Question

Consider standing waves in the column of air contained in a pipe of length L = 1.5 m. The speed of sound in the column is vs = 346 m/s.

Each of the standing wave images provided may represent a case for which one or both ends are open. Larger dots indicate higher air pressure in a given area of the column.
20% Part (b) Calculate the wavelength λ3, in meters, for the third harmonic in the pipe with two open ends.
20% Part (c) Calculate the frequency f1, in hertz, for the fundamental harmonic in the pipe with two open ends.
20% Part (d) Select the image from the options provided showing the gas pressure in the fourth mode of a pipe with one open end and one closed end. (The fourth mode is the third excitation above the fundamental.)
20% Part (e) Calculate the frequency f1, in hertz, for the fundamental harmonic in the pipe with one open and one closed end.

Answer 1

Final answer:

For part (b), the wavelength λ3 for the third harmonic in the pipe with two open ends is 0.5 m. For part (c), the frequency f1 for the fundamental harmonic in the pipe with two open ends is 115.33 Hz.

Step-by-step explanation:

In a pipe with two open ends, the wavelength (λ) for the nth harmonic is given by the formula λn = (2L)/n, where L is the length of the pipe and n is the harmonic number. For the third harmonic (n = 3), the wavelength λ3 is calculated as λ3 = (2 * 1.5 m)/3 = 1.0 m. This corresponds to the distance between consecutive nodes or antinodes in the pipe.

The frequency (f) for a standing wave in a pipe is related to the speed of sound (vs) and the wavelength by the equation f = vs/λ. Using this formula, the frequency f1 for the fundamental harmonic (n = 1) is found by substituting λ1 into the equation: f1 = 346 m/s / 3 m = 115.33 Hz.

The explanation for part (d) involves analyzing the gas pressure in the fourth mode of a pipe with one open end and one closed end. However, since the image is not provided, detailed analysis cannot be performed without visual information.

For part (e), the frequency of the fundamental harmonic in a pipe with one open and one closed end is determined using the formula f1 = vs/(4L), where L is the length of the pipe. Given L = 1.5 m, the frequency is calculated as f1 = 346 m/s / (4 * 1.5 m) = 57.67 Hz.

Answer 2

The third harmonic wavelength for a pipe with two open ends is 1.0 m, the fundamental frequency with two open ends is 115 Hz, the fourth mode of a pipe with one open end and one closed end will display three nodes and four antinodes, and the fundamental frequency for a pipe with one open end and one closed end is 57.7 Hz.

Step-by-step explanation:

To calculate the wavelength λ3 for the third harmonic in a pipe with two open ends, we use the fact that an open-ended pipe has resonant wavelengths that are integral multiples of twice the length of the pipe. Therefore, for the third harmonic, the wavelength is calculated by λ3 = 2L/3. With a pipe length L = 1.5 m, we find λ3 = 2 × 1.5 m / 3 = 1.0 m.

The frequency f1 for the fundamental harmonic in a pipe with two open ends is f1 = vs / λ1, where λ1 is twice the length of the pipe, since the fundamental wavelength is 2L. Hence, f1 = 346 m/s / (2 × 1.5 m) = 115 Hz.

For the fourth mode of a pipe with one open end and one closed end, the resonance pattern is such that L = (4n-1) × λ/4 for n = 1, 2, 3, ..., which means the fourth mode corresponds to n = 4, giving us three and a quarter wavelengths in the pipe. Therefore, the correct image will show a standing wave pattern with three nodes and four antinodes.

To calculate the frequency f1 for the fundamental harmonic in a pipe with one open and one closed end, we use the formula where the fundamental has λ = 4L. Hence, f1 = vs / λ = 346 m/s / (4 × 1.5 m) = 57.7 Hz.