Question

Draw a graph over a whole period of each of the following combinations of a fundamental musical tone and some of its overtones: $$ \sin t-\frac{1}{9} \sin 3 t $$

Short answer

Plot the function \(\sin t - \frac{1}{9} \sin 3t\) from \(\0 \leq t \leq 2\pi\) and ensure it repeats every \(\2\pi\).

Step-by-step solution

- Identify the Components

Recognize that the function to be graphed is a combination of sine waves: o \(\sin t\) represents the fundamental toneo \(\frac{1}{9} \sin 3t\) represents an overtone

- Determine the Period

The periods of the individual components must be considered:o The period of \(\sin t\) is \(\2\pi\)o The period of \(\sin 3t\) is \(\frac{2\pi}{3}\)The overall period of the combined function is the least common multiple (LCM) of these two periods, which is \(\2\pi\).

- Create a Table of Values

Calculate the values of the function for points within one period (from \(\0\) to \(\2\pi\)). Sample points typically include integer multiples of \(\pi/6\):o \(\t = 0\), \(\sin(0) - \frac{1}{9} \sin(0) = 0\)o \(\t = \pi/6\), \(\sin(\frac{\pi}{6}) - \frac{1}{9}\sin(3\frac{\pi}{6}) = \frac{1}{2} - \frac{1}{9}\frac{1}{2} = \frac{1}{2} - \frac{1}{6}\)o Continue this across multiple points up to \(\2\pi\)

- Plot the Points

Using graphing paper or graphing software, plot the points calculated in the table of values from step 3. Each point will have coordinates \(\t, y = \sin t - \frac{1}{9} \sin 3t\).

- Connect the Points

Once the points are plotted, connect them smoothly to display the waveform of the combined function across one period, \(\0 \leq t \leq 2\pi\).

- Verify the Graph

Ensure that the periodicity and amplitude align with the expected characteristics:o The function should repeat after \(\2\pi\)o Peaks and troughs should match theoretical values at certain points like \(\pi, \frac{3\pi}{2}\), and \(\2\pi\)

Key concepts

sine wave

A sine wave is one of the simplest and most important types of waveforms that occur in trigonometry and signal processing. It represents a smooth, periodic oscillation that is continuous and symmetric.
A typical sine wave can be described using the function \(\text{y = sin(t)}\), where \(t\) is time or the angle in radians.
Sine waves have several key characteristics:

• Amplitude: This is the maximum value the wave achieves from its central axis, often denoted as A in the function \(y = A \sin(t)\).
• Period: The duration it takes for the sine wave to complete one full cycle. For \(\text{y = sin(t)}\), the period is \(2\pi\) radians. After completing a cycle, the wave starts to repeat itself.
• Frequency: This is the number of cycles the wave completes in a given unit of time. It is the reciprocal of the period.
• Phase Shift: This indicates a horizontal shift along the x-axis, changing where the wave starts its cycle.

The sine function's waveform is crucial in physics, engineering, and music because of its periodic nature.

periodic functions

Periodic functions are those that repeat their values at regular intervals or periods. This characteristic is fundamental in many realms, including electronics, sound, and light.
The function \(\text{f(t)}\) is periodic if there exists a positive number \(P\) such that \(\text{f(t + P) = f(t)}\) for all t.
Some features of periodic functions include:

• Period: The smallest positive integer “P” for which the function value repeats.
• Cycle: The interval length over which the function completes one full oscillation.
• Amplitude: For sinusoidal functions, it’s the peak value from the mean position.

In the given problem, we deal with two periodic functions: \(sin t\) and \(\frac{1}{9} \sin 3t\). Though each has different periods: \(2\pi \) and \( \frac{2\pi}{3} \), they can be combined due to their periodic nature. The overall period of the combined function is the Least Common Multiple (LCM) of individual periods, which in this case is \(2\pi \).
This combination results in an interesting waveform that inherits the periodic properties of its components.

overtones

In music and acoustics, overtones are natural resonant frequencies higher than the fundamental tone.
They occur when a musical instrument produces a sound, creating a complex waveform that's a combination of the fundamental frequency and its harmonics or overtones.

• Fundamental Tone: The lowest frequency produced by an instrument, perceived as the primary note.
• Overtones: Higher frequencies produced simultaneously, which contribute to the instrument’s overall sound.
• Harmonics: Specific overtones that are integer multiples of the fundamental frequency. For instance, if the fundamental frequency is \(f\), its first harmonic is \(2f\), its second harmonic is \(3f\), and so on.

In the exercise given, \(\text{sin t}\) represents the fundamental tone, whereas \(\text{\frac{1}{9} sin 3t}\) is an overtone. The combination of these waves gives a more complex waveform because of their interaction.
Not only does it affect the sound quality, creating what musicians call a “timbre,” but it also results in a signal with different peaks and troughs than a simple sine wave.
Such interactions demonstrate the inherently rich and vibrant nature of sound produced by musical instruments.