Laplace Transform Function Pair Visualization

Introduction: This application demonstrates the correspondence of various time-domain functions with their frequency-domain Laplace pair function.

Directions:

• Function Pair: Select a function pair from the Function Pairs drop-down. Some functions have parameters such as "A" and "B" that may be altered using the corresponding sliders. The Up/Down cursor keys can also be used to alter the "A" parameter, and the Left/Right cursor keys can be used to alter the "B" parameter.
• Streamlines: Adjusting the "Streamline Density" slider alters the incremental density of the streamlines. Adjusting the "Streamline Offset" slider shifts the increment of the streamline densities.
• Dashes: Checking and clearing "Dashes" shows the streamlines as moving dashes or as solid lines.
• Bubbles: Checking and clearing "Bubbles" shows or hides bubbles that move along the streamlines to indicate a direction of travel. Altering the "Bubble Density" slider changes the density of the bubbles.
• Dead Zones: Checking and clearing "Dead Zones" shows or hides small circles that show areas where the Laplace transform is zero.

More Info:

Each of the function pairs includes a parameter A representing the amplitude, which is initially zero, but can be altered by using the up and down keys. Some of the function pairs also include a second parameter B that allows further adjustment of the functions, such as sin(Bt), which allows for adjusting the time-domain sine frequency. This parameter can be altered by using the left and right keys.

Plotting the time-domain function is easy: a function f(t) = y accepts a real number (t) and outputs a real number (y), and this function can be plotted as a typical graph of t vs. y.

Plotting the frequency domain function is more difficult, since such functions F(s) = z accept a complex number (s) and output a complex number (z). The easiest way to present this on a two-dimensional graph is as a vector plot - at each point (r₁, i₁) on the complex plane, compute the output and plot a vector pointing to (r₂, i₂). However, this plot gets noisy very fast, and plotting only a few points does not accurately convey the shape of the plot. Instead, the output is plotted on the complex plane using streamlines, with dead zones (illustrated as big circles) wherever z = 0. The direction of each streamline can be illustrated either by drawing the streamlines as moving dashes, or by drawing moving bubbles along the streamlines flowing in the direction of the streamline.

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