# Sinusoidal motion

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 4 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

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

 \displaystyle \begin{align} &e^{i\theta }{\ =\ cos\ }\theta {\ +\ }i{\rm \ sin\ }\theta . \end{align} (5)

For negative $\displaystyle \theta$ , this is

 \displaystyle \begin{align} &e^{-i\theta }{\rm =\ cos\ }\theta -i{\rm \ sin\ }\theta . \end{align} (6)

Here, $\displaystyle i{\rm =}\sqrt{-{\rm l}}$ . The two versions of Euler’s equation above give the following expressions for the cosine and the sine:

 \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} (7)

We now introduce the continuous time variable t, and we let $\displaystyle \theta {\ =\ }\omega \ t$ in Euler’s equation. The result is

 \displaystyle \begin{align} &e^{i\omega t}{\ =\ cos\ }\omega t{\ +\ }i{\rm \ sin\ }\omega t. \end{align} (8)

The function $\displaystyle e^{i\omega t}$ represents the rotating vector shown in Figure 2a. The angular frequency is $\displaystyle \omega$ , which for this discussion, we take to be an intrinsically positive number. The cyclical frequency is $\displaystyle f{\rm =}\omega {\rm /2}\pi$ . As t increases, the vector $\displaystyle e^{i\omega t}$ 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 $\displaystyle e^{i\omega t}$ for fixed $\displaystyle \omega$ and t represents a vector whose projection on the x-axis is $\displaystyle {\rm \ cos\ }\omega t$ and whose projection on the y-axis is $\displaystyle {\rm \ sin\ }\omega t$ . The angle of this vector is $\displaystyle \omega t$ , and the length of this vector is one. The (x,y)-plane is called the complex z-plane, where $\displaystyle z{\rm =}x{\rm +}iy$ . 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 $\displaystyle {\rm \ cos\ }\omega t$ on the x-axis traces out a cosine curve, and its projection $\displaystyle {\rm \ sin\ }\omega t$ on the y-axis traces out a sine curve. Both $\displaystyle {\rm \ cos\ }\omega$ and $\displaystyle {\rm \ sin\ }\omega t$ represent sinusoidal motion at a fixed frequency $\displaystyle \omega$ . 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 $\displaystyle \Delta t$ and then defining the time index n so that time is given by $\displaystyle t{\rm =}n\Delta t$ . A typical time increment is 0.004 s.

The two vectors in Figure 2a show that an angle of $\displaystyle \omega n\Delta t$ radians is swept out in $\displaystyle t{\rm =}n\Delta t$ 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 $\displaystyle \omega$ (Figure 2b).

Figure 2.  (a) Spoke of wheel rotating at angular velocity $\displaystyle \omega$ . (b) Vector rotating at angular velocity $\displaystyle \omega$ .

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 $\displaystyle \omega n\Delta t$ radians with the horizontal axis. The projections of this vector on the x- and y-axes give the horizontal and vertical components, respectively: $\displaystyle \left({\rm \ cos\ }\omega n\Delta t{\rm ,\ \ sin\ }\omega n\Delta t\right)$ . Let the vertical axis be the imaginary axis (i.e., a unit distance on the vertical axis is $\displaystyle i{\rm =}\sqrt{-{\rm 1}}$ ). We can represent the vector at time $\displaystyle n\Delta t$ in terms of its components $\displaystyle {\rm \ cos\ }\omega n\Delta t$ and $\displaystyle {\rm \ sin\ }\omega n\Delta t$ as follows:

 \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} (9)

This equation, which is a form of Euler’s equation, shows that the exponential $\displaystyle e^{i\omega n\Delta t}$ represents a unit vector (or wheel) rotating at constant angular velocity $\displaystyle \omega$ . The components of this vector represent simple harmonic motion.