The fundamental frequency of a string fixed at both ends is 325 Hz. How long does it take for a wave to travel the length of this string?
To calculate the time it takes for a wave to travel the length of a string, you need to know the speed of the wave. The speed of a wave on a string depends on its tension (T) and mass per unit length (μ). The equation to calculate the speed (v) of the wave on the string is:
v = sqrt(T/μ)
Given that the string is fixed at both ends, we can assume it is under a high tension and has a uniform mass per unit length. Therefore, the tension (T) and mass per unit length (μ) can be considered constants.
Now, we are given the fundamental frequency of the string, which is 325 Hz. The fundamental frequency (f) of a vibrating string is related to its speed (v) and length (L) by the equation:
f = v / (2L)
By rearranging the equation, we can solve for the length (L):
L = v / (2f)
To find the time it takes for the wave to travel the length of the string, we need to know its length. Plugging in the values we have:
L = v / (2f)
L = sqrt(T/μ) / (2f)
Now, to calculate the time (t) it takes for the wave to travel the length (L) of the string, we can use the equation:
t = L / v
Substituting the expression for L we found earlier:
t = (sqrt(T/μ) / (2f)) / sqrt(T/μ)
t = 1 / (2f)
By plugging in the value of the fundamental frequency (f = 325 Hz), we can calculate the time it takes for the wave to travel the length of the string:
t = 1 / (2 * 325 Hz)
t ≈ 0.001538 s
Therefore, it takes approximately 0.001538 seconds for a wave to travel the length of the given string.