The fundamental frequency of a string fixed at both ends is 388 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, we need to know the speed of the wave. The speed of a wave on a string is determined by the tension in the string and its linear density.
The formula for the speed of a wave on a string is:
v = sqrt(Tension / linear density)
where v is the wave speed, Tension is the tension in the string, and linear density is the mass per unit length of the string.
In this case, we are given the fundamental frequency of the string, which is the frequency of the lowest mode of vibration. The fundamental frequency is related to the wave speed and the length of the string by the formula:
f = (v / 2L)
where f is the fundamental frequency and L is the length of the string.
Rearranging this formula, we can solve for the wave speed:
v = 2Lf
Now, we can substitute this expression for the wave speed into the formula for the time it takes for a wave to travel the length of the string:
t = L / v
Let's substitute the given values into these formulas:
Fundamental frequency (f) = 388 Hz
Length of the string (L) = ?
To find the length of the string, we need to solve for L in the formula for the fundamental frequency.
f = (v / 2L)
Rearranging the formula, we get:
L = v / (2f)
Let's substitute the given values for the fundamental frequency and solve for L:
L = v / (2f) = (2Lf) / (2f) = v / f = v / (388 Hz)
Since we know the fundamental frequency and the wave speed, we can now calculate the length of the string.
Once you have the value of the length (L), you can use the formula t = L / v to calculate the time it takes for a wave to travel the length of the string.