Problem 64 - University Physics with Modern Physics

Question

Ep A New Musical Instrument. You have designed a new musical instrument of very simple construction. Your design consists of a metal tube with length \(L\) and diameter \(L / 10 .\) You have stretched a string of mass per unit length \(\mu\) across the open end of the tube. The other end of the tube is closed. To produce the musical effect you're looking for, you want the frequency of the third-harmonic standing wave on the string to be the same as the fundamental frequency for sound waves in the air column in the tube. The speed of sound waves in this air column is \(v_{\mathrm{s}}\) . (a) What must be the tension of the string to produce the desired effect? (b) What happens to the sound produced by the instrument if the tension is changed to twice the value calculated in part (a)? (c) For the tension calculated in part (a), what other harmonics of the string, if any, are in resonance with standing waves in the air column?

Step-by-Step Solution

Answer

(a) Calculate using tension formula. (b) Pitch increases. (c) Ninth harmonic resonates.

Step 1: Understanding Harmonics in a Tube

The metal tube is closed at one end, creating a situation where there are standing waves with one node and one antinode. This means only odd harmonics are present. The fundamental frequency (or first harmonic) for the sound wave is given by \( f_1 = \frac{v_s}{4L} \).

Step 2: Understanding String Harmonics

For a string fixed at both ends, like ours, the frequency of the third harmonic is given by \( f_3 = \frac{3}{2L} \sqrt{\frac{T}{\mu}} \), where \( T \) is the tension, \( L \) is the length, and \( \mu \) is the mass per unit length of the string.

Step 3: Equating Frequencies

We want the third harmonic frequency of the string to equal the fundamental frequency of the tube: \( \frac{3}{2L} \sqrt{\frac{T}{\mu}} = \frac{v_s}{4L} \). Simplifying this gives \( T = \frac{(v_s \cdot 2L)^2 \cdot \mu}{36} \).

Step 4: Effect of Doubling Tension

Doubling the tension leads to a new frequency \( f_3' = \frac{3}{2L} \sqrt{\frac{2T}{\mu}} = \sqrt{2} \cdot f_3 \). This means that the new frequency will be \( \sqrt{2} \times \) the original frequency, leading to a higher pitched sound that is no longer in resonance with the fundamental frequency.

Step 5: Identifying Other Resonant Harmonics

The closed tube supports odd harmonics (1st, 3rd, 5th, ...). Since we tuned the third harmonic of the string to the fundamental of the tube resonator, only the odd harmonics of the tube will align with every third harmonic of the string (_n_ = 3, 9, etc.). Thus, the ninth harmonic of the string might be another harmonic in resonance.

Key Concepts

Musical Instrument Design

Musical instrument design is a creative process that combines understanding of physics with artistry. In this exercise, the design focuses on a metal tube paired with a stretched string, each contributing to the instrument's sound. The metal tube is characterized by a length \( L \) and a diameter of \( L/10 \), serving as a resonant cavity for sound waves. By closing one end of the tube, specific patterns of standing waves are formed, influencing how sound is produced and perceived.
The string plays a critical role, and its properties, such as mass per unit length \( \mu \) and tension \( T \), significantly impact the sound. Tension in the string directly affects the frequency of the standing waves, determining the pitch of the sound produced. Getting the design just right requires understanding how these physical properties interact with waves to create musical notes.

Harmonics

Harmonics are the fundamental building blocks of musical sound. They refer to the different frequencies at which a string or air column can naturally vibrate. Each harmonic represents a distinct frequency pattern. For a tube closed at one end, like in the exercise, only odd harmonics are present.
In a closed tube, the fundamental frequency (first harmonic) is given by \( f_1 = \frac{v_s}{4L} \), where \( v_s \) is the speed of sound and \( L \) is the length of the tube. This equation outlines how the properties of the air column dictate the fundamental pitch.

• Even Harmonics: In this scenario, even harmonics (2nd, 4th, etc.) are absent.
• Odd Harmonics: Frequencies like the 3rd and 5th harmonics are prominent.

The string's third harmonic frequency is represented as \( f_3 = \frac{3}{2L} \sqrt{\frac{T}{\mu}} \). Aligning this with the tube's harmonic creates resonance, harmonizing the two frequencies.

Standing Waves

Standing waves are essential in musical instruments for producing consistent tones. These are waves that remain in a fixed position, creating nodes where there is no movement, and antinodes, where maximum movement occurs.
In a tube closed at one end, a standing wave is characterized by:

One Node: A point of zero amplitude, at the closed end.

One Antinode: A point of maximum amplitude, at the open end.

This setup in the tube supports odd harmonics, which influences the sound pattern significantly. In the exercise, the desired effect is for the wave patterns in the string and tube to "match" at certain frequencies, allowing them to resonate at harmonics that complement each other. This matching is what creates a rich and complete sound.

Frequency Resonance

Frequency resonance in musical instruments is a crucial concept where two systems can oscillate at the same frequency, amplifying sound. For the instrument in question, resonance occurs when the third harmonic of the string coincides with the fundamental frequency of the air column.
Resonance is achieved by adjusting the string tension to exactly match these frequencies. The formula \( T = \frac{(v_s \cdot 2L)^2 \cdot \mu}{36} \) helps calculate the precise tension required. This alignment boosts the sound's volume and quality. When tension is changed, as in part (b) where the tension is doubled, resonance is disrupted. The string's frequency is altered to \( \sqrt{2} \times \) the original frequency, resulting in a mismatch with the tube's resonance, hence changing the instrument's tone.

• Impact: When resonance is lost, the harmonics that once complemented each other become discordant.
• Optimal Performance: Ensuring that resonance is sustained maintains the instrument's intended sound quality.