Question

/ a "c" flute creates the note c4 when it is played at its resonance frequency, that is 261 ... Question: A "C" flute creates the note C4 when it is played at its resonance frequency, that is 261 Hz on t... A "C" flute creates the note C4 when it is played at its resonance frequency, that is 261 Hz on the evenly tempered scale. Flute frequencies are changed by opening holes in its tube, creating a node at that hole and effectively shortening the 66 cm long tube. The flute and the guitar make a good instrumental combination, and for that they must be tuned together. You have seen that a guitar's 6 strings are usually tuned to (E) 329.63 Hz E4 (B) 246.94 Hz B3 (G) 196.00 Hz G3 (D) 146.83 Hz D3 (A) 110.00 Hz A2 (E) 82.41 Hz E2

1. Is there a way to lower the frequency of the flute so that it matches one of these open strings at resonance? Explain your answer.

2. Given the length of its tube and the frequency of the flute sound, what is the speed of sound in air it has been made for?

3. At what temperature would the air be in °C to give this speed? What is that in °F?

4. At this temperature, what is the most probable speed of a nitrogen molecule in m/s?

Answer 1

A flute works similar to a open ended cylinder. Consider a flute without any of the register holes and only the embouchure hole remains. When you blow into it, the air inside the flute starts to vibrate. A longitudinal compression standing wave is created. In this wave, the air is free to move at the ends hence creating anti-nodes (Places of maximum displacement), and nodes are created in the middle (places of no displacement). This leads to a wave that is, antinode-node-antinode, meaning it contains exactly half of the standing wave.

So, the wavelength of the wave is given by,

where, = Length of the tube,

This is called the fundamental wavelength, conversely the fundamental frequency. It is the lowest resonant frequency one can obtain with a particular length.

We can also find the velocity of the sound in an open tube using the frequency-wavelength relation,

When a hole is drilled into a flute at some distance, (say for instance, we want to play C5, we can add a hole at the mid-point of the tube and create a node over there, essentially, making a standing wave with the wavelength equal to the length of the tube.

So the frequency becomes,

Which is twice the original frequency. Which is called the first overtone, or the second harmonic.

(A) So, we cannot really decrease the frequency of the flute to produce anything lower than its fundamental resonant frequency, which is C4 = 261 Hz in our case. It can only be achieved if we were somehow to increase the length of the tube, since . We can indeed increase it to E4 by creating a hole in the flute such that the frequency of the wave inside the tube is 329.63 Hz when the hole is open.

(B) We have seen the formula of velocity in equation (2). We know the length of the flute. It's easy to calculate the velocity of sound,

(C) The relation between the velocity of sound and temperature can obtained. The velocity is given by,

where, is the adiabatic ratio, P is the pressure and, is the density.

Suppose we consider air to be an ideal gas, then,

Subsstituting this equation in our velocity we get,

Now, the molar density is just the number of moles per unit volume, , where M is the mean molar mass of air which is around 0.0289 kg/mol. Substituting which gives us,

R = gas constant = 8.3145 J/(mol.K) and = 1.4 for an ideal diatomic gas,

Temperatures in degrees Celsius and Fahrenheit will be,

(D) The most probable velocity of a molecule at a temperature T is the most likley velocity the molecule will possess if it obeys Maxwell's Speed Distribution.

k_B = Boltzmann's Constant. m = molecular mass of a single molecule (in our case N2),

m = 14.0067*2 = 28.0134 amu = 4.6517 e-26 kg