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 determine the time it takes for a wave to travel the length of a string, we need to calculate the wave speed using the fundamental frequency and then divide the length of the string by the wave speed.
The wave speed can be determined using the formula:
Wave speed = frequency * wavelength
However, we need to determine the wavelength first. For a string fixed at both ends, the fundamental frequency corresponds to the first harmonic, which has a wavelength equal to twice the length of the string (λ = 2L).
So, we can rewrite the wave speed formula as:
Wave speed = frequency * 2 * length of the string
Plugging in the values given in the question:
Wave speed = 388 Hz * 2 * length of the string
Now, we need to calculate the length of the string. Since it is fixed at both ends, the fundamental frequency corresponds to the first harmonic, where the string appears as a full wavelength. Thus, we can rearrange the formula to solve for the length:
Length of the string = λ / 2
Substituting the wavelength formula (λ = 2L), we get:
Length of the string = 2L / 2 = L
Therefore, the length of the string is equal to L.
Now, let's come back to the wave speed formula:
Wave speed = 388 Hz * 2 * L
Finally, to determine the time it takes for the wave to travel the length of the string, we use the equation:
Time = distance / speed
In this case, the distance is equal to the length of the string (L) and the speed is the wave speed we just calculated.
Time = L / (388 Hz * 2 * L)
Simplifying the equation:
Time = 1 / (388 Hz * 2)
Let's calculate it:
Time = 1 / (776 Hz)
Time = 0.00129 seconds (rounded to 5 decimal places)
Therefore, it takes approximately 0.00129 seconds (or 1.29 milliseconds) for a wave to travel the length of this string.