Determine the first three standing wave frequencies of a 25.0-cm-long open pipe. Assume the speed of sound in air is 340 m/s. Do the same for a 25.0-cm-long closed pipe.
To find the standing wave frequencies of an open pipe and a closed pipe, we can use the formula:
f = (n * v) / (2L)
where:
f is the frequency of the standing wave
n is the harmonic number (1, 2, 3, ...)
v is the speed of sound in air
L is the length of the pipe
For an open pipe:
L = 25 cm = 0.25 m
v = 340 m/s
For the first three harmonics, n = 1, 2, 3.
Let's calculate the frequencies:
For n = 1:
f1 = (1 * 340) / (2 * 0.25) = 680 Hz
For n = 2:
f2 = (2 * 340) / (2 * 0.25) = 1360 Hz
For n = 3:
f3 = (3 * 340) / (2 * 0.25) = 2040 Hz
So, the first three standing wave frequencies of a 25.0-cm-long open pipe are 680 Hz, 1360 Hz, and 2040 Hz.
For a closed pipe, the only difference is that the harmonics are odd multiples of the fundamental frequency (n = 1). Let's calculate the frequencies:
For n = 1:
f1 = (1 * 340) / (4 * 0.25) = 425 Hz
For n = 3:
f2 = (3 * 340) / (4 * 0.25) = 1275 Hz
For n = 5:
f3 = (5 * 340) / (4 * 0.25) = 2125 Hz
So, the first three standing wave frequencies of a 25.0-cm-long closed pipe are 425 Hz, 1275 Hz, and 2125 Hz.
To determine the first three standing wave frequencies of a 25.0-cm-long open pipe, we can use the formula:
\( f_n = \frac {nv}{2L} \),
where \( f_n \) is the frequency of the nth harmonic, n is the harmonic number, v is the speed of sound in air, and L is the length of the pipe.
For the open pipe:
1. First harmonic (n=1):
Plug in the values into the formula:
\( f_1 = \frac {1 \cdot 340 \, \text{m/s}} {2 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_1 = 340 \, \text{Hz} \)
2. Second harmonic (n=2):
Plug in the values into the formula:
\( f_2 = \frac {2 \cdot 340 \, \text{m/s}} {2 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_2 = 1360 \, \text{Hz} \)
3. Third harmonic (n=3):
Plug in the values into the formula:
\( f_3 = \frac {3 \cdot 340 \, \text{m/s}} {2 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_3 = 2040 \, \text{Hz} \)
Now, to determine the first three standing wave frequencies of a 25.0-cm-long closed pipe, we use the same formula:
\( f_n = \frac {nv}{4L} \),
where n is the harmonic number, v is the speed of sound in air, and L is the length of the pipe.
For the closed pipe:
1. First harmonic (n=1):
Plug in the values into the formula:
\( f_1 = \frac {1 \cdot 340 \, \text{m/s}} {4 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_1 = 170 \, \text{Hz} \)
2. Second harmonic (n=2):
Plug in the values into the formula:
\( f_2 = \frac {2 \cdot 340 \, \text{m/s}} {4 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_2 = 340 \, \text{Hz} \)
3. Third harmonic (n=3):
Plug in the values into the formula:
\( f_3 = \frac {3 \cdot 340 \, \text{m/s}} {4 \cdot 0.25 \, \text{m}} \)
Simplify the expression:
\( f_3 = 510 \, \text{Hz} \)
Therefore, the first three standing wave frequencies for the open pipe are 340 Hz, 1360 Hz, and 2040 Hz, while for the closed pipe, they are 170 Hz, 340 Hz, and 510 Hz.
Formula for open-pipe:
f = nv / (2L)
Formula for closed-pipe:
f = (2n-1)v / (4L)
where
n = harmonic number
v = speed
L = length of pipe
Just substitute the given values to these equations for n = 3.
Units in Hz