·A 1.50 -m-long rope is stretched between two supports with...

$\cdot$ A 1.50 -m-long rope is stretched between two supports with a tension that makes the speed of transverse waves 48.0 $\mathrm{m} / \mathrm{s} .$ What are the wavelength and frequency of (a) the fundamental tone? (b) the second overtone? (c) the fourth harmonic?

Key Concepts

Overtones

Overtones are the vibrational frequencies higher than the fundamental frequency. The first overtone is the second harmonic (since the fundamental is counted as the zeroth overtone), the second overtone is the third harmonic, and so on. They are important in determining the richness of the sound and are directly related to the harmonic series in a fixed string.

Harmonics

Harmonics are the higher frequency modes that occur in integer multiples of the fundamental frequency. Each harmonic corresponds to a standing wave pattern with additional nodes and antinodes. The nth harmonic has a wavelength that is a fraction of the fundamental wavelength, and its frequency is n times the fundamental frequency.

Wave Speed, Frequency, and Wavelength Relationship

The basic relationship between wave speed, frequency, and wavelength is given by the equation v = f?. This equation allows one to calculate the frequency of the wave if the speed and wavelength are known, or vice versa, and is central to analyzing standing waves in strings or ropes.

Standing Waves

Standing waves are the patterns of vibration that occur when waves of equal amplitude and frequency travel in opposite directions and interfere. In strings fixed at both ends, the boundary conditions force the formation of nodes and antinodes, creating discrete vibration modes.

Fundamental Frequency

The fundamental frequency, or the first harmonic, is the lowest frequency mode of vibration. In a string fixed at both ends, this mode has a wavelength that is twice the length of the string, and it represents the simplest standing wave pattern with only one antinode between the endpoints.

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