(a) 34.6 Hz
The fundamental frequency of a pipe closed at one end is given by
[tex]f_1 = \frac{v}{4 L}[/tex]
where
v = 343 m/s is the speed of the sound in air
L is the length of the pipe
In this problem,
L = 248 cm = 2.48 m
So, the fundamental frequency is
[tex]f_1 = \frac{343 m/s}{4 (2.48 m)}=34.6 Hz[/tex]
(b) 103.8 Hz
In a open-closed pipe, only odd harmonics are produced; therefore, the frequency of the first overtone (second harmonic) is given by:
[tex]f_2 = 3 f_1[/tex]
where
[tex]f_1 = 34.6 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_2 = 3 (34.6 Hz)=103.8 Hz[/tex]
(c) 173 Hz
The frequency of the second overtone (third harmonic) is given by:
[tex]f_3 = 5 f_1[/tex]
where
[tex]f_1 = 34.6 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_3 = 5 (34.6 Hz)=173 Hz[/tex]
(d) 242.2 Hz
The frequency of the third overtone (fourth harmonic) is given by:
[tex]f_4 = 7 f_1[/tex]
where
[tex]f_1 = 34.6 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_4 = 7 (34.6 Hz)=242.2 Hz[/tex]
(e) 69.2 Hz
The fundamental frequency of a pipe open at both ends is given by
[tex]f_1 = \frac{v}{2 L}[/tex]
where
v = 343 m/s is the speed of the sound in air
L is the length of the pipe
In this problem,
L = 248 cm = 2.48 m
So, the fundamental frequency is
[tex]f_1 = \frac{343 m/s}{2 (2.48 m)}=69.2 Hz[/tex]
(f) 138.4 Hz
In a open-open pipe, both odd and even harmonics are produced; therefore, the frequency of the first overtone (second harmonic) is given by:
[tex]f_2 = 2 f_1[/tex]
where
[tex]f_1 = 69.2 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_2 = 2 (69.2 Hz)=138.4 Hz[/tex]
(g) 207.6 Hz
The frequency of the second overtone (third harmonic) in an open-open pipe is given by:
[tex]f_3 = 3 f_1[/tex]
where
[tex]f_1 = 69.2 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_3 = 3 (69.2 Hz)=207.6 Hz[/tex]
(h) 276.8 Hz
The frequency of the third overtone (fourth harmonic) in an open-open pipe is given by:
[tex]f_4 = 4 f_1[/tex]
where
[tex]f_1 = 69.2 Hz[/tex] is the fundamental frequency.
Substituting into the equation,
[tex]f_4 = 4 (69.2 Hz)=276.8 Hz[/tex]
