Problem 62 - College Physics

Question

The fundamental frequency of a stretched string is \(150 \mathrm{~Hz}\). What are the frequencies of (a) the second harmonic and (b) the third harmonic?

Step-by-Step Solution

Answer

Second harmonic: 300 Hz; Third harmonic: 450 Hz.

Step 1: Understand the Question

The fundamental frequency of a stretched string is given as \(150\ \mathrm{Hz}\). We need to find the frequencies of the second and third harmonics. The frequency of each harmonic is a multiple of the fundamental frequency.

Step 2: Identify the Formula

Recall that the frequency of the \(n^{th}\) harmonic is given by \(f_n = n \times f_1\), where \(n\) is the harmonic number and \(f_1\) is the fundamental frequency.

Step 3: Calculate the Second Harmonic

The second harmonic corresponds to \(n = 2\). Use the formula:\[ f_2 = 2 \times f_1 \]Substitute \(f_1 = 150\ \mathrm{Hz}\):\[ f_2 = 2 \times 150 = 300\ \mathrm{Hz} \]

Step 4: Calculate the Third Harmonic

The third harmonic corresponds to \(n = 3\). Apply the same formula:\[ f_3 = 3 \times f_1 \]Substitute \(f_1 = 150\ \mathrm{Hz}\):\[ f_3 = 3 \times 150 = 450\ \mathrm{Hz} \]

Key Concepts

Fundamental Frequency

The fundamental frequency is the lowest natural frequency at which a system like a stretched string vibrates. Think of it as the root or base note when a string is plucked. Understanding this is crucial, as it forms the basis of harmonic frequencies.
The fundamental frequency provides the starting point for calculating harmonics, which are simply multiples of this base frequency. In mathematical terms, if the fundamental frequency is denoted as \(f_1\), then this frequency serves as a building block for other harmonics. For example, if a string has a fundamental frequency of \(150\ \mathrm{Hz}\), it oscillates at this rate when not influenced by overtones.
Additionally, the fundamental frequency depends on several parameters, including the length of the string, its tension, and its linear density. By adjusting these parameters, the pitch produced by the string can be modified. However, regardless of adjustments, the harmonics always remain multiples of the base frequency.

Stretched String

A stretched string is an essential concept when exploring harmonics and wave behavior. When a string is stretched and fixed at both ends, its ability to vibrate at various frequencies is changed.
This vibration is a result of the wave that travels along the string when it is plucked or struck. The string’s fixed points create boundaries where wave reflections occur, generating specific patterns known as standing waves. These standing waves are responsible for the creation of harmonics.
The properties of a stretched string, including its length and tension, significantly influence the sound it produces. For instance:

• Length: A longer string will have a lower fundamental frequency compared to a shorter one of the same tension and mass per unit length.
• Tension: Increasing the tension raises the fundamental frequency, making the string produce a higher pitch.
• Mass per Unit Length: A lighter string will vibrate faster than a heavier one, altering the sound's pitch.

By understanding these properties, we can predict and calculate the vibrations and frequencies one expects from a specific string setup.

Frequency Calculation

Calculating frequencies of harmonics is straightforward once the fundamental frequency is known. Each harmonic is a specific multiple of this fundamental frequency.
Harmonics are essential in the study of sound waves, where understanding their calculation helps in a variety of applications, from music to acoustics. The formula used for finding the frequency of the \(n^{th}\) harmonic is \(f_n = n \times f_1\). Here, \(n\) represents the harmonic number:\

• For the second harmonic: Substitute \(n = 2\) into the formula, resulting in \(f_2 = 2 \times 150 = 300\ \mathrm{Hz}\).
• For the third harmonic: Substitute \(n = 3\) into the formula, resulting in \(f_3 = 3 \times 150 = 450\ \mathrm{Hz}\).

This method shows that harmonics are linear multiples of the fundamental frequency, making frequency prediction straightforward once \(f_1\) is known. This repeatability is what solidifies harmonic study as a foundational topic in understanding wave phenomena.