WALL SHAPE, TAPER, FLARE, ACOUSTIC VELOCITY, DISCONTINUITIES: Theoretical Aspects of Trumpet Design © Part 1 of 21

An Excerpt from the Book: Trumpet Construction & Performance

“A cylindrical wall shape has a constant radius and a straight axis.  The first and second derivatives of radius are zero for such a shape.  A conical wall shape has a constant radial taper and a straight axis.  The first derivative of radius is constant and the second derivative of radius is zero for such a shape.  The second derivative of radius is the flare rate.  Only cylindrical and conical wall shapes have a zero-flare rate. Within cylindrical and conical wall shapes, the acoustic velocity remains constant and is equal to the velocity of sound in still air for all frequencies.  Within all other wall shapes, the acoustic velocity varies with both the flare rate and frequency.

An acoustic air column is defined by the wall shape.  An abrupt change in the radius, taper, or centerline direction of the air column comprises a discontinuity. Discontinuities interfere with an attack, the starting of the note, and the ability to play softly. A large discontinuity returns a premature echo, or reflected pulse, to the lips.  All discontinuities are to be avoided.

When an abrupt change in radius, taper, or centerline direction exists, the discontinuity is eliminated by using a transitional wall shape.  At each end of the transitional wall shape, the radius, the first derivative of radius, and the second derivative of radius exactly match the corresponding values of the adjacent wall shape.  The second derivative of radius is zero at each end of the transitional wall shape, it changes continuously over the length, and it reaches a maximum or minimum value at or near to the mid-length.  The magnitude of the second derivative at or near to the mid-length is reduced as the length of the transitional wall shape is increased.  It is necessary to use a transitional wall shape in the design of the mouthpiece and the leadpipe.”