Question

Draw a graph over a whole period of each of the following combinations of a fundamental musical tone and some of its overtones: $$ \sin \pi t+\sin 2 \pi t+\frac{1}{3} \sin 3 \pi t $$

Short answer

To draw the combined waveform, sketch each sine function separately and then add them over the period from 0 to 2.

Step-by-step solution

- Identify the Components

First, identify each term in the combination of the fundamental tone and overtones. The given function is \(\sin \pi t + \sin 2 \pi t + \frac{1}{3} \sin 3 \pi t\). These are the fundamental tone and overtones.

- Determine the Period

Determine the period of each sine function. \(\sin \pi t\) has a period \(T_1 = \frac{2\pi}{\pi} = 2\), \(\sin 2\pi t\) has a period \(T_2 = \frac{2\pi}{2\pi} = 1\), and \(\frac{1}{3} \sin 3\pi t\) has a period \(T_3 = \frac{2\pi}{3\pi} = \frac{2}{3}\). The fundamental period is the least common multiple (LCM) of these periods, which is 2.

- Sketch the Individual Components

Sketch each component separately over one period \(T = 2\). Draw \(\sin \pi t \), \(\sin 2\pi t \), and \(\frac{1}{3} \sin 3\pi t \) on the same graph, focusing on one cycle from \(t = 0\) to \(t = 2\).

- Combine the Components

Add the functions graphically to get the combined waveform. At each point in time \(t\), sum the values of \(\sin \pi t \), \(\sin 2\pi t\), and \( \frac{1}{3} \sin 3\pi t\). Plot the resulting points to form the combined waveform.

- Label the Graph

Label important features on the graph such as peaks, troughs, and zero crossings. Make sure to indicate the period on the x-axis, which could be labeled from 0 to 2.

Key concepts

Fundamental Tone

The fundamental tone is the basic frequency of a sound, often called the first harmonic. It defines the pitch of the sound and is the lowest frequency component in any periodic wave. In the given exercise, the fundamental tone is represented by the function \(\sin \pi t\).
When graphing, this tone outlines the pure, unmodified waveform without any additional harmonics. Think of it as the simplest form of the sound wave, providing the base frequency.
Understanding fundamental tones is essential because they serve as the reference point for adding more complex components like overtones.

Overtones

Overtones are the additional harmonic frequencies that overlay the fundamental tone, adding richness and complexity to the sound. In the exercise, these are captured by \(\sin 2\pi t\) and \(\frac{1}{3} \sin 3\pi t\).
- \(\sin 2\pi t\): This is the first overtone, doubling the frequency of the fundamental tone. It contributes to higher-pitched sounds.
- \(\frac{1}{3} \sin 3\pi t\): This second overtone triples the fundamental frequency and is scaled by \(\frac{1}{3}\), slightly mellowing its effect.
By adding these overtones to the fundamental tone, you enrich the waveform, creating a more complex and natural sound. Each overtone plays a crucial role in shaping the final soundwave.

Periodicity

Periodicity refers to the repeating nature of a function over a specific interval called the period. In this context, we deal with the periodicity of sine functions like \(\sin\pi t\), \(\sin 2\pi t\), and \(\frac{1}{3} \sin 3\pi t\).
- The period of \(\sin\pi t\): Calculated as \(\frac{2\pi}{\pi} = 2\).
- The period of \(\sin 2\pi t\): Calculated as \(\frac{2\pi}{2\pi} = 1\).
- The period of \(\frac{1}{3} \sin 3\pi t\): Calculated as \(\frac{2\pi}{3\pi} = \frac{2}{3}\).
The overall period of the combined wave is the least common multiple (LCM) of these individual periods, which is 2. Being aware of periodicity helps in predicting where the wave will repeat, simplifying both analysis and graphing.

Sine Functions

Sine functions describe smooth, periodic oscillations. They are crucial in graphing trigonometric functions because they model natural phenomena like sound waves.
Some important properties of sine functions include:

• The amplitude, which determines the height of the wave from the center line.
• The period, the length of one complete cycle.
• The phase shift, which moves the wave left or right.

For the given function \(\sin \pi t + \sin 2\pi t + \frac{1}{3} \sin 3\pi t\), sine functions play a primary role in defining the waveform's shape. Each sine component oscillates at different frequencies and amplitudes, creating a unique, complex wave when combined.