Question

The second harmonic in sound is given by \(f(x)=\sin 2 x,\) while the third harmonic is given by \(f(x)=\sin 3 x .\) Sketch the curves and compare the graphs of the second and third harmonics for \(-2 \pi \leq x \leq 2 \pi\)

Short answer

The second harmonic \(f(x) = \sin 2x\) has 2 cycles and the third harmonic \(f(x) = \sin 3x\) has 3 cycles from \(-2\pi\) to \(2\pi\).

Step-by-step solution

Understanding the Functions

The second harmonic is given by the function \(f(x) = \sin 2x\), and the third harmonic is provided by \(f(x) = \sin 3x\). These functions represent sine waves with different frequencies.

Determine the Domain

The interval given is \(-2\pi \leq x \leq 2\pi\). This means that we need to sketch the graphs for values of \(x\) ranging from \(-2\pi\) to \(2\pi\).

Sketching the Second Harmonic

For \(f(x) = \sin 2x\), the sine wave completes one cycle from 0 to \(2\pi\). Within the interval \(-2\pi\) to \(2\pi\), the function will complete 2 cycles. Sketch the graph of \(\sin 2x\) showing these cycles.

Sketching the Third Harmonic

For \(f(x) = \sin 3x\), the sine wave completes one cycle from 0 to \(2\pi\) but it will cover 3 cycles in the interval \(-2\pi\) to \(2\pi\). Sketch the graph of \(\sin 3x\) showing these 3 cycles.

Compare the Graphs

Compare the graphs by observing the frequency and amplitude. The second harmonic has 2 cycles in the given interval, while the third harmonic has 3 cycles. Both functions maintain an amplitude of 1 but differ in the frequency of oscillations.

Key concepts

second harmonic

The second harmonic refers to the trigonometric function given by the equation \( f(x) = \sin 2x \). This harmonic is called the 'second' because it oscillates at twice the frequency of the fundamental sine wave \( \sin x \).
The sine wave function repeats itself every \( 2\pi \) radians.
Let's break this down further:

• When we say \( \sin 2x \), we're indicating that the wave will complete two full cycles in the interval between 0 and \( 2\pi \).
• This means, on the interval from \( -2\pi \) to \( 2\pi \), you'll see four complete cycles.

By plotting \( \sin 2x \) from \( -2\pi \) to \( 2\pi \), you'll observe these repeated oscillations that double the frequency of the basic sine function, making it a crucial concept in understanding harmonic motion.

third harmonic

The third harmonic is a trigonometric function represented by \( f(x) = \sin 3x \). This function oscillates at three times the frequency of the fundamental sine wave, \( \sin x \).
Here's how it plays out:

• With \( \sin 3x \), the function completes three full cycles within the interval from 0 to \( 2\pi \).
• Thus, for the interval \( -2\pi \leq x \leq 2\pi \), you'll observe six complete cycles of the wave.

When you plot the graph of \( \sin 3x \), you'll see an increase in frequency compared to both \( \sin x \) and \( \sin 2x \).
The third harmonic, therefore, provides a higher frequency oscillation that is essential for analyzing more complex wave patterns in various fields such as acoustics and signal processing.

trigonometric functions

Trigonometric functions are mathematical functions of an angle and are an essential part of trigonometry.
The most common trigonometric functions include \( \sin \), \( \cos \), and \( \tan \). These functions help describe the relationships between the angles and sides of right-angled triangles.
For waves:

• The sine function, \( \sin x \), is particularly useful because it describes smooth periodic oscillations.
• When frequency of the sine function is altered as in \( \sin 2x \) and \( \sin 3x \), it leads to higher harmonics.

Each harmonic enables a deeper understanding of complex waveforms by breaking them down into simpler sinusoidal components.
Fundamentally, recognizing these trigonometric functions and their harmonics is indispensable for fields like physics, engineering, and even music theory, ensuring a comprehensive grasp of periodic phenomena.