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.

(a) What is the wavelength of the sound waves in the tube? _____ m.

(b) If the 200·Hz vibration respresents the fundamental standing wave for the tube, how long is the tube? _____________ m.

(c) If the air in the tube is replaced with a different gas, the frequency of the fundamental standing wave changes to 160·Hz. What is the speed of sound in this new gas? ___________m/s.

To find the answers to the given questions, we can use the formula for wavelength:

Wavelength = Speed of Sound / Frequency

Let's calculate the answers step-by-step:

(a) To find the wavelength of the sound waves in the tube, we need to use the formula:

Wavelength = Speed of Sound / Frequency

Given: Speed of Sound in air = 340 m/s, Frequency = 200 Hz

Wavelength = 340 m/s / 200 Hz
Wavelength = 1.7 m

Therefore, the wavelength of the sound waves in the tube is 1.7 meters.

(b) If the 200 Hz vibration represents the fundamental standing wave for the tube, then the length of the tube can be calculated using the formula:

Length of tube = (1/2) * Wavelength

Given: Wavelength = 1.7 m

Length of tube = (1/2) * 1.7 m
Length of tube = 0.85 m

Therefore, the length of the tube is 0.85 meters.

(c) If the frequency of the fundamental standing wave changes to 160 Hz when the air in the tube is replaced with a different gas, we can find the speed of sound in this new gas using the formula:

Speed of Sound = Wavelength * Frequency

Given: Wavelength = 1.7 m, Frequency = 160 Hz

Speed of Sound = 1.7 m * 160 Hz
Speed of Sound = 272 m/s

Therefore, the speed of sound in the new gas is 272 m/s.

To find the answers to the given questions, we can use the formulas related to wave properties and the characteristics of standing waves in tubes.

(a) The wavelength of a sound wave can be calculated using the formula:

wavelength = speed of sound / frequency

Given that the speed of sound is 340 m/s and the frequency is 200 Hz, we can substitute these values into the formula to find the wavelength:

wavelength = 340 m/s / 200 Hz = 1.7 m

Therefore, the wavelength of the sound waves in the tube is 1.7 meters.

(b) For a tube closed at both ends, the length of the tube corresponds to half of the wavelength of the fundamental standing wave. Therefore, to find the length of the tube, we can use the formula:

length of tube = wavelength / 2

From part (a), we know that the wavelength is 1.7 meters. Substituting this value into the formula, we get:

length of tube = 1.7 m / 2 = 0.85 m

Therefore, the length of the tube is 0.85 meters.

(c) If the frequency of the fundamental standing wave changes to 160 Hz when the air in the tube is replaced with a different gas, we can use the relationship between frequency, wavelength, and the speed of sound to find the speed of sound in the new gas.

Given that the frequency is now 160 Hz, and the wavelength can be derived from the previous parts as 1.7 m, we can rearrange the formula to solve for the speed of sound:

speed of sound = frequency * wavelength

Substituting the values we have:

speed of sound = 160 Hz * 1.7 m = 272 m/s

Therefore, the speed of sound in the new gas is 272 m/s.