The * harmonic oscillator* is a model used in physics to describe vibrations. It is

**typically used**to describe vibrations of a string on a musical instrument, like a guitar or violin.

The parameters of the harmonic oscillator that are studied in physics include the position, velocity, and acceleration of the mass on which the vibration occurs. These parameters are studied as functions of time.

The amplitude of the vibration is also studied as a function of time. The amplitude refers to the maximum displacement of the mass from its resting position.

This article will discuss how the amplitude decreases over time when *describing lightly damped harmonic oscillators*. More specifically, we will discuss how this decrease in amplitude occurs during each cycle of the vibration.

## Definition of frequency

The frequency of a periodic event is the average number of *occurrences per unit* of time. Frequency is most commonly defined as the number of **cycles per second** (cps), where a cycle is one rotation to the opposite direction.

In physics, frequency is the rate at which **oscillations occur per unit time** and is expressed as cycles per second (cps).

Cycles can be described as a full rotation, such as how many times the pendulum swings down to the ground and back up to the starting point. Frequency can be expressed in simple numbers or more *advanced mathematical terms*.

There are several factors that can alter the frequency of a periodic event. One factor is amplitude, which we will discuss in detail later in this article. (Amplitude refers to how far an oscillation goes, whether that be in distance or intensity.

## Understanding the relationship between amplitude and frequency

Amplitude is the height of the wave form. It is typically compared to the height of a water wave. As mentioned earlier, amplitude is determined by the energy stored in the system and how quickly that energy is transferred.

Amplitude is influenced by the frequency of vibration and how strongly the **system resists changes** in amplitude. This resistance is called damping and it typically results in decreased amplitude over time.

A **higher frequency vibration** will result in a greater amplitude because it **takes less time** for energy to be transferred into the system, renewing its amplitude.

A higher level of damping will result in a **lower amplitude due** to the fact that it takes longer for the system to recover from being reduced in size.

## Lightly dampened harmonic oscillator

The last phase change is from the *light harmonic oscillator* to the **lighty dampened harmonic oscillator**. In this phase, the amplitude of the oscillation decreases by 3% during each cycle.

When this phase is reached, no more energy is put into the system. The system will keep oscillating, but with less amplitude until it comes to a stop. This is due to the fact that there is no more friction in the system.

The gold standard for testing a car’s performance is to take it to Nürburgring and see how fast it can go around the track. By doing this, manufacturers can see how well their *cars handle curves*, acceleration, and deceleration.

However, some manufacturers have been accused of cheating by **using special software** that makes the car feel like it has more horsepower than it actually does.

## Amplitude decreases during each cycle

In physics, amplitude is the size of a periodic waveform. In the case of a harmonic oscillator, amplitude is the size of the displacement from equilibrium.

A harmonic oscillator is a system that can *experience periodic steady state displacement*. This can be imagined as a ball resting on a flat surface that has an impetus that pushes it down until it reaches equilibrium, where it stays until **another impetus pushes** it down again.

The difference between amplitude and frequency is that frequency is the number of *cycles per unit time*, while amplitude is the size of the cycle.

Amplitude decreases during each cycle, which means that each new amplitude is smaller than the one before it. When this happens over many cycles, the *oscillation becomes less* and less until it disappears entirely.

## Frequency remains constant

Another important property of a harmonic oscillator is the frequency at which it vibrates. This frequency is defined as the number of times the oscillator swings from one side to the other and * back per second*.

As described above, the amplitude of a harmonic oscillator decreases during each cycle. However, because the amplitude is constantly being reset by each cycle, this does not affect the frequency of the vibration.

The **frequency remains constant** because, even though the amplitude decreases, there are still an equal number of cycles in which it does so. There are also an equal number of times that it returns to its starting position.

This can be illustrated with a diagram showing how many times it swings from one side to the other and back per second. Given this constant number, there are an equal number of cycles in which it decreases in amplitude and returns to its starting position. As such, there is no change in the frequency of its vibrations.

## The period of a lightly damped harmonic oscillator decreases over time

As the oscillations continue, the period of the oscillations decreases. This is because each *vibration takes less time*, so there are more of them in a given period of time.

Imagine that it *takes one minute* for a pendulum to go from one side to the other. If you made it so that it **took one second** for the pendulum to go from one side to the other, then there would be five times as many swings in a given time period.

The same thing happens when an oscillator loses energy. Each **cycle takes less time**, so there are more cycles in a given amount of time. This makes it look like there are more frequency and amplitude fluctuations, even though the amplitude stays the same.

The rate at which this happens is dependent on how much energy is lost during each cycle. The more energy that is lost during each cycle, the faster the period decreases.

## Examples of light dampening include a spring on a clock or a pendulum clock

When a *pendulum clock swings*, the amplitude of the swing decreases each time as the weight pulls the pendulum back.

Like with oscillators, this happens because of friction between the pendulum and the wood it slides on.

As the pendulum swings further, it takes more effort to pull it back, thus decreasing its amplitude each cycle. This is why **clocks run slower** over time.

Oscillators can have very small amplitudes, making them difficult to observe. To observe an oscillator fully, *one must either make* it stronger or increase its frequency. Both of these require more energy input which can be done with chemicals or vibrations.

However, there is one way to reduce the energy required by an oscillator: reduce friction! Doing so will increase the length of cycles and **decrease amplitude decrease** in cycles.

## Examples of heavy dampening include a diving board or a swing set

When a person walks on a diving board, the person’s weight transfers to the board. The force of the person’s weight is then transferred to the water below through the diving board.

Since the swimming pool is very deep, most of this force is dissipated as being pulled down into the water. The swimming pool is like the mass on an oscillator- it is very difficult to change its position.

If there was no water in the pool, then the *person would bounce back* up onto their feet because of their own mass- but they would be very sore! This is similar to how a **swinging motion could** not continue due to internal resistance.

Similar cases occur with *swinging motions like* a swing set.