“The acoustics and the transient process incurred in playing the instrument are now examined in detail. The player is assumed to play a Trumpet High C (acoustic resonance mode 8) at full dynamic volume.
The vibrating lips create a disturbance that contains many harmonics. At the start of a transient, only the fundamental is present in the pressure wave. The pressure wave travels from the lips toward the bell at a wave velocity determined by the fundamental frequency and the flare rate. Within the flaring bell shape, the pressure wave encounters an acoustic barrier and some of the pressure wave is reflected to the lips.
The time for such a round-trip is about seven milliseconds. The oscillation period for a trumpet High C is about one millisecond. For this note, the lips go through about seven oscillations before the lips receive any feedback from the instrument. For a Double C, the lips go through about fourteen oscillations before the lips receive any feedback. Upon receiving feedback, the player may adjust the lip vibration frequency if required.
At this point in the transient, the fundamental has excited air column mode 8 into resonance, and a standing wave has been established. The standing wave combines with the disturbance being produced by the lips, and the amplitude of the pressure wave is acoustically amplified.
In the next step in the transient, the second harmonic is added to the pressure wave. The pressure wave travels at the wave velocity, but the second harmonic travels at the group velocity. The group velocity depends upon the frequencies of the harmonics present and the flare rate. The pressure wave encounters an acoustic barrier closer to the large end of the flaring bell and some of the pressure waves are reflected back to the lips. The second harmonic has excited acoustic resonance mode 16 into resonance, and the amplitude of the standing wave has been increased. Air column modes 8 and 16 simultaneously resonate.
In the next step, the third harmonic is added to the pressure wave. The pressure wave travels at the wave velocity, but the second and third harmonics travel at the group velocity. The group velocity is changed because of the higher frequency of the third harmonic. Air column mode 24 is excited into resonance, and modes 8, 16 and 24 simultaneously resonate.
If ten harmonics are produced by the lips, the transient is completed in ten round trips, and air column modes 8, 16, 24, 32, 40, 48, 56, 64, 72, and 80 all simultaneously resonate.
The group velocity, affected by the frequencies of all the harmonics, is different from the wave velocity, causing the harmonics to have a different transit time than the fundamental. This difference in transit times is referred to as transit time dispersion.
Nevertheless, a bell shape that has near perfect harmonicity causes the reflection phase shifts at the open bell to vary with frequency in such way as to cancel the transit time dispersion.
Trumpet air columns that lack near-perfect harmonicity facilitate transit time dispersion and thereby create an incoherent reflected pressure wave. All commercially available trumpets have increasingly large harmonicity deviations beginning at about mode 30.
In this example, the reflected pressure wave is partially incoherent. The three lowest harmonics (with the largest acoustic energy) do not cause transit time dispersion, and the seven higher harmonics (with much smaller acoustic energy) cause transit time dispersion. The reflected pressure wave gets progressively more incoherent for higher notes. For a Double C, the reflected pressure wave is very incoherent because nine of the ten harmonics cause transit time dispersion.”
