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

Brass Instruments

This look into brass instrument acoustic behavior begins our study of instruments which initiate their vibrations directly in air, rather than via mechanical vibrations which are later transferred to air. These ``wind'' instruments include brasses, woodwinds, and the voice.

The three principle components of brass instruments are given by the air column (waveguide), 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 first two transverse modes in a pipe of 7.5 millimeter radius occur at frequencies of 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. For a pipe 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 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. Brass Instrument Air Columns:
   
   
   • Because brass instrument air columns are typically constructed of a short conical section, a long cylindrical section, and a flaring end, a simple formula for the air column resonances is not possible. Adjustments are carried out in the course of the design to produce a mode series approximating (0.7, 2, 3, 4, ...)$f_{0}$.

3. Impedance:
   
   
   • Acoustic impedance: $Z = P/U$, where $P$ and $U$ are sinusoidal quantities of pressure and volume flow, respectively.
   • Characteristic or wave impedance in pipes: $Z_{0} = \frac{\rho c}{S}$, 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.
   • 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.

4. 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.

5. Wall Damping:
   
   
   • Wall material generally has negligible effect on the playing behavior of a wind instrument.
   • Various materials, however, have different 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 16: 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 Raditaion

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 air 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.