A String Under Tension A string under tension F^tension oscillates in the third harmonic at fre

step 1 So in the first part of this problem, we have to calculate the frequency of the oscillations in

A String Under Tension A string under tension F^tension ...

A String Under Tension A string under tension $F^{\text {tension }}$ oscillates in the third harmonic at frequency $f_{3}$, and the waves on the string have wavelength $\lambda_{3} .$ If the tension is increased to $\tau_{f}=4 \tau_{i}$ and the string is again made to oscillate in the third harmonic, what then are (a) the frequency of oscillation in terms of $f_{3}$ and (b) the wavelength of the waves in terms of $\lambda_{3} ?$

Key Concepts

Wave Speed and Tension Relationship

The speed of a wave on a string is determined by the square root of the tension divided by the linear mass density. This relationship shows that if the tension is increased, the wave speed increases proportionally to the square root of the change in tension. In the context of the problem, increasing the tension by a factor of 4 results in the wave speed doubling, since ?4 = 2.

Harmonic Modes on a Stretched String

A string fixed at both ends supports standing waves at discrete frequencies, known as harmonics. The harmonics represent the natural modes of vibration where the string length accommodates an integer number of half-wavelengths. For any given harmonic, the wavelength is determined solely by the length of the string and the mode number, and does not change when only the tension is varied. Therefore, when the tension is increased, while the wave speed and consequently the frequency change, the wavelength for a specific harmonic (e.g., the third harmonic) remains constant.

Frequency-Wavelength Relationship for Waves

The frequency of a wave is related to its propagation speed and wavelength by the relation f = v/?. With an increase in the tension leading to a higher wave speed, even though the harmonic mode (and thus the wavelength) remains unchanged for a fixed string length, the frequency increases. In this problem, doubling the wave speed in the third harmonic yields a frequency that is twice the original frequency while the wavelength remains the same.

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