**Natural frequency** is the rate at which an object vibrates when it is disturbed (e.g. plucked, strummed, or hit). A vibrating object may have one or multiple natural frequencies. Simple harmonic oscillators can be used to model the natural frequency of an object.

### Key Takeaways: Natural Frequency

- Natural frequency is the rate at which an object vibrates when it is disturbed.
- Simple harmonic oscillators can be used to model the natural frequency of an object.
- Natural frequencies are different from forced frequencies, which occur by applying force to an object at a specific rate.
- When the forced frequency equals the natural frequency, the system is said to experience resonance.

## Waves, Amplitude, and Frequency

In physics, frequency is a property of a wave, which consists of a series of peaks and valleys. A wave’s frequency refers to the number of times a point on a wave passes a fixed reference point per second.

Other terms are associated with waves, including amplitude. A wave’s amplitude refers to the height of those peaks and valleys, measured from the middle of the wave to the maximum point of a peak. A wave with a higher amplitude has a higher intensity. This has a number of practical applications. For example, a sound wave with a higher amplitude will be perceived as louder.

Thus, an object that is vibrating at its natural frequency will have a characteristic frequency and amplitude, among other properties.

## Harmonic Oscillator

Simple harmonic oscillators can be used to model the natural frequency of an object.

An example of a simple harmonic oscillator is a ball on the end of a spring. If this system has not been disturbed, it is at its equilibrium position – the spring is partially stretched out due to the weight of the ball. Applying a force to the spring, like pulling the ball downward, will cause the spring to start oscillating, or go up and down, about its equilibrium position.

More complicated harmonic oscillators can be used to describe other situations, such as if the vibrations are “damped” slow down due to friction. This type of system is more applicable in the real world – for instance, a guitar string will not keep vibrating indefinitely after it has been plucked.

## Natural Frequency Equation

The natural frequency f of the simple harmonic oscillator above is given by

f = ω/(2π)

where ω, the angular frequency, is given by √(k/m).

Here, k is the spring constant, which is determined by the stiffness of the spring. Higher spring constants correspond to stiffer springs.

m is the mass of the ball.

Looking at the equation, we see that:

- A lighter mass or a stiffer spring increases natural frequency.
- A heavier mass or a softer spring decreases natural frequency.

## Natural Frequency vs. Forced Frequency

Natural frequencies are different from **forced frequencies**, which occur by applying force to an object at a specific rate. The forced frequency can occur at a frequency that is the same as or different from the natural frequency.

- When the forced frequency is not equal to the natural frequency, the amplitude of the resulting wave is small.
- When the forced frequency equals the natural frequency, the system is said to experience “resonance”: the amplitude of the resulting wave is large compared to other frequencies.

## Example of Natural Frequency: Child on a Swing

A child sitting on a swing that is pushed and then left alone will first swing back and forth a certain number of times within a specific timeframe. During this time, the swing is moving at its natural frequency.

To keep the child swinging freely, they must be pushed at just the right time. These “right times” should correspond to the natural frequency of the swing to make the swing experience resonance, or yield the best response. The swing receives a little more energy with each push.

## Example of Natural Frequency: Bridge Collapse

Sometimes, applying a forced frequency equivalent to the natural frequency is not safe. This can happen in bridges and other mechanical structures. When a poorly designed bridge experiences oscillations equivalent to its natural frequency, it can violently sway, becoming stronger and stronger as the system gains more energy. A number of such “resonance disasters” have been documented.

## Sources

- Avison, John.
*The World of Physics*. 2nd ed., Thomas Nelson and Sons Ltd., 1989. - Richmond, Michael.
*An Example of Resonance*. Rochester Institute of Technology, spiff.rit.edu/classes/phys312/workshops/w5c/resonance_examples.html. *Tutorial: Fundamentals of Vibration*. Newport Corporation, www.newport.com/t/fundamentals-of-vibration.