Problem 22 - College Physics

Question

\(\bullet\) Guitar string. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) frequency 245 Hz \()\) when vibrating in its fundamental mode. (a) Find the speed of transverse waves on this string. (b) If the tension in this string is increased by \(1.0 \%,\) what will be the new fundamental frequency of the string? (c) If the speed of sound in the surrounding air is 344 \(\mathrm{m} / \mathrm{s}\) , find the frequency and wavelength of the sound wave produced in the air by the vibration of the \(\mathrm{B}_{3}\) string. How do these compare to the frequency and wavelength of the standing wave on the string?

Step-by-Step Solution

Answer

(a) 311.65 m/s. (b) 246.22 Hz. (c) Frequency in air: 245 Hz; wavelength in air: 1.404 m. String wavelength: 1.27 m.

Step 1: Convert Length to Meters

First, we must convert the string's length from centimeters to meters. The given length is 63.5 cm. We know that 1 meter is equal to 100 centimeters. Therefore, the length of the string in meters is:\[ 63.5 \text{ cm} = \frac{63.5}{100} \text{ m} = 0.635 \text{ m} \]

Step 2: Calculate Speed of Waves on String (Part a)

The speed of waves on a string can be found using the equation for the fundamental frequency of a string vibrating in its fundamental mode:\[ v = 2 \times L \times f \]where \( v \) is the speed of the wave, \( L \) is the length of the string, and \( f \) is the frequency. Given \( L = 0.635 \text{ m} \) and \( f = 245 \text{ Hz} \), substitute the values:\[ v = 2 \times 0.635 \times 245 = 311.65 \text{ m/s} \]

Step 3: Calculate New Frequency with Increased Tension (Part b)

When the tension of a string is increased, the speed of the wave increases as well, according to:\[ v = \sqrt{\frac{T}{\mu}} \]If tension \( T \) is increased by 1%, the new tension \( T' \) is given by:\[ T' = 1.01 \times T \]The new speed \( v' \) is:\[ v' = \sqrt{\frac{1.01 T}{\mu}} = \sqrt{1.01} \times v \]Thus, the new frequency \( f' \) is:\[ f' = \frac{v'}{2L} = \sqrt{1.01} \times \frac{v}{2L} = f \times \sqrt{1.01} \]Calculate:\[ f' = 245 \times \sqrt{1.01} = 245 \times 1.004987 = 246.22 \text{ Hz} \]

Step 4: Calculate Sound Wave Properties in Air (Part c)

When the string vibrates as \( B_{3} \), it generates sound waves in the air with the same frequency as the standing wave on the string. The frequency in the air is thus \( f = 245 \text{ Hz} \).To find the wavelength \( \lambda \) of the sound wave in air, use:\[ v_{\text{air}} = f \times \lambda \]where \( v_{\text{air}} = 344 \text{ m/s} \). Solve for \( \lambda \):\[ \lambda = \frac{v_{\text{air}}}{f} = \frac{344}{245} \approx 1.404 \text{ m} \]

Step 5: Compare Frequency and Wavelengths

The frequency of the sound wave in air is 245 Hz, the same as the standing wave's frequency on the string. However, the wavelength differs. On the string, the wavelength of the standing wave for fundamental mode is twice the length of the string:\[ \lambda_{\text{string}} = 2L = 1.27 \text{ m} \]In comparison, the wavelength in the air is approximately 1.404 m.

Key Concepts

Guitar String Frequency

The frequency of a guitar string is a fundamental concept in understanding sound production in string instruments. When a guitar string vibrates in its fundamental mode, it creates a specific note characterized by its frequency. For instance, a string vibrating at 245 Hz produces the note B3. This frequency is intrinsic to the string's physical properties and determines the pitch of the note we hear.

In musical terms, frequency refers to how often the string vibrates per second, measured in Hertz (Hz). The higher the frequency, the higher the pitch of the sound. Guitarists often tune their strings to specific frequencies to achieve the desired notes.

• The relationship between frequency and pitch is direct; higher frequency creates a higher pitch sound.
• The specific note produced by a string in its fundamental frequency is unique to that string's length, tension, and material.

Understanding this relationship is crucial for musicians to ensure their instruments are tuned correctly.

Speed of Sound

The speed of sound refers to how fast sound waves travel through a medium, such as air. When a guitar string vibrates, it creates sound waves that propagate through the surrounding air at a given speed. In a typical environment, the speed of sound in air is approximately 344 m/s. This value can be influenced by factors such as temperature and humidity.

When considering the relationship between frequency and wavelength, the formula for the speed of sound is helpful:\( v = f \cdot \lambda \)where:

• \( v \) is the speed of sound
• \( f \) is the frequency
• \( \lambda \) is the wavelength

This relationship implies that for a fixed speed of sound, as the frequency increases, the wavelength decreases, and vice versa.

Standing Wave on String

A standing wave is formed on a guitar string when it vibrates between its fixed endpoints - at the bridge and the nut. This creates a pattern of nodes and antinodes on the string. In its simplest form, also called the fundamental mode or first harmonic, the string vibrates as a single segment between these two points.

The fundamental wavelength of the wave on the string is twice the length of the string, as described by this formula:\[ \lambda = 2L \]

• Here, \( L \) is the length of the vibrating part of the string.

This condition of standing waves results in nodes at each end, where the displacement is zero, and an antinode in the center, where the displacement is maximum.

This phenomenon is essential for producing clear and consistent notes, as it determines the characteristic tone quality of the instrument.

Tension Effect on Frequency

Increasing the tension in a guitar string causes an increase in the frequency of the vibration, which raises the pitch of the sound the string produces. This effect is due to the relationship between tension and wave speed:

\[ v = \sqrt{\frac{T}{\mu}} \]

where:

• \( T \) is the tension in the string
• \( \mu \) is the mass per unit length of the string

When tension increases by a factor, say 1%, the speed of the wave increases slightly, as observed from multiplying the original speed by the square root of the tension change factor. This change in speed results in a new frequency:\[ f' = f \times \sqrt{1.01} \]Understanding this relationship is crucial for tuning instruments, as simply adjusting string tension allows the musician to achieve the desired note accurately.