Question

For each of the following combinations of a fundamental musical tone and some of its overtones, make a computer plot of individual harmonics (all on the same axes) and then a plot of the sum. Note that the sum has the period of the fundamental (Problem 5).

$\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}$

Short answer

The plot of individual harmonics is given below as follows:

Step-by-step solution

Given data

A fundamental musical tone and some of its overtones of harmonics are $2\cos t+\cos 2t$.

Concept of Sinusoidal sound wave

A sinusoid has a specific functional form that is described using the trigonometric cosine function, and we can write the most general sinusoid as:

$\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathbf{\left(}\mathbf{2}\mathit{\pi }\mathbf{\right(}\mathit{f}\mathit{t}\mathbf{+}\mathit{\varphi }\mathbf{\left)}\mathbf{\right)}$

Equate the individual harmonics with different tones

To make a computer plot of individuals harmonics, firstly will equate the individual harmonics with functions f(t) for musical tone and g(t) for overtones and the sum of both the harmonics is equal to the h(t).

$f\left(t\right)=2\cos t$[Musical tone harmonic]

$g\left(t\right)=\cos 2t$[Over tone harmonic]

$h\left(t\right)=f\left(t\right)+g\left(t\right)$[Sum of individuals harmonic]

Plot the individual harmonics

Now, make a computer plot of individual harmonics and there sum are as follows:

The sum has the period of the fundamental tone is cos t .