The number of vibrations per second in the note of a whistle or other “closed pipe” depends on its depth.  The theory of acoustics shows that the length of each complete vibration is four times that of the depth of the closed pipe, and since experience proves that all sound, whatever may be its pitch, is propagated at the same rate, which under ordinary conditions of temperature and barometric pressure may be taken at 1120 feet, or 13,440 inches per second,—­it follows that the number of vibrations in the note of a whistle may be found by dividing 13,440 by four times the depth, measured in inches, of the inner tube of the whistle.  This rule, however, supposes the vibrations of the air in the tube to be strictly longitudinal, and ceases to apply when the depth of the tube is less than about one and a half times its diameter.  When the tube is reduced to a shallow pan, a note may still be produced by it, but that note has reference rather to the diameter of the whistle than to its depth, being sometimes apparently unaltered by a further decrease of depth.  The necessity of preserving a fair proportion between the diameter and the depth of a whistle is the reason why these instruments, having necessarily little depth, require to be made with very small bores.

The depth of the inner tube of the whistle at any moment is shown by the graduations on the outside of the instrument.  The lower portion of the instrument as formerly made for me by the late Mr. Tisley, optician, Brompton Road,[28] is a cap that surrounds the body of the whistle, and is itself fixed to the screw that forms the plug.  One complete turn of the cap increases or diminishes the depth of the whistle, by an amount equal to the interval between two adjacent threads of the screw.  For mechanical convenience, a screw is used whose pitch is 25 to the inch; therefore one turn of the cap moves the plug one twenty-fifth of an inch, or ten two-hundred-and-fiftieths.  The edge of the cap is divided into ten parts, each of which corresponds to the tenth of a complete turn; and, therefore, to one two-hundred-and-fiftieth of an inch.  Hence in reading off the graduations the tens are shown on the body of the whistle, and the units are shown on the edge of the cap.

The scale of the instrument having for its unit the two-hundred-and-fiftieth part of an inch, it follows that the number of vibrations in the note of the whistle is to be found by dividing (13440 x 250)/4 or 84,000, by the graduations read off on its scale.

A short table is annexed, giving the number of vibrations calculated by this formula, for different depths, bearing in mind that the earlier entries cannot be relied upon unless the whistle has a very minute bore, and consequently a very feeble note.