# 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:

**(**)

For negative , this is

**(**)

Here, . The two versions of Euler’s equation above give the following expressions for the cosine and the sine:

**(**)

We now introduce the continuous time variable *t*, and we let in Euler’s equation. The result is

**(**)

The function represents the rotating vector shown in Figure 2a. The angular frequency is , which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is . As *t* increases, the vector 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 for fixed and *t* represents a vector whose projection on the *x*-axis is and whose projection on the *y*-axis is . The angle of this vector is , and the length of this vector is one. The (*x,y*)-plane is called the *complex z-plane*, where . 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 on the *x*-axis traces out a cosine curve, and its projection on the *y*-axis traces out a sine curve. Both and represent sinusoidal motion at a fixed frequency . 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 and then defining the time index *n* so that time is given by . A typical time increment is 0.004 s.

The two vectors in Figure 2a show that an angle of radians is swept out in 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 (Figure 2b).

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 radians with the horizontal axis. The projections of this vector on the *x-* and *y*-axes give the horizontal and vertical components, respectively: . Let the vertical axis be the imaginary axis (i.e., a unit distance on the vertical axis is ). We can represent the vector at time in terms of its components and as follows:

**(**)

This equation, which is a form of Euler’s equation, shows that the exponential represents a unit vector (or wheel) rotating at constant angular velocity . The components of this vector represent simple harmonic motion.

## Continue reading

Previous section | Next section |
---|---|

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