Problem 80 - Fundamentals of Physics

Question

When played in a certain manner, the lowest resonant frequency of a certain violin string is concert \(\mathrm{A}(440 \mathrm{~Hz}) .\) What is the frequency of the (a) second and (b) third harmonic of the string?

Step-by-Step Solution

Answer

(a) 880 Hz; (b) 1320 Hz.

Step 1: Understanding Harmonics

In any vibrating string, harmonics are integer multiples of the fundamental frequency. The lowest resonant frequency is the fundamental frequency, often referred to as the first harmonic.

Step 2: Defining the Frequency of the Second Harmonic

The second harmonic is the first overtone of the fundamental frequency. It is calculated as twice the fundamental frequency. So, if the fundamental frequency is 440 Hz, then the second harmonic is: \[2 \times 440 \, \text{Hz} = 880 \, \text{Hz}\]

Step 3: Defining the Frequency of the Third Harmonic

The third harmonic is the second overtone, calculated as three times the fundamental frequency. So, the frequency of the third harmonic will be: \[3 \times 440 \, \text{Hz} = 1320 \, \text{Hz}\]

Step 4: Final Consideration

Analyze if any additional mode frequencies are requested or if a description of overtones is needed, otherwise, conclude the task based on the calculated harmonics.

Key Concepts

Fundamental Frequency

The fundamental frequency, also known as the first harmonic, is the lowest frequency at which a system like a musical instrument can naturally vibrate. In the case of a vibrating string, this is the simplest vibration pattern, often referred to as the "basic" tone. Imagine plucking a violin string; the fundamental frequency is the first and most prominent sound you hear. This frequency is crucial because it sets the pitch of the note. Without it, the sounds we associate with musical notes wouldn't have a consistent pitch.
Factors influencing the fundamental frequency include:

• String tension: Tightening a string raises its frequency.
• Mass per unit length: Heavier strings vibrate at lower frequencies.
• Length of the string: Shorter strings produce higher frequencies.

For example, in our exercise, the fundamental frequency of the violin string is 440 Hz, corresponding to the musical note A.

Overtone

Overtones are frequencies higher than the fundamental frequency. They are also known as harmonics, but it's important to note that the overtone numbering differs slightly from harmonic numbering. While the first harmonic is the fundamental frequency, the first overtone is the second harmonic. When we talk about overtones on a violin string, these are the subsequent frequencies that follow the fundamental. Each overtone represents a different mode of vibration.

• First overtone = second harmonic
• Second overtone = third harmonic

In the provided exercise, finding the harmonics of a string vibrating at a fundamental frequency of 440 Hz, the first overtone (second harmonic) is at 880 Hz and the second overtone (third harmonic) is at 1320 Hz. These overtone frequencies add richness and complexity to the sound of the string.

Vibrating String

A vibrating string serves as a perfect example of simple harmonic waves. This is a common scenario in musical instruments like guitars and violins. When a string vibrates, it does so at various frequencies simultaneously, yet these frequencies are dictated by the string's fundamental characteristics. A string's ability to vibrate stems from:

• The force applied, like plucking or bowing.
• The boundaries of the string which are typically fixed, allowing only specific modes of vibration.
• The physical properties such as tension, length, and mass per unit length.

The interaction between these factors creates a set of harmonics, including the fundamental frequency and overtones. These harmonics combine to create the musical tone we hear. Therefore, understanding vibrating strings is foundational to acoustics and instrument design. When you understand how vibrating strings work, you unlock the science behind the creation of musical notes.