A uniform cylindrical steel wire, 55.0 cm long and 1.14 mm...

A uniform cylindrical steel wire, 55.0 cm long and 1.14 mm in diameter, is fixed at both ends. To what tension must it be adjusted so that, when vibrating in its first overtone, it produces the note D-sharp of frequency 311 Hz? Assume that it stretches an insignificant amount. ($Hint:$ See Table 12.1.)

Key Concepts

Standing Waves on a Fixed String

When a string is fixed at both ends, only specific patterns called standing waves can exist due to the boundary conditions. These standing waves arise from the constructive and destructive interference of waves traveling in opposite directions along the string. The allowed modes, or harmonics, are determined by the length of the string and the requirement that the ends remain stationary, resulting in nodes (points of zero amplitude) at both ends.

Harmonics and Overtones

Harmonics are the discrete set of frequencies at which a string can vibrate when it supports standing waves. The fundamental frequency, or first harmonic, is the lowest frequency mode, and the higher modes are integer multiples of this frequency. The first overtone is the second harmonic - it is the next higher frequency in the series, having a frequency that is twice that of the fundamental when the string vibrates in a perfectly idealized mode.

Relationship between Tension, Mass Density, and Wave Speed

The speed at which waves propagate along a string is governed by the tension in the string and its linear mass density (mass per unit length). The wave speed is given by the formula v = ?(T/?), where T is the tension and ? is the linear mass density. This concept is crucial because the tension directly influences the characteristics of the standing waves and therefore the frequencies produced.

Frequency of a Vibrating String

The frequency at which a string vibrates in a particular harmonic is determined by both the wave speed and the mode of vibration. For a string fixed at both ends, the frequency of the nth harmonic is given by f = n/(2L) ?(T/?), where L is the length of the string. This relationship shows how adjustments to tension or the mass per unit length affect the vibrational frequency and is key when tuning strings in musical instruments.

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