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

For negative ${\displaystyle \theta }$, this is

 {\displaystyle {\begin{aligned}&e^{-i\theta }{\rm {=\ cos\ }}\theta -i{\rm {\ sin\ }}\theta .\end{aligned}}} (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{aligned}&{\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{aligned}}} (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{aligned}&e^{i\omega t}{\ =\ cos\ }\omega t{\ +\ }i{\rm {\ sin\ }}\omega t.\end{aligned}}} (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{aligned}{\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{aligned}}} (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.