Question

A standing wave in a pipe with both ends open has a frequency of 440. Hz. The next higher overtone has a frequency of 660. Hz. a) Determine the fundamental frequency. b) How long is the pipe?

Short answer

a) The fundamental frequency is 220 Hz. b) The length of the pipe is approximately 0.7727 meters.

Step-by-step solution

Determine the difference in frequencies

The difference between the next higher overtone and the provided frequency can tell us the number of overtones separating them. Calculate the difference in frequencies: Difference in frequencies = Next higher overtone frequency - Given frequency = 660 Hz - 440 Hz = 220 Hz

Find the number of overtones

Now, we need to find the number of overtones separating the given frequency (440 Hz) and the next higher overtone (660 Hz). Since the difference in frequencies is 220 Hz, and the fundamental frequency must be a divisor of this difference, it is safe to assume that the fundamental frequency might be 220 Hz itself.

Calculate the fundamental frequency

Based on the assumption in step 2, we can now calculate the fundamental frequency (f1): Fundamental frequency (f1) = Difference in frequencies = 220 Hz

Determine the length of the pipe

To calculate the length of the pipe, we need to use the following formula for a pipe with both ends open: f1 = (v / (2 * L)) Where, f1 = fundamental frequency v = speed of sound in air (approximately 340 m/s) L = length of the pipe From step 3, we know the fundamental frequency (f1) is 220 Hz. Now, we can simply solve for L: L = v / (2 * f1) L = 340 m/s / (2 * 220 Hz) L = 340 m/s / 440 Hz L ≈ 0.7727 m The length of the pipe is approximately 0.7727 meters.