀: natural frequency (Hz)
- f: excitation frequency imposed from outside (Hz)
- k: spring stiffness constant (N·m⁻¹)
- m: mass (kg)
- L: length of the pendulum or the tube (m)
- R: radius of the tube (m)
- g: gravitational field strength, approximately 9.81 m·s⁻²
- v: speed of sound, approximately 343 m·s⁻¹ in air at 20 °C
- V: volume of the cavity (m³)
- S: cross-section of the neck (m²)
- Q: quality factor (dimensionless)
- Δf: width of the resonance peak at half power (Hz)
Application examples
- A 75 cL bottle with a neck 2 cm in diameter and 5 cm long resonates around 110 to 130 Hz. Filling the bottle halfway divides V by two and f₀ is multiplied by √2, approximately 1.41: the sound rises by about a tritone (augmented fourth), the ratio of a fifth being 1.5.
- A closed tube 1 m long resonates at f₁ = 343 / 4 = approximately 86 Hz, then at 257 Hz and 429 Hz: the odd harmonics only. The same tube open at both ends would give 172 Hz, 343 Hz, 515 Hz.
- The formants of the voice are resonances of the vocal tract, approximately 17 cm long. It is their position, and not the frequency of the vocal cords, that distinguishes an “a” from an “i”.
- The stemmed glass shattered by the voice: you must hit exactly its natural frequency (often 500 to 800 Hz), with a very high level and a glass of large Q. The demonstration is real but much harder than television makes it look.
- Skyscrapers and bridges are dimensioned so that their natural frequencies are far from those of wind and earthquakes; tuned mass dampers are added, such as the 660-tonne sphere of the Taipei 101 tower.
- A tuned RLC circuit selects a radio station: only the frequency close to its resonance produces a significant voltage.
FAQ
Q: What is the difference between natural frequency and resonance frequency? A: The natural frequency f₀ is the one at which the system oscillates on its own when displaced from equilibrium: it depends only on the system. The resonance frequency is the excitation frequency that produces the greatest amplitude. The two are practically identical when damping is small, which is the case in almost all high school experiments.
Q: Does the amplitude become infinite at resonance? A: No, except in the ideal frictionless model. In reality, the amplitude stabilizes at the value for which the energy dissipated by friction exactly compensates that supplied by the excitation. The less damping there is, the larger this limiting amplitude.
Q: Did the Tacoma bridge really collapse by resonance? A: No, this is a very widespread error, including in some textbooks. The wind was steady and provided no periodic excitation tuned to a natural frequency of the bridge. The cause is aeroelastic flutter: the deck, as it twisted, modified the airflow, which then amplified its motion. The energy came from a feedback loop, not from an external periodic excitation.
Q: Why does the note of a bottle rise when you fill it? A: Because you reduce the volume V of air in the cavity. And f₀ varies as 1/√V: dividing the volume by four doubles the frequency, that is one octave. Beware, it is the opposite if you strike the bottle with a spoon instead of blowing into it: you then make the glass and the water vibrate, and the note goes down as you fill.
Q: Why does the microwave oven heat water, if it is not by resonance? A: The oven emits at 2.45 GHz, a frequency very far from the resonance frequencies of water molecules. The water molecules, which are dipoles, try to follow the alternating electric field; friction between molecules then dissipates this agitation as heat. It is heating by dielectric losses, not resonance.
Related concepts
Harmonic Oscillator - Damping - Standing Wave - Fundamental Frequency - Timbre