A steel tube, which is closed at both ends, is filled with air. The air in the tube is vibrating at a frequency of 200·Hz. Use 340·m/s for the speed of sound in air.

If the 200·Hz vibration respresents the fundamental standing wave for the tube, how long is the tube?

nm figured it out

To determine the length of the tube, we need to consider the relationship between the wavelength, frequency, and speed of sound in air. The speed of sound in air is given as 340 m/s.

In a closed tube, the fundamental frequency, or the first harmonic, is generated by a half-wavelength standing wave pattern. This means that there is a node (a point of minimum displacement) at each end of the tube. In this case, the tube is closed at both ends.

The formula for the frequency of a standing wave mode is given by:
f = (n * v) / (2 * L)

Where:
- f is the frequency of vibration (200 Hz in this case)
- v is the speed of sound in air (340 m/s in this case)
- n is the harmonic number (in this case, n = 1 for the fundamental frequency)
- L is the length of the tube (unknown)

Rearranging the formula, we can solve for L:
L = (n * v) / (2 * f)

Substituting the given values into the formula, we have:
L = (1 * 340 m/s) / (2 * 200 Hz)

Simplifying the equation:
L = 1.7 m

Therefore, the length of the tube is 1.7 meters.