# Definition:Simple Harmonic Motion

*This page is about Simple Harmonic Motion. For other uses, see Harmonic.*

## Definition

Consider a physical system $S$ whose motion can be expressed in the form of the following equation:

- $x = A \map \sin {\omega t + \phi}$

where $A$ and $\phi$ are constants.

Then $S$ is in a state of **simple harmonic motion**.

### Amplitude

The parameter $A$ is known as the **amplitude** of the motion.

### Period

The **period** $T$ of the motion of $S$ is the time required for one complete cycle:

- $T = \dfrac {2 \pi} \omega$

### Frequency

The **frequency** $\nu$ of the motion of $S$ is the number of complete cycles per unit time:

- $\nu = \dfrac 1 T = \dfrac \omega {2 \pi}$

## Also defined as

**Simple harmonic motion** can also be characterised in the form:

- $x = A \map \cos {\omega t + \phi}$

From Sine of Angle plus Right Angle:

- $\map \sin {\omega t + \phi + \dfrac \pi 2} = \map \cos {\omega t + \phi}$

the two forms can be seen to be equivalent.

## Also known as

**Simple harmonic motion** can also be referred to as **simple harmonic oscillation** or **simple harmonic vibration**.

Some sources abbreviate it to **SHM**.

## Sources

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3.20$: Vibrations in Mechanical Systems