In the fundamental mode of vibration, also known as the first harmonic, a stretched string produces a standing wave pattern with a single antinode in the center and two nodes at the ends. The wavelength (λ) of this standing wave pattern can be determined using the length of the string (L) and the velocity (v) of the wave on the string.
In general, the velocity of a wave on a string depends on the tension (T) in the string and the linear mass density (μ) of the string material. However, since the tension and linear mass density are not provided in your question, I will assume that the string is uniform and under constant tension.
In this case, the velocity of the wave on the string can be approximated using the following formula:
v = √(T/μ)
Assuming the tension and linear mass density remain constant, the velocity can be considered a constant value for a given string.
Since the string is stretched to a length of 2m, and the fundamental mode of vibration has a single antinode in the center, the distance between adjacent nodes or antinodes is equal to half the wavelength. Therefore:
L = (λ/2)
Rearranging the equation, we find:
λ = 2L
Substituting L = 2m, we get:
λ = 2 * 2m = 4m
So, the wavelength of the wave pattern created in the string in the fundamental mode is 4 meters.