Problem 49 - University Physics with Modern Physics

Question

Guitar String. One of the 63.5 -cm-long strings of an ordinary guitar is tuned to produce the note \(B_{3}(\) freguency 245 \(\mathrm{Hz})\) 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

Wave speed: 311.15 m/s. New frequency with tension: 246.15 Hz. Sound wavelength: 1.404 m.

Step 1: Identify the Parameters

First, note the given parameters for the string. The length of the string is \( L = 63.5 \) cm, which is equal to \( 0.635 \) m. The frequency of the note \( B_{3} \) is \( 245 \) Hz. The fundamental mode implies one half-wavelength fits onto the string.

Step 2: Calculate the Wave Speed on the String

In the fundamental mode, the wave speed \( v \) on a string is given by the equation: \[ v = f \times \lambda \]where \( f \) is the frequency and \( \lambda \) is the wavelength. For the fundamental frequency, \( \lambda = 2L = 2 \times 0.635 \text{ m} = 1.27 \text{ m} \).Substitute the values to get: \[ v = 245 \times 1.27 = 311.15 \text{ m/s} \].

Step 3: Calculate New Fundamental Frequency with Increased Tension

Tension affects the wave speed, and thus the frequency. If tension increases by 1%, the speed increases by the square root of 1% change.The new wave speed \( v_{new} \) is: \[ v_{new} = v \times \sqrt{1.01} \approx 311.15 \times 1.00498 = 312.69 \text{ m/s} \]The new frequency \( f_{new} \) is: \[ f_{new} = \frac{v_{new}}{\lambda} = \frac{312.69}{1.27} \approx 246.15 \text{ Hz} \].

Step 4: Calculate Frequency and Wavelength in Air

The frequency of sound created in air by the vibrating string is the same as the string's frequency. Therefore, the frequency is \( 245 \text{ Hz} \). Using the speed of sound \( v_{air} = 344 \text{ m/s} \), calculate the wavelength in air: \[ \lambda_{air} = \frac{v_{air}}{f} = \frac{344}{245} \approx 1.404 \text{ m} \].

Step 5: Compare the Standing Wave Values

On the string, the frequency remains the same at \( 245 \text{ Hz} \) until tension is changed, and the wavelength in the fundamental mode is \( 1.27 \text{ m} \).The wavelength of sound in air is \( 1.404 \text{ m} \), which is greater than the wavelength on the string. The frequencies match.

Key Concepts

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a string vibrates. For guitar strings, this occurs when they vibrate in their simplest form, which is known as the fundamental mode. In this mode, there is a single antinode at the midpoint of the string and nodes at both ends.
The frequency of this vibration is determined by the length of the string and the speed of the wave moving along it. For the guitar string in question, this frequency is 245 Hz as indicated by the note B₃.

• The fundamental frequency gives rise to a half-wavelength fitting across the length of the string.
• For a string length of 0.635 m, the wavelength in its fundamental mode becomes twice the length of the string.

This property is key in musical instruments to produce distinct notes by altering string length, tension, or both.

Tension Effects on Wave Speed

Tension in a string is crucial in determining the speed of wave travel. When the tension in a string is increased, the wave speed also increases, allowing the frequency (note) produced by the string to rise.
This relationship is particularly important for musical instruments such as guitars. An increase in tension by 1% in our scenario slightly boosts the wave speed.

• The wave speed initially calculated is 311.15 m/s for the starting tension.
• With an increase in tension, the new wave speed becomes approximately 312.69 m/s.

The underlying physics is that the wave speed on a string is proportional to the square root of the tension: more tension results in quicker propagation of the wave. This change in speed directly influences the frequency, which gives the new fundamental frequency as approximately 246.15 Hz due to the tension increase.

Wavelength in Air vs String

Wavelength, in various media, may differ due to differing wave speeds. For the note produced by the guitar string, the wavelength in air wouldn’t match the string’s wavelength due to differing medium characteristics.
The adjacent air, differing from the string, has a speed of sound of 344 m/s.

• While the wavelength along the string is 1.27 m in its fundamental frequency mode, the wave travels through air as a sound wave.
• In air, the calculated wavelength is longer, around 1.404 m.

This longer wavelength in air still corresponds to the original frequency of 245 Hz, showcasing how speed differences in the medium affect wavelength but not frequency. This shows how strings give rise to sound that travels through air with different characteristics.

Transverse Waves on String

Transverse waves are characterized by particles of the medium moving perpendicular to wave direction. In the context of guitar strings, these waves dictate how notes are produced.
Add a pluck of the string, and a series of oscillations occur. The string’s ends, where it’s fixed, form nodes, and these form standing wave patterns.

• The speed of these waves along the string is tied to both string tension and mass per unit length.
• Understanding wave speed using both these facets allows calculation of fundamental frequencies important for note production.

Such transverse waves allow a string to be tuned to produce precise notes by changing either tension or string length. Consequently, they are foundational to playing and understanding stringed instruments.