Question

Draw a graph over a whole period of each of the following combinations of a fundamental musical tone and some of its overtones: $$ 2 \cos t+\cos 2 t $$

Short answer

Plot the function \( 2 \cos(t) + \cos(2t) \) over \([0, 2\pi]\).

Step-by-step solution

Identify the Components

The given combination is \( 2 \cos(t) + \cos(2t) \). Identify \( \cos(t) \) as the fundamental tone and \( \cos(2t) \) as the first overtone (2nd harmonic).

Determine the Period of Each Component

The period of \( \cos(t) \) is \( 2\pi \). The period of \( \cos(2t) \) is \( \frac{2\pi}{2} = \pi \).

Find the Least Common Multiple

To determine the period of the combination, find the least common multiple of the individual periods. The LCM of \(2\pi \) and \(\pi \) is \(2\pi\).

Sketch the Graph

Plot the function \( 2 \cos(t) + \cos(2t) \) over the interval \([0, 2\pi]\). First, graph \( 2 \cos(t) \) and \( \cos(2t) \) separately. Then, add these graphs point by point to get the resultant graph.

Key concepts

fundamental tone

In the world of music and sound waves, the term 'fundamental tone' refers to the lowest frequency produced by a vibrating object, such as a musical instrument. It is often called the first harmonic. In mathematical terms, if we consider the cosine function, the fundamental tone is represented by \(\cos(t)\). For example, in the exercise given, the fundamental tone is \(2 \cos(t)\). This is the base frequency around which the other tones, known as overtones or harmonics, are built.

The fundamental tone determines the pitch of the sound. In simpler terms, it is the main note you hear when a musical instrument is played. All other tones, which are multiples of this base frequency, enrich the sound but do not change its basic pitch.

overtones

Overtones add depth and richness to the fundamental tone. They are higher frequency components that occur naturally when a musical note is played. Mathematically, overtones are represented as harmonics. In our example, \( \cos(2t)\) is considered the first overtone, also known as the second harmonic. It is twice the frequency of the fundamental tone.

Understanding overtones is crucial for graphing trigonometric functions. While the fundamental tone determines the base pitch, overtones add texture and complexity. Combining these tones, such as in the given function \(2 \cos(t) + \cos(2t)\), produces a richer sound wave that requires careful plotting over its entire period to fully understand its behavior.

least common multiple

To graph a combination of trigonometric functions accurately, it is essential to find the Least Common Multiple (LCM) of their periods. The period is the length of one full cycle of the function. For \( \cos(t) \), the period is \(2\pi \). For \( \cos(2t) \), it is \ \pi \.

The LCM helps determine the smallest interval over which the combined function repeats. In the given exercise, the LCM of \( 2\pi \) and \ \pi \ is \ 2\pi \. This means we should graph the function \( 2 \cos(t) + \cos(2t) \) over the interval \[0, 2\pi \]. Looking for the LCM ensures that we capture the complete behavior of the combined function within this interval, which is critical for accurate representation.

Finding the LCM simplifies complex problems, enabling easier computation and more precise graphs.