Problem 47 - Fundamentals of Physics

Question

One of the harmonic frequencies for a particular string under tension is \(325 \mathrm{~Hz}\). The next higher harmonic frequency is \(390 \mathrm{~Hz}\). What harmonic frequency is next higher after the harmonic frequency \(195 \mathrm{~Hz}\) ?

Step-by-Step Solution

Answer

The next higher harmonic frequency after 195 Hz is 260 Hz.

Step 1: Identify the Harmonic Numbers

The problem provides two consecutive harmonic frequencies: \(325 \mathrm{~Hz}\) and \(390 \mathrm{~Hz}\). To find the harmonic numbers \(n\) and \(n+1\) corresponding to these frequencies, we use the formula for harmonic frequencies \(f_n = n \cdot f_1\), where \(f_1\) is the fundamental frequency.

Step 2: Determine Frequency Difference

Calculate the difference between the two given frequencies: \(390 \mathrm{~Hz} - 325 \mathrm{~Hz} = 65 \mathrm{~Hz}\). This difference equals the fundamental frequency \(f_1\), since it is the difference between consecutive harmonics \((f_{n+1} - f_n = f_1)\).

Step 3: Calculate Fundamental Frequency

From Step 2, we established that \(f_1 = 65 \mathrm{~Hz}\).

Step 4: Calculate Harmonic Number for 195 Hz

To find what harmonic number corresponds to \(195 \mathrm{~Hz}\), use \(f_n = n \times 65\). Solving \(n = \frac{195}{65} = 3\). Thus, \(195 \mathrm{~Hz}\) is the 3rd harmonic.

Step 5: Find the Next Higher Harmonic after 195 Hz

Since \(195 \mathrm{~Hz}\) is the 3rd harmonic, the next higher harmonic is the 4th harmonic. Calculate the 4th harmonic: \(f_4 = 4 \times 65 = 260 \mathrm{~Hz}\).

Key Concepts

Fundamental Frequency

The fundamental frequency, often denoted as \( f_1 \), is the lowest frequency produced by a vibrating string or instrument. It serves as the base frequency upon which other harmonic frequencies build. Like a foundation in a building, the fundamental frequency determines the overall structure or sound profile of the object.

The equation used to calculate any harmonic frequency is \( f_n = n \cdot f_1 \), where \( n \) represents the harmonic number. This means each harmonic frequency is a multiple of the fundamental frequency:

• First harmonic (fundamental frequency): \( f_1 \)
• Second harmonic: \( 2 \times f_1 \)
• Third harmonic: \( 3 \times f_1 \)

Understanding the fundamental frequency is crucial because it allows for the prediction and analysis of the entire harmonic sequence for a given system, such as a musical string or column of air.

Harmonic Number

Harmonic numbers label the sequence of frequencies generated by the harmonics of a vibrating object. The first harmonic corresponds to the fundamental frequency, while each subsequent harmonic frequency is a multiple of the fundamental frequency.

The harmonic number, \( n \), indicates how many times the fundamental frequency is multiplied to reach a particular harmonic frequency. For example, if \( n = 3 \), the frequency is the third harmonic, calculated with \( f_3 = 3 \times f_1 \). This concept allows us to

• Identify the position of a frequency within the harmonic series
• Determine the sequence of available harmonic frequencies

To find the next higher harmonic after a given frequency, simply increment the harmonic number by 1 and multiply it by the fundamental frequency. This rhythmic increase in frequency defines the sequence and allows musicians and engineers to thoroughly understand and manipulate sound.

Frequency Difference

The difference in frequency between consecutive harmonics provides valuable insights into the properties of the system. Often, this difference is precisely equal to the fundamental frequency itself.

When you have two consecutive harmonic frequencies, you can calculate this difference by subtracting the lower frequency from the higher one: \( f_{n+1} - f_n \). In a practical example:

• Given two harmonic frequencies: \( 325 \) Hz and \( 390 \) Hz
• The frequency difference is: \( 390 \text{ Hz} - 325 \text{ Hz} = 65 \text{ Hz} \)

This difference tells us that \( 65 \text{ Hz} \) is the fundamental frequency, reaffirming how it guides the spacing of all subsequent harmonics. Recognizing these differences can allow musicians to tune instruments accurately or acousticians to adjust for sound quality and clarity in various environments.