# Sinusoidal motion

Series | Geophysical References Series |
---|---|

Title | Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing |

Author | Enders A. Robinson and Sven Treitel |

Chapter | 4 |

DOI | http://dx.doi.org/10.1190/1.9781560801610 |

ISBN | 9781560801481 |

Store | SEG Online Store |

At this point, we introduce an important relation that is familiar to us from elementary calculus — Euler’s equation:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &e^{i\theta }{\ =\ cos\ }\theta {\ +\ }i{\rm \ sin\ }\theta . \end{align} }****(**)

For negative **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta }**
, this is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &e^{-i\theta }{\rm =\ cos\ }\theta -i{\rm \ sin\ }\theta . \end{align} }****(**)

Here, **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i{\rm =}\sqrt{-{\rm l}}}**
. The two versions of Euler’s equation above give the following expressions for the cosine and the sine:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &{\rm \ cos\ }\theta {\ =\ }\frac{e^{i\theta }{\rm +}e^{-i\theta }} {{\rm 2}}{\rm ,\ \ sin\ }\theta {\ =\ }\frac{e^{i\theta }-e^{-i\theta }}{{\rm 2}i}. \end{align}}****(**)

We now introduce the continuous time variable *t*, and we let **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta {\ =\ }\omega \ t}**
in Euler’s equation. The result is

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} &e^{i\omega t}{\ =\ cos\ }\omega t{\ +\ }i{\rm \ sin\ }\omega t. \end{align}}****(**)

The function **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{i\omega t}}**
represents the rotating vector shown in Figure 2a. The angular frequency is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega }**
, which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f{\rm =}\omega {\rm /2}\pi }**
. As *t* increases, the vector **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{i\omega t}}**
rotates in the counterclockwise direction. In Cartesian coordinates, we let the *x*-axis represent the real axis and the *y*-axis represent the imaginary axis (Figure 2b). Then the quantity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{i\omega t}}**
for fixed **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega }**
and *t* represents a vector whose projection on the *x*-axis is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ cos\ }\omega t}**
and whose projection on the *y*-axis is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ sin\ }\omega t}**
. The angle of this vector is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega t}**
, and the length of this vector is one. The (*x,y*)-plane is called the *complex z-plane*, where **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z{\rm =}x{\rm +}iy}**
. As time *t* increases, this vector rotates in a counterclockwise direction, and the tip of the vector traces out a circle. Because this circle has unit radius, it is called the unit circle in this complex *z*-plane.

As the vector rotates, its projection **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ cos\ }\omega t}**
on the *x*-axis traces out a cosine curve, and its projection **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ sin\ }\omega t}**
on the *y*-axis traces out a sine curve. Both **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ cos\ }\omega}**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ sin\ }\omega t}**
represent sinusoidal motion at a fixed frequency **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega }**
. Such motion is called either sinusoidal motion or *simple harmonic motion*. Instead of using a continuous time scale *t* for the signal, digital processing requires first choosing a time increment **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta t}**
and then defining the time index *n* so that time is given by **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t{\rm =}n\Delta t}**
. A typical time increment is 0.004 s.

The two vectors in Figure 2a show that an angle of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega n\Delta t}**
radians is swept out in **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t{\rm =}n\Delta t}**
seconds. The lower vector corresponds to time index *n* = 0 and the upper vector to time index *n*. Instead of considering a rotating wheel, we simply can think of a single vector that rotates at a constant angular velocity

At time *n* = 0, the vector lies in the positive direction along the horizontal coordinate axis. Then at some arbitrary time index *n*, the vector will make an angle of **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega n\Delta t}**
radians with the horizontal axis. The projections of this vector on the *x-* and *y*-axes give the horizontal and vertical components, respectively: **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \left({\rm \ cos\ }\omega n\Delta t{\rm ,\ \ sin\ }\omega n\Delta t\right)}**
. Let the vertical axis be the imaginary axis (i.e., a unit distance on the vertical axis is **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i{\rm =}\sqrt{-{\rm 1}}}**
). We can represent the vector at time **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle n\Delta t}**
in terms of its components **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ cos\ }\omega n\Delta t}**
and **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {\rm \ sin\ }\omega n\Delta t}**
as follows:

**Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} {\rm vector}\ ({\rm at}\ {\rm time}\ n\Delta t)\ {\rm =\ cos\ }\omega n\Delta t{\ +\ }i{\rm \ sin\ }\omega n\Delta t{\ =\ }e^{i\omega n\Delta t} \end{align}}****(**)

This equation, which is a form of Euler’s equation, shows that the exponential **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle e^{i\omega n\Delta t}}**
represents a unit vector (or wheel) rotating at constant angular velocity **Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega}**
. The components of this vector represent simple harmonic motion.

## Continue reading

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Frequency | Aliasing |

Previous chapter | Next chapter |

Visualization | Filtering |

## Also in this chapter

- Time series
- The wavelet
- Digitization
- Frequency
- Aliasing
- The Nyquist frequency
- Sampling geophysical data
- Appendix D: Exercises