The trombone's low B flat 2 is played with the slide in first position. What is its effective length, that is the length of an idealized closed tube with the same second mode frequency? Estimate how far the slide should be moved to reach second position in order to play A2, remembering that each cm of slide motion adds 2cm to the tube length. If already in sixth position(F2), how much farther must you move the slide to reach seventh position (E2)?
To find the effective length of the trombone's low B flat 2, we need to determine the length of an idealized closed tube with the same second mode frequency as the trombone's low B flat 2.
The second mode frequency of a closed tube (such as an idealized closed tube) is three times the fundamental frequency. Therefore, the fundamental frequency of the trombone's low B flat 2 corresponds to the second mode frequency of the idealized closed tube.
To find the effective length, we'll start by calculating the fundamental frequency of the trombone's low B flat 2. The fundamental frequency of a vibrating string or an air column (in a tube) is given by the formula:
f = v / λ
where:
- f is the frequency
- v is the speed of sound
- λ is the wavelength
The speed of sound is approximately 343 meters per second (m/s) at room temperature.
To determine the wavelength, we can use the formula:
λ = 2L
where L is the length of the vibrating air column.
Since the trombone's low B flat 2 is played with the slide in the first position, we consider the length of the vibrating air column as the length of the trombone slide in the first position.
Now, let's calculate the effective length of the trombone's low B flat 2:
1. Find the wavelength (λ) using the formula λ = 2L.
λ = 2 * L_first_position
2. Calculate the fundamental frequency (f) using the formula f = v / λ.
f = 343 / λ
3. The effective length of the idealized closed tube is then given by the formula:
Effective Length = λ / 2
So, Effective Length = λ / 2 = (2 * L_first_position) / 2 = L_first_position
Therefore, the effective length of the trombone's low B flat 2 is equal to the length of the trombone slide in the first position.
Now, let's estimate how far the slide should be moved to reach the second position (to play A2). It is given that moving the slide by 1 cm adds 2 cm to the tube length.
1. Calculate the difference between the effective length of the trombone's low B flat 2 (in the first position) and the length needed for A2 (in the second position).
Difference = Effective Length_A2 - Effective Length_Bb2
2. Convert the difference in effective length to the difference in slide position.
Slide Position Difference = Difference / 2
So, Slide Position Difference = (Effective Length_A2 - Effective Length_Bb2) / 2
Lastly, if you are already in the sixth position (F2) and want to move to the seventh position (E2), you need to determine the additional distance to move the slide.
1. Calculate the difference between the effective length of the trombone's F2 (in the sixth position) and the effective length of E2 (in the seventh position).
Difference = Effective Length_E2 - Effective Length_F2
2. Convert the difference in effective length to the difference in slide position.
Slide Position Difference = Difference / 2
So, Slide Position Difference = (Effective Length_E2 - Effective Length_F2) / 2
Please note that these calculations are based on approximations and assumptions, and might not be exact.