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Magazine Chapter 11: Problem 51
A particular string resonates in four loops at a frequency of 240 Hz. Give at least three other frequencies at which it will resonate. What is each called?
Short Answer
Expert verified
Resonate frequencies: 120 Hz (first harmonic), 360 Hz (third harmonic), and 480 Hz (fourth harmonic).
Step by step solution
01
Understanding the concept
When a string resonates, it undergoes standing wave formation. Each loop mentioned in the problem corresponds to a half-wavelength. A four-loop standing wave means the string is vibrating at its second harmonic, or the first overtone.
02
Calculate additional resonance frequencies
The fundamental frequency (first harmonic) is half of the frequency at four loops: \( f_1 = \frac{f_4}{2} = \frac{240\, \text{Hz}}{2} = 120\, \text{Hz} \). This is because the fundamental frequency corresponds to the first loop.
03
Calculating higher harmonics
The second and third overtones (third and fourth harmonics) can be calculated by multiplying the fundamental frequency by the respective harmonic number. For the third harmonic (fifth loop), \( f_3 = 3 \times 120\, \text{Hz} = 360\, \text{Hz} \). For the fourth harmonic (sixth loop), \( f_4 = 4 \times 120\, \text{Hz} = 480\, \text{Hz} \).
04
Assignment of names to each harmonic
The fundamental frequency is called the first harmonic. The frequency at 240 Hz in the four-loop scenario is the second harmonic. The next is the third harmonic at 360 Hz, followed by the fourth harmonic at 480 Hz.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Standing Wave
Standing waves are fascinating phenomena that occur when a wave is confined within a certain space and reflects back and forth. In a standing wave, parts of the wave, known as nodes, remain consistently still, while other parts, called antinodes, continue to oscillate between maximum and minimum displacement. This setup is crucial in understanding the behavior of musical instruments, especially strings. The formation of standing waves leads to distinct patterns that are integral in determining the sound produced. For a string fixed at both ends, like a guitar string, the number of loops or antinodes represents half-wavelengths, clarifying the conditions necessary for resonance.
Resonance
Resonance occurs when a system is driven at its natural frequency, leading to large amplitude oscillations. Imagine pushing a swing; if you push it at just the right times, it goes higher with less effort. This is resonance in action. For strings, resonance is achieved when they vibrate at specific frequencies that match their natural harmonics. This results in the formation of standing waves, where the energy efficiently travels back and forth along the string without excessive loss. As resonance sets in, it allows the string to produce clear and lasting notes, a key reason behind the rich sounds of string instruments.
Fundamental Frequency
The fundamental frequency is the lowest frequency at which a string, or any object for that matter, naturally vibrates, also known as the first harmonic. It's the simplest mode of vibration where the string forms just one loop between the fixed ends. This frequency is pivotal because it sets the stage for higher harmonics or overtones. It can be understood as the building block for other resonant frequencies. In musical contexts, the fundamental frequency determines the pitch of the note being played. For our string scenario, it resonates at 120 Hz, which means that 120 complete wave cycles occur each second, creating the first harmonic.
Overtone
Overtones are frequencies higher than the fundamental frequency; they are essentially the next possible standing wave patterns after the fundamental has been established. These are integral to the richness and complexity of sound. The first overtone is also known as the second harmonic, and it occurs at a frequency where two loops or antinodes form on the string. For the given problem, the first overtone, or second harmonic, is at 240 Hz. Subsequent overtones are termed as the third harmonic (360 Hz) and fourth harmonic (480 Hz), contributing further layers to the sound, each progressively adding more energy and complexity to the overall vibration of the string.
String Vibration
String vibration is the process by which strings move to produce sound. When a string is plucked, struck, or otherwise disturbed, it experiences tension and displacement, initiating vibrations that propagate as waves. These waves reflect at the fixed ends, creating standing waves if the frequency matches one of the string鈥檚 natural harmonics. The vibration is characterized by the tension of the string, its length, and mass per unit length, which collectively determine its frequency characteristics. As these factors interact, they define the fundamental frequency and harmonics of the string, which musicians then manipulate to create music by adjusting tension and length.
