One method for measuring the speed of sound uses standing waves. A cylindrical tube is open at both ends, and one end admits sound from a tuning fork. A movable plunger is inserted into the other end at a distance L from the end of the tube where the tuning fork is. For a fixed frequency, the plunger is moved until the smallest value of L is measured that allows a standing wave to be formed. Suppose that the tuning fork produces a 449-Hz tone, and that the smallest value observed for L is 0.202 m. What is the speed of sound in the gas in the tube?
The open end is antinode for the standing wave in air column, the closed end (plunger) is nod, => L= λ/4 => λ = 4•L.
λ=v/f.
4•L = v/f.
v = 4•L• f =4v0.202•449 =362.8 m/s
To find the speed of sound in the gas in the tube, we can use the formula:
v = f * λ
Where:
v is the speed of sound
f is the frequency of the tuning fork
λ is the wavelength of the standing wave.
In this case, the frequency (f) of the tuning fork is given as 449 Hz.
To find the wavelength (λ) of the standing wave, we need to first determine the distance between the two nodes of the standing wave. This can be done by considering the length (L) of the tube and the boundary conditions for an open-open tube.
In an open-open tube, the distance between two adjacent nodes is equal to one-half of the wavelength. So, for the smallest value of L, we can say:
L = λ/2
Rearranging the equation, we get:
λ = 2L
Substituting the value of L (0.202 m) into the equation, we find:
λ = 2 * 0.202 m = 0.404 m
Now, we can substitute the values of f and λ into the formula for the speed of sound:
v = f * λ
v = 449 Hz * 0.404 m
v = 181.546 m/s
Therefore, the speed of sound in the gas in the tube is approximately 181.546 m/s.
To find the speed of sound in the gas in the tube, we can use the formula:
v = f λ,
where v is the speed of sound, f is the frequency, and λ is the wavelength of the sound wave.
In this case, the frequency f is given as 449 Hz. To find the wavelength, we need to determine the value of L in terms of the wavelength.
For a standing wave in a tube that is open at both ends, the wavelength λ can be related to the length of the tube L by the equation:
2L = n λ,
where n is the number of nodes in the standing wave pattern. In this case, since the tube is open at both ends, we have n = 1.
Substituting the given value of L = 0.202 m into the equation, we have:
2 * 0.202 m = 1 * λ,
0.404 m = λ.
Now we can substitute the values of f = 449 Hz and λ = 0.404 m into the speed of sound formula:
v = f λ = (449 Hz) * (0.404 m) = 181.096 m/s.
Therefore, the speed of sound in the gas in the tube is approximately 181.096 m/s.