which of the following is true about the possible frequencies of standing waves allowed inside an instrument of a given length (like the pipes of an organ, or the strings on a guitar)?
a. all possible frequencies of standing waves are allowed: the pipes and strings can play any note that we desire as long as we excite them the right way.
b. each length of pipe or string can play a particular frequency of standing wave, known as the fundamental frequency, as well as multiples of that frequency (called the harmonics).
c. each length of pipe or string corresponds to exactly one possible frequency of standing wave. for instance, the E string on a guitar only plays the single E frequency.
The correct answer and explanation is:
The correct answer is b. each length of pipe or string can play a particular frequency of standing wave, known as the fundamental frequency, as well as multiples of that frequency (called the harmonics).
Explanation:
In a musical instrument like an organ pipe or a guitar string, the length of the pipe or string determines the possible frequencies of standing waves that can form. The fundamental frequency, often referred to as the “first harmonic,” is the lowest frequency at which a standing wave can form. This fundamental frequency is determined by the length of the pipe or string, the tension (in the case of strings), and the properties of the material.
For example, on a guitar, the open E string produces a fundamental frequency (the lowest pitch). When you pluck the string, you can excite the string to vibrate in ways that produce not only this fundamental frequency but also higher frequencies, which are multiples of the fundamental. These higher frequencies are called harmonics or overtones.
The harmonics are integer multiples of the fundamental frequency. For instance, the second harmonic is twice the fundamental frequency, the third harmonic is three times the fundamental, and so on. The relationship between the fundamental frequency and its harmonics creates the complex sound that is characteristic of a particular instrument.
The phenomenon of harmonics occurs because the boundaries of the string or pipe (where it is fixed) create points where the wave has zero displacement (called nodes). The points in between, where the displacement is maximum, are called antinodes. The standing wave pattern that forms depends on how the wave fits within the length of the string or pipe. Longer pipes or strings tend to produce lower fundamental frequencies, while shorter pipes or strings produce higher ones. The harmonics provide the richness of sound that gives each instrument its unique timbre.
Thus, the instrument is capable of producing many frequencies, but they are restricted to the fundamental and its harmonics based on the instrument’s length and structure.