Problem 63 - University Physics with Modern Physics

Question

CP A person is playing a small flute 10.75 \(\mathrm{cm}\) long, open at one end and closed at the other, near a taut string having a fundamental frequency of 600.0 Hz. If the speed of sound is \(344.0 \mathrm{m} / \mathrm{s},\) for which harmonics of the flute will the string resonate? In each case, which harmonic of the string is in resonance?

Step-by-Step Solution

Answer

The flute's 3rd harmonic (2400 Hz) resonates with the string's 4th harmonic.

Step 1: Identify the Conditions

The flute is a closed pipe; it has one closed end and one open end. The fundamental frequency is the lowest frequency at which the system naturally resonates, and a closed pipe resonates at the odd harmonics (1st, 3rd, 5th, etc.). The speed of sound in air is given as 344.0 m/s.

Step 2: Calculate the Fundamental Frequency of the Flute

The fundamental frequency for a closed pipe is given by the formula \( f_n = \frac{v}{4L} \), where \( v \) is the speed of sound, and \( L \) is the length of the flute. Substituting the given values, \( f_1 = \frac{344.0\, \mathrm{m/s}}{4 \times 0.1075\, \mathrm{m}} = 800.0\, \mathrm{Hz} \).

Step 3: Find Subsequent Harmonics of the Flute

The harmonics for a closed flute will be the odd harmonics: \( f_1 = 800.0 \text{ Hz} \), \( f_3 = 3 \times 800.0 \text{ Hz} = 2400.0 \text{ Hz} \), \( f_5 = 5 \times 800.0 \text{ Hz} = 4000.0 \text{ Hz} \), and so on.

Step 4: Match Flute Harmonics to String Harmonics

The string has a fundamental frequency of 600.0 Hz, and its harmonics will be \( f_{n} = 600 \times n \), where \( n \) is an integer (1st harmonic: 600 Hz, 2nd harmonic: 1200 Hz, etc.). Compare these with the harmonics calculated for the flute.

Step 5: Determine Resonance Conditions

600 Hz (string's 1st harmonic) does not match any flute harmonics. 1200 Hz (string's 2nd harmonic) also does not match. 2400 Hz (string's 4th harmonic) matches the flute's 3rd harmonic (2400 Hz). Continuing, analyze each possible frequency: 3600 Hz (6th harmonic, no match), 4800 Hz (8th harmonic, no match), etc.

Step 6: Draw Conclusions

The string resonates at the 3rd harmonic of the flute (2400 Hz), which corresponds to the 4th harmonic of the string. Therefore, the only harmonic where both resonate is when the flute plays its 3rd harmonic and the string plays its 4th harmonic.

Key Concepts

Closed Pipe Resonance

The concept of closed pipe resonance is fascinating. In musical instruments like flutes, closed pipe resonance occurs because one end of the pipe is open and the other is closed. This creates unique standing wave patterns, causing the pipe to resonate at specific frequencies. Unlike open pipes, which resonate at all harmonics (1st, 2nd, 3rd, etc.), closed pipes resonate only at odd harmonics (1st, 3rd, 5th, etc.).
Understanding this concept is fundamental when dealing with musical instruments and acoustics. The length of the pipe and the speed of sound play crucial roles in determining the resonance frequencies. Specifically, a closed pipe resonates at a fundamental frequency that can be calculated using the formula:
\[ f_n = \frac{v}{4L} \]
Where \( f_n \) is the fundamental frequency, \( v \) is the speed of sound, and \( L \) is the length of the pipe. By employing this formula, you can determine the specific resonant frequencies of a closed pipe, ensuring you know which harmonics are produced.

Fundamental Frequency

The fundamental frequency is the lowest frequency at which a system naturally resonates. It is the cornerstone of understanding how different systems, including strings and pipes, produce tones.
In the context of the exercise, the fundamental frequency of a closed pipe, such as our flute example, is determined using the formula mentioned earlier. For a string, like the one in the example given, the fundamental frequency is the first harmonic, calculated by the equation:
\[ f_1 = \frac{v}{2L} \]
Where \( v \) is the wave speed on the string and \( L \) its length. However, in acoustical situations using sound waves, like in the problem, the speed of sound in air (344 m/s) is considered instead.
Knowing the fundamental frequency is essential because it acts as the building block for all other harmonics in the system. Once the fundamental is determined, subsequent harmonics are simple multiples of this frequency. For strings, these harmonics are integral multiples, whereas for closed pipes, only odd multiples form the harmonics.

Speed of Sound

The speed of sound is a critical factor when dealing with resonating systems like pipes and strings. It is defined as the speed at which sound waves travel through a medium. At room temperature, the speed of sound in air is approximately 344 m/s. This velocity can change with conditions like temperature and pressure.
In our exercise focusing on flutes and strings, the speed of sound is necessary to calculate the fundamental frequency and harmonics. For example, a closed pipe's resonant frequencies depend significantly on this speed. The formula for calculating the pipe's fundamental frequency and harmonics utilizes the speed of sound divided by four times the pipe length.
It's important to note that while the speed of sound in air is standard, different mediums will change the speed. Water, for example, transmits sound much faster than air. So, when practicing these calculations, always ensure you're using the correct speed for the medium you're dealing with.

String Vibrations

String vibrations are a classic example of how materials can resonate at specific frequencies. When a string vibrates, it produces waves that travel along its length, and these waves reflect back upon reaching the fixed ends. These reflections establish standing waves.
The frequency at which these standing waves form is the fundamental frequency. This frequency corresponds to the lowest possible energy state of vibration for the string. The string can also vibrate at higher energies, forming harmonic frequencies. Each subsequent harmonic is a multiple of the fundamental frequency. For example, if the fundamental frequency is 600 Hz, the second harmonic is 1200 Hz, the third is 1800 Hz, and so forth.
Stringed instruments, like violins and guitars, use this principle to produce musically pleasing notes. By altering the tension, length, and mass of the string, musicians can adjust the frequency and harmonics produced. Understanding string vibrations and their harmonics is key in acoustics and string instrument design.