Air is vibrating in a tube (closed at both ends) at its fundamental frequency of 160 Hz. If the tube is filled with helium instead, what is its fundamental frequency? Use 1020 m/s for the speed of sound in helium.
v(air) = lambda*fair
where lambda is the wavelength, fair is the frequency in air
v(helium) = lambda*fhelium
v(air)/v(helium) = f(air)/f(helium)
Look up the speed of sound in air and solve for f(helium)
To find the fundamental frequency of the tube filled with helium, we need to use the formula for the speed of sound in a medium:
v = f * λ
where:
v is the speed of sound in the medium (helium in this case),
f is the frequency,
and λ is the wavelength.
Since we know the speed of sound in helium (v = 1020 m/s) and the frequency of the tube with air (f = 160 Hz), we can find the wavelength of the air vibration.
First, rearrange the formula to solve for the wavelength:
λ = v / f
Substituting the values:
λ = 1020 m/s / 160 Hz
λ ≈ 6.375 m
Now we can use the wavelength of air to find the fundamental frequency of helium. The fundamental frequency occurs when the length of the tube is equal to one-half of the wavelength. In other words, the length of the tube corresponds to half of a wavelength for the fundamental frequency.
Therefore, the fundamental frequency of the tube filled with helium can be calculated as:
f = v / λ
Substituting the values:
f = 1020 m/s / (2 * 6.375 m)
f ≈ 80 Hz
So, the fundamental frequency of the tube filled with helium is approximately 80 Hz.