A particularly beautiful note reaching your ear from a rare...

A particularly beautiful note reaching your ear from a rare Stradivarius violin has a wavelength of $39.1 \mathrm{cm} .$ The room is slightly warm, so the speed of sound is $344 \mathrm{m} / \mathrm{s}$. If the string's linear density is $0.600 \mathrm{g} / \mathrm{m}$ and the tension is $150 \mathrm{N},$ how long is the vibrating section of the violin string?

Key Concepts

Wave Speed on a String

This concept involves determining the speed at which disturbances travel along a string. The speed is calculated using the formula v = ?(T/?), where T is the tension in the string and ? is its linear density. This relationship helps in understanding how changes in the string’s physical properties affect the propagation of waves along it.

Standing Waves on a String

When a string is fixed at both ends, waves traveling along it reflect and interfere to form standing waves. The simplest, or fundamental, mode of vibration occurs when the length of the string corresponds to half the wavelength of the standing wave. This principle is essential for analyzing the pitch produced by stringed instruments, as the vibrating length directly influences the frequency.

Wave Properties of Sound

Sound waves are characterized by their wavelength, frequency, and speed. The relationship ? = v/f links these properties, indicating that the frequency of a sound produced is determined by the wave's speed in the medium and its wavelength. Understanding this relationship is crucial in connecting the characteristics of the sound (such as pitch) with the physical properties of the instrument producing it.

Fundamental Frequency Relationship

In the case of a vibrating string, the fundamental frequency is defined by the length of the string, the speed of the wave on the string, and the boundary conditions (fixed at both ends). The formula f = v/(2L) shows that as the vibrating length L changes, so does the frequency of the note produced. This relationship is central to tuning musical instruments and understanding how different notes are generated.

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