A violinist places her finger so that the vibrating section...

A violinist places her finger so that the vibrating section of a $1.0 \mathrm{g} / \mathrm{m}$ string has a length of $30 \mathrm{cm},$ then she draws her bow across it. A listener nearby in a $20^{\circ} \mathrm{C}$ room hears a note with a wavelength of $40 \mathrm{cm} .$ What is the tension in the string?

Key Concepts

Wave Propagation on a Stretched String

This concept refers to the idea that waves travel along a string with a speed that is determined by the tension in the string and its linear mass density. The relationship v = ?(T/?) shows that increasing the tension or decreasing the mass per unit length will result in a higher wave speed. This fundamental formula is a cornerstone in understanding how vibrations propagate in stretched strings.

Standing Waves and Harmonics

When a string is fixed at both ends, the waves that reflect from the boundaries can interfere constructively, creating standing waves. The lowest mode, or fundamental frequency, occurs when the length of the vibrating portion of the string is half of the wavelength (? = 2L). Higher harmonic modes correspond to integer multiples of this fundamental frequency. This concept is essential in explaining the vibration patterns in musical instruments like violins.

Frequency, Wavelength, and Wave Speed Relationship

The relationship f = v/? connects the frequency, wavelength, and speed of a wave through any medium. In musical acoustics, this equation is used to relate the frequency of the sound produced by a vibrating string (through its standing wave pattern) to the wavelength of the sound in air. This concept provides the bridge between the physical properties of the string and the tones heard by the listener.

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