The second harmonic dominates, however, because the rest of the string is free to vibrate. Δ x = 1/2 λ 4 = 16.0 cm Δ x = 2 ( λ 4 /2 ) = 32.0 cm Δ x = 3 ( λ 4 /2 ) = 48.0 cm LEARN MORE REMARKS Placing a finger at the position Δ x = 32.0 cm damps out the fundamental and odd harmonics, but not all the higher even harmonics. The fourth harmonic, of wavelength λ 4 = 1/2 L 0 = 32.0 cm, has three nodes between the endpoints. The distance from nut to node corresponds to half a wavelength. (B) Find the locations of nodes for the second and fourth harmonics. Subtract this length from the original Δ x = L 0 − L = 64.0 cm − 60.3 cm = 3.7 cm length L 0 to find the distance from the nut to the first fret. Substitute L 0 = 0.640 m and f 1 = 329 f 1 = 2 0 v Hz into the equation, finding the wave speed on the string. (A) Find the distance from the nut to the first fret. The distance from the nut to the bridge (below the sound hole) is the lenpth of the string. Note: The nut is a small piece of wood or ebony at, the top of the fret board. Calculate the wavelength, divide it in two, and locate the nodes, which are integral numbers of half-wavelengths from the nut. In part (b) remember that the distance from node to node is half a wavelength. Solve the equation for the new length L, using the new fundamental frequency, and subtract this length from the original length to find the distance from the nut to the first fret. Shortening the string by playing a higher note doesn't affect the wave speed, which depends orily on the tension and linear density of the string (which are unchanged). Where on the guitar string relative to the nut should the finger be lightly placed so as to hear the second harmonic of the high E string? The fourth harmonic? (This is equivalent to finding the location of the nodes in each case.) SIRAIEGY For part (a) use the equation corresponding to the fundamental frequency to find the speed of waves on the string. (Given the width of a finger, pressing too hard will damp out higher hamonics as well.) The fundamental frequency is thereby suppressed, making it possible to hear overtones. The string should be touched, but not depressed against a fret. (a) How far is the fret from the nut? (b) Overtones can be produced on a guitar string by gently placing the index finger in the location of a node of a higher harmonic. a), the string is shortened so that it plays an F note that has a frequency of 349 Hz. When a guitarist presses down so that the string is in contact with the first fret (Fig. PROBLEM The high E string on a certain guitar measures 64.0 cm in length and has a fundamental frequency of 329 Hz. Apply standing-wave concepts to a stringed instrument.
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