Problem 69 - Calculus Early Transcendentals: Pearson New International Edition

Question

. Plot Lissajous figures for the following combinations of \(a\) and \(b\) for \(0 \leq t \leq 2 \pi\) : (a) \(a=1, b=2\) (b) \(a=4, b=8\) (c) \(a=5, b=10\) (d) \(a=2, b=3\) (e) \(a=6, b=9\) (f) \(a=12, b=18\)

Step-by-Step Solution

Answer

Use equations \(x = \sin(at)\) and \(y = \sin(bt)\) to plot each figure for given \((a, b)\) pairs.

Step 1: Understand the Lissajous Figure Formula

A Lissajous figure is represented by the parametric equations \( x = A \sin(at + \delta) \) and \( y = B \sin(bt) \), where \(A\) and \(B\) are amplitudes, \(a\) and \(b\) are frequencies, and \(\delta\) is the phase difference. Here, we assume \(A = B = 1\) and \(\delta = 0\).

Step 2: Set the Range for t

Set \(t\) to range from 0 to \(2\pi\). This will cover one full period of the figure for each of the frequency combinations \(a\) and \(b\).

Step 3: Generate the Coordinates for Each Combination

Plug the given values of \(a\) and \(b\) into the Lissajous equations to get the coordinates \((x, y)\) for each point in the range of \(t\), using the equations \(x = \sin(at)\) and \(y = \sin(bt)\).

Step 4: Plot Lissajous Figure for Each (a, b) Pair

For each pair \((a, b)\):- (a) Use \(a = 1\) and \(b = 2\) to plot the figure.- (b) Use \(a = 4\) and \(b = 8\) to plot the figure.- (c) Use \(a = 5\) and \(b = 10\) to plot the figure.- (d) Use \(a = 2\) and \(b = 3\) to plot the figure.- (e) Use \(a = 6\) and \(b = 9\) to plot the figure.- (f) Use \(a = 12\) and \(b = 18\) to plot the figure.For each plot, graph \(x\) on the horizontal axis and \(y\) on the vertical axis.

Key Concepts

Parametric Equations

Lissajous figures are beautiful patterns that can be plotted using parametric equations. These equations express the coordinates of the figures in terms of an independent parameter, usually denoted as \( t \). For Lissajous figures, the parametric equations are \( x = A \sin(at + \delta) \) and \( y = B \sin(bt) \). Here, \( A \) and \( B \) represent the amplitudes of the x and y axes, \( a \) and \( b \) are the frequencies, and \( \delta \) is the phase difference.

Parametric equations allow us to describe complex motion in a simple way. By varying \( t \), we can plot points along the figure, making it a powerful tool for visualizing motion and shapes. In our case, we'll use \( A = B = 1 \) and \( \delta = 0 \), which simplifies the equations to \( x = \sin(at) \) and \( y = \sin(bt) \). This setup is ideal for creating symmetrical and satisfying Lissajous curves.

Frequency Combinations

The frequency combinations in Lissajous figures determine their intricate patterns. Frequencies, \( a \) and \( b \), dictate how many times the sine wave oscillates in each direction as \( t \) varies from 0 to \( 2\pi \).

When you choose different values for \( a \) and \( b \), the figure changes its shape and complexity. If \( a \) and \( b \) have a simple ratio, like 1:2, it results in a relatively simple pattern. However, as the ratio gets more complex, the figure becomes more intricate.

• A ratio of 1:2 (e.g., \( a = 1, b = 2 \)) produces simpler, repeating shapes.
• For higher ratios like 4:8 or 5:10, the figures become more detailed and elaborate.
• The variety in frequency combinations explains why Lissajous figures are both mathematically interesting and visually striking.

By experimenting with different frequencies, you gain insight into how oscillations interact to create unique geometric shapes.

Period of the Figure

The period of a Lissajous figure refers to the duration required for the figure to complete one full cycle and return to its starting point. It's the least common multiple (LCM) of the periods of the two component sine waves, \( \frac{2\pi}{a} \) and \( \frac{2\pi}{b} \).

Understanding the period is crucial because it tells you how long you need to follow the parametric path to see one complete figure. For example, if you have \( a = 2 \) and \( b = 3 \), the period of the figure is the LCM of \( 2\pi/2 \) and \( 2\pi/3 \), which is \( 2\pi \), thereby ensuring that one full cycle is viewed as \( t \) varies between 0 and \( 2\pi \).

• Equal period lengths for \( x \) and \( y \) result in simple figures.
• More complex periods often lead to intricate and fascinating Lissajous figures.

This reflection on periods helps predict the figure's repetition without needing to plot it entirely.

Graph Plotting

Plotting a Lissajous figure involves graphing its x and y coordinates obtained from the parametric equations over a specified range of \( t \). Generally, the range from 0 to \( 2\pi \) is sufficient to see the complete pattern.

To plot these figures effectively, follow these steps:

• Choose the values for \( a \) and \( b \), which define the Lissajous figure you want to plot.
• Calculate the \( x \) and \( y \) coordinates using \( x = \sin(at) \) and \( y = \sin(bt) \) for successive values of \( t \).
• Use a graphing tool or software that accepts parametric equations to visualize the result.

As you plot each figure, the unique interaction between the frequencies becomes apparent. Graph plotting not only showcases the mathematical beauty of Lissajous figures but also helps verify theoretical predictions about their shape and symmetry.