Nylon Guitar String A nylon guitar string has a linear...

Nylon Guitar String A nylon guitar string has a linear density of $7.2 \mathrm{~g} / \mathrm{m}$ and is under a tension of $150 \mathrm{~N}$. The fixed supports are $90 \mathrm{~cm}$ apart. The string is oscillating in the standing wave pattern shown in Fig. 17-35. Calculate the (a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.

Nylon Guitar String A nylon guitar string has a linear density of $7.2 \mathrm{~g} / \mathrm{m}$ and is under a tension of $150 \mathrm{~N}$. The fixed supports are $90 \mathrm{~cm}$ apart. The string is oscillating in the standing wave pattern shown in Fig. 17-35. Calculate the
(a) speed, (b) wavelength, and (c) frequency of the traveling waves whose superposition gives this standing wave.

Key Concepts

Frequency Calculation

The frequency of the waves on a string is calculated using the relationship f = v/?, where v is the wave speed and ? is the wavelength. This concept connects the measurable quantities of speed and wavelength to the perceived pitch in musical instruments, demonstrating how physical parameters determine the tone produced by vibrating strings.

Relationship between String Length and Wavelength

For a standing wave on a string with fixed supports, the wavelength is directly related to the length of the string and the number of half-wavelengths that fit into that length. Specifically, if the string resonates in the n-th harmonic, the relationship is L = n(?/2), meaning the wavelength can be determined by the length of the string divided by half the number of humps or segments in the mode.

Wave Speed on a String

This concept involves determining the speed of a transverse wave propagating along a string. The speed is calculated using the formula v = ?(T/?), where T is the tension in the string and ? is the linear mass density. This relationship shows how the physical properties of the string (mass per unit length and applied tension) govern the speed of wave motion.

Standing Wave Patterns

Standing waves are formed by the superposition of two waves traveling in opposite directions along a string. In a string with fixed ends, the boundary conditions force the wave to have nodes at the endpoints. The resulting interference creates a pattern of nodes (points of no displacement) and antinodes (points of maximum displacement), which correspond to specific harmonic modes.

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