Final answer:
The wavelength of the standing wave is 1.60 m, and its speed is 640 m/s on an 80 cm long guitar string with a fundamental frequency of 400 Hz. When the string length is altered to 2/3 of the original length, the new fundamental frequency becomes approximately 601.13 Hz, while the wave speed remains 640 m/s.
Step-by-step explanation:
To calculate both the wavelength and the speed of the standing wave in an 80 cm long guitar string that has a fundamental frequency of 400 Hz:
a) Wavelength: In the fundamental frequency, the wavelength is twice the length of the string since it forms one loop or antinode. Thus, λ = 2 * L = 2 * 0.80 m = 1.60 m.
b) Speed of the standing wave: The wave speed can be found using the formula Vw = fλ, where f is the frequency and λ is the wavelength. So, Vw = 400 Hz * 1.60 m = 640 m/s.
If the string length is reduced to 2/3 L:
- The new string length is L' = 2/3 * 0.80 m = 0.533 m.
- Fundamental frequency of string at new length f' is found by creating a ratio between the original and new frequencies based on the inversely proportional relationship of length and frequency: f' = f * L/L'. Therefore, f' = 400 Hz * 0.80/0.533 = 601.13 Hz approximately.
- The wave speed remains constant in this scenario because the tension in the string and the string’s mass per unit length (linear density) are unchanged. Thus, the wave speed is still 640 m/s.