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

A particularly beautiful note reaching your car 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

Vibrating String’s Fundamental Frequency

For a string fixed at both ends, the lowest (fundamental) mode of vibration has a frequency that depends on the length of the vibrating section as well as the wave speed along the string. This frequency is given by the formula f = (1/(2L)) times the wave speed along the string, where L is the length of the string. This concept bridges the gap between the vibrations of the string and the acoustical properties of the sound produced.

Tension and Linear Density in String Vibration

The wave speed along a stretched string is determined by the tension in the string and its linear mass density, described by the relation v = ?(T/?). This connection indicates that increasing the tension or decreasing the linear density results in a faster wave speed, which in turn affects the vibrational frequency of the string. It is a crucial concept for understanding how physical properties of a string influence sound production.

Wave Speed and Frequency Relationship

A fundamental concept in wave physics is the relationship between speed, frequency, and wavelength, where the speed of a wave is equal to the product of its frequency and wavelength. This relationship is used to connect the properties of sound waves propagating through a medium with those of the source wave, and forms the basis of understanding how the frequency of a sound is determined by its wavelength in a given medium.

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