Answer :
Final answer:
The first three harmonics of a 0.31 m long violin string with a wave speed of 274.4 m/s have frequencies of 442.58 Hz, 885.16 Hz, and 1327.74 Hz corresponding to the first, second, and third harmonics respectively.
Explanation:
The question asks about the first three harmonics of a note produced on a violin string which is 0.31 meters long with a wave speed of 274.4 m/s. In physics, particularly the study of waves and sound, harmonics are the integer multiples of the fundamental frequency. Since the string is fixed at both ends, it forms standing waves at specific frequencies known as the harmonics.
The first harmonic (fundamental frequency) is where the string vibrates in its simplest mode - half a wavelength fits into the length of the string. For the first harmonic, the wavelength (λ) is twice the length of the string, so λ1 = × 0.31 m = 0.62 m. The frequency can be calculated using the wave speed (v) and wavelength: f = v / λ. For the first harmonic: f1 = 274.4 m/s / 0.62 m = 442.58 Hz.
The second harmonic features a full wavelength within the string length, so λ2 = 0.31 m. The frequency of the second harmonic: f2 = 274.4 m/s / 0.31 m = 885.16 Hz.
The third harmonic will have three halves of a wavelength along the string length, so λ3 = (2/3) × 0.31 m = 0.2067 m. The frequency for the third harmonic: f3 = 274.4 m/s / 0.2067 m = 1327.74 Hz.
Answer:
442 Hz, 882 Hz, 1330 Hz
Explanation:
Given data: length of the string= 0.31 m = 31 cm
speed of the waves = 274.4 M/s or 2740 cm/s
Wavelengths are
62 cm
31 cm and
62 / 3 = 20.67 cm
The frequencies associated with these wavelengths ( which are the 1st, 2nd and 3rd harmonics )
are
27440/ 62 = 442.6 = 442 Hz
27440/31 = 885.2 = 882 Hz
27440 / 20.67 = 1328 = 1330 Hz
