Final answer:
The guitar string is vibrating in its fundamental mode with nodes at each end. When the length of the string is reduced to 2/3 L, the fundamental frequency of the string is 329.63 Hz and the wave speed is 253.10 m/s.
Step-by-step explanation:
The guitar string is vibrating in its fundamental mode, which means it is vibrating at its lowest possible frequency. In this mode, there are nodes at each end of the string. The length of the segment that is free to vibrate is 0.384 m.
To calculate the fundamental frequency of the string, we can use the formula:
Frequency (f) = Wave speed (v) / Wavelength (λ)
Since the string is vibrating in its fundamental mode, the wavelength (λ) will be twice the length of the segment:
λ = 2 x 0.384 m = 0.768 m
The wave speed (v) can be calculated using the formula:
Wave speed (v) = Frequency (f) x Wavelength (λ)
Using the given frequencies in the Table 17.8:
High E = 329.63 Hz, B = 246.94 Hz, G = 196.00 Hz, D = 146.83 Hz, A = 110.00 Hz, low E = 82.41 Hz
Calculating the wave speed for each frequency:
High E: v = 329.63 Hz x 0.768 m = 253.10 m/s
B: v = 246.94 Hz x 0.768 m = 189.79 m/s
G: v = 196.00 Hz x 0.768 m = 150.05 m/s
D: v = 146.83 Hz x 0.768 m = 112.79 m/s
A: v = 110.00 Hz x 0.768 m = 84.48 m/s
Low E: v = 82.41 Hz x 0.768 m = 63.28 m/s
Therefore, the fundamental frequency of the string when the length is reduced to 2/3 L would be 329.63 Hz, and the wave speed present would be 253.10 m/s.