Question

Create a series of Lissajous figures as follows: For each, let our reference wave be \(y_{1}=\sin t .\) Then for \(y_{2}\) choose (a) \(y_{2}=2 \sin t\) (double the amplitude of the reference wave) (b) \(y_{2}=\sin 2 t\) (double the frequency of the reference wave) (c) \(y_{2}=\sin 3 t\) (triple the frequency of the reference wave) (d) \(y_{2}=\sin 4 t\) (quadruple the frequency of the reference wave) (e) \(y_{2}=\sin \left(t+45^{\circ}\right)\) (a phase shift of \(45^{\circ}\) from the reference wave) (f) \(y_{2}=\sin \left(t+135^{\circ}\right)\) (a phase shift of \(135^{\circ}\) from the reference wave) (g) \(y_{2}=\sin \left(t+90^{\circ}\right)\) (a phase shift of \(90^{\circ}\) from the reference wave) (h) \(y_{2}=\cos t\) (the cosine function) Plot each pair of parametric equations. Can you draw any conclusions about how to interpret a Lissajous figure on an oscilloscope?

Short answer

Lissajous figures depict the amplitude, frequency, and phase relationship between two periodic signals. Doubling the amplitude results in vertical stretching, increasing frequency results in more loops, and phase shifts result in rotations of the figure. A sine and cosine wave pair will give an undistorted circle or ellipse.

Step-by-step solution

Reference Wave Equation

Establish the reference wave equation, which in this case is given as: \( y_{1} = \text{sin}(t) \). This represents the base function for all the Lissajous figures to be compared against.

Plotting the Lissajous Figures

Create separate plots for each Lissajous figure by graphing the pair of parametric equations defined by the reference wave \( y_{1} \) and each different \( y_{2} \) stated in parts (a) through (h). Use a graphing tool or software that allows parametric plotting, ensuring that the domain of \( t \) is sufficient to show the figure's behavior, typically from \( 0 \) to \( 2\pi \) or more.

Interpreting the Figures

After plotting each figure, observe the patterns and use these observations to draw conclusions about the relationship between amplitude, frequency, phase shifts, and the resulting Lissajous figure. Direct variations in these parameters will be visible as changes in the figures' shapes, which indicate how one wave relates to the other in terms of amplitude, frequency, and phase.

Drawing Conclusions

Examine all figures and take note of the following: the doubling of amplitude leads to a figure that stretches vertically by a factor of two; Doubling, tripling, and quadrupling the frequency creates more complex figures with additional loops; Phase shifts rotate the figure around the origin, and a phase shift of \( 90^\circ \) turns the sine wave into a cosine wave, demonstrated by a figure rotated by \( 90^\circ \) compared to the base figure. Overall, Lissajous figures on an oscilloscope can illustrate the relationship between two periodic signals, showing their relative amplitude, frequency, and phase.

Key concepts

Parametric Equations

Parametric equations define a set of related quantities as explicit functions of an independent variable, typically time. In the context of Lissajous figures, the equations use the sine and cosine functions to represent the coordinates of points in a two-dimensional space as a function of time. For example:
\( x(t) = A_x \sin(\omega_x t + \phi_x) \) and \( y(t) = A_y \sin(\omega_y t + \phi_y) \),
where \( A_x \) and \( A_y \) are the amplitudes, \( \omega_x \) and \( \omega_y \) are the frequencies, and \( \phi_x \) and \( \phi_y \) are the phase shifts of the respective x and y waves. These equations allow us to create a graphical representation of the waveforms as they evolve over time, resulting in various intricate patterns of Lissajous figures based on the chosen parameters.

Sine and Cosine Functions

The sine and cosine functions in mathematics describe oscillatory and wave-like behavior. These functions are periodic, meaning they repeat their values in regular intervals. These functions are central to the creation of Lissajous figures, whose shapes are influenced by the amplitude, frequency, and phase of the sine and cosine waves. Sine and cosine functions are expressed as:

\( \sin(t) \) and \( \cos(t) \).
The sine function represents a wave that starts from zero, rises to a peak, descends back to zero, falls to a trough, and then returns to zero, completing one cycle. In contrast, the cosine function starts at a peak, decreases to zero, goes to a trough, comes back to zero, and finishes the cycle at the starting peak. By manipulating these basic waves in Lissajous figures, students can visualize how these functions interact when combined at different frequencies and phase shifts.

Amplitude and Frequency Relationship

The amplitude and frequency of a waveform measure how high the peaks are and how rapidly the waves occur, respectively. When dealing with Lissajous figures, understanding the relationship between amplitude and frequency is key to interpreting their patterns. A higher amplitude means a larger wave, which will make the Lissajous figure taller or wider. Changing the frequency, however, results in a more significant number of oscillations in a given time frame, which in turn transforms the simple Lissajous figure into a more complex and intricate pattern. This relationship between amplitude and frequency is pivotal in signal analysis and helps users of oscilloscopes to understand the nature of the electrical signals they observe.

Phase Shifts

Phase shifts indicate a change in the starting point of a wave relative to its standard or reference position. They will cause the characteristic Lissajous figures to rotate about the origin because one wave is 'delayed' in respect to the other. For example, a phase shift of \( 90^\circ \) in one wave, like in the exercise where we had \( y_{2}=\sin(t+90^\circ) \) compared to the reference wave without a phase shift, results in one wave leading or lagging behind the other by a quarter of a cycle. This phenomenon can affect the interpretation of waveforms on an oscilloscope, as it may impact the timing and interaction of signals, causing the standard sine wave to resemble a cosine wave for a \( 90^\circ \) shift.

Oscilloscope Waveform Interpretation

An oscilloscope is a device used to visualize and analyze the waveforms of electrical signals. The Lissajous figures are particularly useful in interpreting these waveforms as they provide insights into the relationship between two periodic signals, such as those from alternating currents. When viewed on an oscilloscope, the figures can tell us whether signals are in phase or out of phase, and by how much. For instance, by examining the shape of the Lissajous figure, we can determine if the signals share the same frequency but differ in phase, or if their amplitudes are different. By understanding these patterns, users of oscilloscopes can extract important information regarding the synchronization and timing of electronic signals, which has practical applications in fields ranging from audio engineering to telecommunications.