A violin string 15.0 cm long and fixed at both ends...

A violin string 15.0 $\mathrm{cm}$ long and fixed at both ends oscillates in its $n=1$ mode. The speed of waves on the string is $250 \mathrm{m} / \mathrm{s},$ and the speed of sound in air is 348 $\mathrm{m} / \mathrm{s} .$ What are the (a) frequency and (b) wavelength of the emitted sound wave?

A violin string 15.0 $\mathrm{cm}$ long and fixed at both ends oscillates in its $n=1$ mode. The speed of waves on the string is $250 \mathrm{m} / \mathrm{s},$ and
the speed of sound in air is 348 $\mathrm{m} / \mathrm{s} .$ What are the (a) frequency and
(b) wavelength of the emitted sound wave?

Key Concepts

Standing Waves on a String

Standing waves occur when waves of the same frequency and amplitude travel in opposite directions, superimposing to create a pattern of nodes and antinodes. On a string fixed at both ends, only specific wavelengths (and hence frequencies) are allowed, leading to discrete modes of vibration. This concept forms the basis for understanding musical instruments like violins, where the length and tension of the string determine its natural frequencies.

Fundamental Mode

The fundamental mode, or first harmonic, is the simplest form of vibration in a fixed string where the wavelength is twice the length of the string. This mode has the lowest possible frequency and serves as the starting point for determining the higher harmonics. The frequency in this mode is calculated by dividing the wave speed on the string by twice its length.

Wave Speed and Frequency Relationship

The relationship between wave speed, frequency, and wavelength is given by the equation v = f?. This fundamental equation applies to all types of waves, including mechanical waves on a string and sound waves in air. It allows one to calculate one variable if the other two are known, making it essential for bridging the properties of vibrations in different media.

Sound Wave Propagation

When a vibrating object, such as a string, perturbs the surrounding medium like air, it produces sound waves that propagate at the speed of sound specific to that medium. Even though the frequency of the emitted sound wave is determined by the vibration source, its wavelength will differ from that on the string because the sound travels at a different speed. This concept explains how a change in medium affects the characteristics (wavelength) of the wave while maintaining the same frequency.

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