A guitar string with a linear density of 2.0 g / m is...

A guitar string with a linear density of $2.0 \mathrm{g} / \mathrm{m}$ is stretched between supports that are $60 \mathrm{cm}$ apart. The string is observed to form a standing wave with three antinodes when driven at a frequency of $420 \mathrm{Hz}$. What are (a) the frequency of the fifth harmonic of this string and (b) the tension in the string?

Key Concepts

Standing Waves

Standing waves occur when two waves of the same frequency and amplitude travel in opposite directions and interfere, resulting in fixed nodes (points of no displacement) and antinodes (points of maximum displacement). In the context of a string fixed at both ends, standing waves are set up when the wave reflects off the boundaries, and the pattern depends on the wavelength and the length of the string.

Harmonics

Harmonics are the specific frequencies at which a system naturally vibrates. For a string with fixed ends, the harmonics are the modes of vibration where the wavelength fits an integer number of half-wavelengths into the string length. Each harmonic is an integer multiple of the fundamental frequency, thereby defining the different possible vibrational patterns.

Wave Speed on a String

The speed of a wave traveling along a string is determined by the tension in the string and its linear density. This relationship is given by the formula v = ?(T/?), where v is the wave speed, T is the tension, and ? is the mass per unit length (linear density). This concept is crucial for linking the mechanical properties of the string to the observed vibrational frequencies.

Frequency-Wavelength Relationship

The relationship between frequency, wavelength, and wave speed is expressed by the equation f = v/?. In the context of a vibrating string, once the mode of vibration (harmonic) is established, the wavelength can be determined by the string length and the number of half-wavelengths fitting into it, allowing one to compute the frequency if the wave speed is known.

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