• Wind Instrument Air Columns
• Excitation: The Brass Player's Lips
• Sound Radiation

Brass Instruments

Wind instruments, or aerophones, produce sounds by directly initiating and maintaining vibrations of air within a confined structure or air column (rather than via mechanical vibrations which are later transferred to air). Members of this group include the brass and woodwind instrument families, as well as the human voice.

The three principal components of a brass instrument are its air column (waveguide), the player's lips/mouthpiece (excitation source), and bell (radiation).

If you wish to pursue a more in-depth analysis of brass instruments, perhaps for your class project, a unique collection of research materials is maintained here at CCRMA. The Musical Acoustics Research Library (MARL) is a collection of independent archives or libraries assembled by distinguished groups or individuals in the field of musical acoustics research. Currently, MARL is comprised of the Catgut Acoustical Society Library, the Arthur H. Benade Archive, the John Backus Archive, and the John W. Coltman Archive. Arthur Benade, in particular, made substantial contributions to our understanding of brass instrument acoustic behavior.

Wind Instrument Air Columns

1. Cylindrical Pipes:
   • Wave motion in a cylindrical pipe can occur along any of three independent coordinate axes.

   • The two lowest frequency transverse modes in a pipe of 7.5 millimeter radius occur at 13.56 kHz and 22.5 kHz. However, they require transverse circular motion and thus are unlikely to be excited with any appreciable magnitude in musical instruments.

   • Wave motion of interest in the study of musical instruments within a cylindrical pipe is primarily planar and along the principal axis of the tube.

   • Pressure distributions along a pipe of length $L$ are dependent on the boundary conditions at each end. An open pipe end is typically approximated as an ideal zero impedance boundary.

   • For a pipe (ideally) open at both ends, the resulting discrete standing-wave frequencies are given by $f_{n} = n c/(2 L)$ for $n = 1, 2, 3, \ldots$, where $c$ is the speed of wave propagation in air.

   • For a pipe (ideally) open at one end and closed at the other, the standing wave frequencies are given by $f_{n} = n c/(4 L)$ for $n = 1, 3, 5, \ldots$.

   • The particular frequency components sustained in the pipe are determined by the excitation mechanism, radiation characteristics, wall damping, etc...

2. Conical Bores: See Woodwinds section.

3. Air Column Discontinuities:
   • Traveling-wave characteristic or wave impedance in cylindrical pipes: $Z_{0} = \rho c / S$, where $\rho$ is the mass density of air, $c$ is the speed of sound in air, and $S$ is the cross-sectional area of the pipe.

   • A change in air column radius causes a corresponding change in wave impedance. At an impedance discontinuity, traveling-wave components are ``scattered'' in a frequency-dependent manner.

   • Low-frequency wave components are more affected by impedance discontinuities than high-frequency components. At the open end of a cylindrical pipe (a very significant impedance discontinuity), low-frequency components are almost completely reflected while high-frequency components will be partly reflected and partly transmitted at the boundary.

   • In an impedance transition region, low-frequency wave components will travel a shorter distance before being reflected than high-frequency compenents. Thus, such a region has a frequency-dependent length.

4. Input Impedance:
   • Acoustic impedance: $Z = P/U$, where $P$ and $U$ are sinusoidal quantities of pressure and volume flow, respectively.

   • An air column's input impedance is defined as its sinusoidal pressure response to a driving sinusoidal volume velocity source at the input to the air column.

   • The input impedance can be determined from an air column's impulse response.

   • The input impedance peaks indicate the frequencies at which a constant volume velocity source will produce the greatest pressure variations at the input to the air column.

   • The input impedance minima indicate the frequencies at which a constant pressure source will produce the greatest volume velocity variations at the input to the air column.

   • For pressure controlled driving mechanisms (such as the brass player's lips), the input impedance peaks indicate the frequencies at which air column vibrations will cooperate with the driving mechanism to sustain steady oscillations.

   • Input impedance is typically measured using a variable-frequency volume velocity source of constant amplitude. However, time-domain measurement techniques have become popular during recent years.

5. Brass Instrument Air Columns:
   • Brass instrument air columns are typically constructed of a short conical section (the mouthpiece pipe), a long cylindrical section, and a flaring end. It is not possible to derive a simple formula for the air column resonances of such a complex system.

   • Adjustments are carried out in the course of the instrument design to produce a mode series approximating (0.7, 2, 3, 4, ...)$f_{0}$.

6. The Mouthpiece:
   • The lowest resonance of the mouthpiece usually occurs in the range 750-850 Hz and is referred to as the popping frequency.

   • The mouthpiece resonance enhances the magnitude of air column resonances (impedance peaks) in its vicinity.

   • The mouthpiece design can also influence the positioning of air column resonances.

7. Valves:
   • Valves are used to vary the acoustical length of the instrument, allowing the brass player to produce a full range of pitches throughout the twelve-tone scale and across several octave ranges.

   • Depressing a valve connects the main air column to an additional piece of tubing.

   • The process of determining valve tube lengths is complicated by the fact that a single tube length will produce different pitch change magnitudes for different main air column lengths and mode numbers.

8. Materials:
   • The thin walls of brass instruments vibrate during playing and contribute to damping, though they radiate a negligible amount of sound.

   • Wall materials have particular thermoviscous characteristics and might have a slight effect on the response of an instrument. This response will most likely be much more apparent to the player than to an audience member.

Excitation: The Brass Player's Lips

Figure 17: The brass player's lips as a mechanical oscillator blown open.

• The lips function as a pressure-controlled valve that admits a puff of air whenever the pressure is high in the mouthpiece.

• Pressure pulses reflected back from the far end of the horn tend to force the player's lips open ... positive feedback.

• The player controls the resonance frequency of his/her lips via tension and mass (position) variations.

• The resonance frequency of the lips is best adjusted to be a little below a horn resonance.

• Oscillations are favored when the air column has one or more resonances that correspond to the harmonics of the fundamental pitch.

Sound Radiation

1. The Bell:
   • The open end of a straight pipe is an inefficient radiator of sound waves, especially at low frequencies.

   • The bell offers a more gradual impedance transition between the high inner tube impedance and the very low outside room impedance.

   • The effective length of the bell increases with frequency. Thus, high frequency input impedance peaks of a pipe and bell combination will be lowered in frequency proportionately more than lower frequency peaks.

   • The radiation/reflection characteristics of the bell have to be carefully balanced to simultaneously produce stable oscillations (via reflections) as well as efficient radiation into the outside environment.

2. Mutes:
   • Mutes modify the radiation characteristics of the bell, reducing low-frequency radiation much more than high-frequency radiation.

   • The various mute styles produce different results, some passing frequencies above a certain limit and others emphasizing a particular band of frequencies.

   • Players may use a hand within the bell to alter the reflection/transmission characteristics.