91Ó°ÊÓ

In physics, resonant frequencies refer to the natural frequencies at which a system tends to oscillate. Resonance occurs when the frequency of externally applied vibrations matches a system's natural frequency. For a stretched string fixed at both ends, like in this exercise, the resonant frequencies correspond to the harmonics or standing wave patterns that can be sustained by the string.

This exercise provides two resonant frequencies, 315 Hz and 420 Hz, with no intermediate frequencies, suggesting they are consecutive harmonics. In other words, these frequencies are the result of two adjacent standing wave patterns on the string.

Understanding resonant frequencies is crucial, as they appear in various fields ranging from music to engineering. Here, discovering that the difference between these consecutive harmonics is 105 Hz helps identify the fundamental frequency.

The fundamental frequency is the lowest frequency at which a system like a string can naturally vibrate. It is also termed the first harmonic. By calculating the difference between the given consecutive resonant frequencies (420 Hz and 315 Hz), we can find this fundamental frequency.

The difference in frequency between consecutive harmonics is equal to the fundamental frequency. For the string in the exercise, the calculation was: \[f_1 = f_{n+1} - f_n = 420 \, \text{Hz} - 315 \, \text{Hz} = 105 \, \text{Hz}\]
This indicates that the string's fundamental frequency is 105 Hz.

Knowing this value is essential because it provides a basis from which all other harmonic frequencies (like the ones given) can be easily identified by simple multiples.

Wave speed for a vibrating string is an important concept in wave mechanics. It is the rate at which a wave propagates along the string. To determine the wave speed, we use the formula:
\[v = f \, \lambda\]
Where \(f\) is the frequency and \(\lambda\) is the wavelength.

For the fundamental frequency, \(\lambda_1\) is twice the length of the string because, at the fundamental frequency, the waveform completes half a wave. In this exercise, the string length is 0.75 m, making \(\lambda_1 = 2 \times 0.75 = 1.5 \, \text{m}\).

Substituting the known values for the fundamental frequency and wavelength into the equation, we find the wave speed:
\[v = 105 \, \text{Hz} \times 1.5 \, \text{m} = 157.5 \, \text{m/s}\]
This calculation shows how quickly waves travel along the string at the fundamental frequency, providing insight into the string's physical characteristics and the tension it might carry.