To calculate the resonant frequency of the flute without any open holes, the formula for the fundamental frequency of a closed cylindrical tube is used. It is:

f=v/2L

Where:

- F is the frequency (in Hz)
- v is the velocity of sound in air (in m/s) = 340m/s
- L is the length of the tube (in meters) = 26 inches

1 inch =0.0254meters

So L = 26*0.0254 meters

Substituting these values into the formula:

F = 340/(2∗26∗0.0254)

=257.417 HZ

The frequency of middle C is 261 HZ. The calculated frequency is approximately equal to this value. So the resonant frequency is close to middle C.

Opening a finger hole on a flute effectively shortens the vibrating air column inside the instrument. The shorter length increases the frequency of the sound produced because the wave now has less distance to travel before being reflected back and forth between the open end and the finger hole. Consequently, the shortened column lengthens the wavelength, resulting in a higher pitch.

For a tube that is open at both ends, the fundamental frequency corresponds to the first harmonic. In this case, the wavelength of the fundamental frequency is equal to twice the length of the tube.

Given that the length of the tube is L = 26 inches or 26×0.025426×0.0254 meters

L=0.6604 meters.

Wavelength of the fundamental frequency:

λ=2L

=2×0.6604λ=2×0.6604 = 1.3208 meters.

Wavelength = 1.3208 meters

Frequency of the fundamental:

V = 340m/s

Wavelength = 1.3208 meters.

f=v/λ = 340/1.3208 = 257.55HZ

Frequency = 257.55HZ.

This frequency is the same as the frequency calculated earlier when considering the flute without open holes, confirming that it represents the fundamental frequency or first harmonic.

The string that cannot produce middle C by depressing the string to end at the fret is the low E string, tuned to 82.41 Hz (E2). This is because:

Middle C is typically around 261.63 Hz. To produce middle C on the guitar, the frequency of the low E string (82.41 Hz) needs to be increased by depressing it against a fret. Each fret increases the frequency by a factor of 2^1/12, due to the equal temperament tuning system used on guitars.

To find how many frets needs to be pressed down on the low E string to reach middle C:

Frequency in in the middle C = 82.41* 2^n/12 where n is the number of frets pressed down.

261.63 = 82.41 * 2^n/12

261.63/82.41=2^n/12

3.17 = 2^n/12

Log2(3.17) = n/12

n =19.2

This means the low E string at the 19^{th} fret needs to be pressed down to reach middle C. However, the guitar has 12 frets and reaching the 19^{th} on this guitar is not possible.