STRINGED & WIND INSTRUMENTS

The association of physics with music is very old and deep. The "Pythagorean School" of ancient Greece were the first, probably, to recognise the relationship of notes to length of string. The "Music of the Spheres" pervaded astronomy in the late C16th . Kepler "composed" music to describe the planetary orbits and died humming these to himself. Newton invented the colour "indigo" to create a musical scale of 7 colours. (Indigo has quietly disappeared from the rainbow again these days.) The quantum mechanics of atoms closely resembles the detailed physics of sound in more complex wind instruments!

STRINGED INSTRUMENTS

Probably all stringed instruments can be traced in history back to the weapon, the bow. Someone realised that the "twang" of the bow string was dependant on tension of the string and thickness of the string. Amplification of this "twang" could be made with a "sound board" or hollow box nearby. The vibrations of the string are transitted to the resonance box via the bridges as well as the sound entering the box.

Contemporary electric guitars make pick up the vibrations in the air generated by the string and amplify them electronically.

Strings are rigidly clamped at both ends so the basic action of the string is to generate sound by the transverse vibrations of standing waves along the string causing longitudinal vibrations in the air of the same frequency.

Newton showed that wave will travel along a string with a velocity given by

v = (T/μ)^1/2 where T = tension or stretching force in the string in newtons, μ = "weightiness" of the string, the linear density or mass per unit length kgm^-1.

Standing waves along a Guitar String.

These strings are fixed at both ends, so both ends are NODES. Antinodes are set up along the string.

The simplest pattern along the string is a single antinode in the centre.

As this node-antinode-node forms half a sine wave, the physical length l of the string equals half a wavelength (λ₁) of this bottom frequency, f₁.

as f = v / λ ,

f₁ = v /λ₁ = v /2l thus f₁ = (T/μ)^1/2 / 2l

This frequency is known as either the FUNDAMENTAL or the FIRST HARMONIC.

The next pattern has a whole standing wave along the string. For the same string, the wavelength λ₂ is now equal to the length, λ₂ = l , so

f₂ = 2 x (T/μ)^1/2 / 2l

This is the first complication so it is called the FIRST OVERTONE and as the frequency is twice the simplest it is also called the SECOND HARMONIC.

Similarly, the SECOND OVERTONE has a frequency of thrice the simplest so is the THIRD HARMONIC.

Generally, f_n = v /λ_n = nv /2l = n(T/μ)^1/2 / 2l

WIND INSTRUMENTS

Wind instruments have longitudinal displacement standing waves set up in them. They can be, very roughly, classified into "open" tubes ( flutes, recorders, penny whistles and some organ pipes) or "closed" tubes ( trumpets, trombones, didgeridoos, clarinets [ and other reed instruments] and some other organ pipes).

These two types of pipes have the displacement standing waves set up in them governed by

"open end equates to antinode",

"closed end equates to node".

( Sound waves in pipes can also be described in terms of PRESSURE waves. Displacement nodes are places of maximum changes of squeeze - the air squeezes up to the spot, a change of velocity implies a force acting, so displacement nodes are pressure antinodes. At displacement antinodes, the change of velocity is actually zero as these points are the centres of oscillations, therefore are places of zero pressure, - pressure nodes. )

A person's lips on a trumpet mouthpiece are clamped closed so this end is a node but for a flute, the lips blow over an open hole - so an antinode of air movement is formed here. Recorders, whistles and organ pipes use a fipple, a sharp edge cutting the air flow into the body of the instrument. This is an antinode area as it is open to the air - the internal air can freely oscillate in and out..

All wind instruments rely on vortices produced by some means - reed, fipple end, lips - which contain as part of their frequency spectrum, the frequency matching the resonant frequency of the body of the instrument. This, in turn, depends on the physical length of the instrument, the holes that have been opened along its length, the strength of the air flow and the shape of the tube. Trumpets are closed tubes BUT because of their conical shape, they behave in many ways like open tubes.

Like strings, OPEN PIPES can fit half a wave in their length in the simplest or Fundamental mode.

For the transverse ( across ) flute, the Fundamental note is governed by the physical length of the pipe with a node in the centre and antinode at each open end - the air is stationary in the very centre and oscillating completely freely at the ends. ( The hole across which wind is blown is open allowing free movement of the air.)

The physical length, l = λ₁ / 2 where λ₁ is the Fundamental wavelength. One half wavelength fits the pipe.

As f₁ = v /λ₁ = v /2l , and v = speed of sound in air = 340ms^-1 at room temperature, the fundamental frequency = first harmonic = 340 /2l

For the transverse flute f₁ = 340 /2 x 0.655 = 260 Hz = the note of C as the flute is 0.655m from end to end.

The First Overtone or Second Harmonic is the next most complex where a single whole standing wave ( two half wavelengths) fits the physical length.

f₂ = v /λ₂ = v /l = 2 x f₁. (Hence the term Second Harmonic)

Similarly, the Second Overtone has 11/2 wavelengths in the pipe = 3 half wavelengths so f₃ = v /λ₃ = v /l = 3 x f₁. (Hence the term Third Harmonic.)

The ratios of frequencies in an "open" pipe is 1 : 2 : 3 : 4........ and f_n = n x f_1.

CLOSED PIPES are characterised by always having stationary air at the closed surface and oscillating air at the open end, a NODE at the closed end and ANTINODE at the open end.

For these, we notice that odd multiples of QUARTERs of wavelengths appear in the pipe.

So the Fundamental or First Harmonic frequency, a quarter of a full standing wave fits the pipe l = λ₁ / 4, so f₁ = v /λ₁ = v /4l

The First Overtone has THREE quarters of a full standing wave l = 3λ₃ / 4, so f₃ = v /λ₃ = 3v /4l

This is 3 x f₁ so is known as the THIRD HARMONIC f₃

GENERALLY for a closed pipe, frequencies are in the ratio of 1 : 3 : 5 : 7................

If the pipe is not a pure cylinder however, like a trumpet, the rules break down quickly and it starts behaving as if it were an open pipe!

PIPE END CORRECTIONS

We are creating an overly simple model for a pipe, whether close or open. Proper models are much more complex. To get closer to a real pipe, it is found necessary to "extend" its length as the antinodes occur just outside the open mouth of the pipe. The corrections are necessary because the pipe radiates from its open ends. This leads to energy dispersion so the antinode in vibration moves out of the mouth slightly.

"effective length" = real length + end correction,

The value of the end correction varies whether the pipe is flanged or not. For a nonflanged pipe, it is approximately 0.58 x radius of pipe at each open end.

Links

University of NSW - a great site on the physics of music