If th' maximum pressure change is determined, the amplitude is given by equation (20), viz.
In both cases the amplitude of the oscillations decreases more or less rapidly.
The component vibrations at P due to the successive zones are thus nearly equal in amplitude and opposite in phase (the phase of each corresponding to that of the infinitesimal circle midway between the boundaries), and the series which we have to sum is one in which the terms are alternately opposite in sign and, while at first nearly constant in numerical magnitude, gradually diminish to zero.
Thus Mach found an amplitude 0 2 cm.
The hyperbolic or Gudermannian amplitude of the quantity x is ta n (sinh x).
If the two forks have the same frequency, it is easily seen that the figure will be an ellipse (including as limiting cases, depending on relative amplitude and phase, a circle and a straight line).
In creased length necessitated an increase in the diameter of the main tube to limit the amplitude of the vibrations caused by being pushed through the water.
But it is very difficult to suppose that the same sensation would be aroused by a truly periodic displacement represented by a smooth curve, and a displacement in which the period is only in the amplitude of the to-and-fro motion, and which is represented by a jagged curve.
Formerly it was generally supposed that the Tartini tone was due to the beats themselves, that the mere variation in the amplitude was equivalent, as far as the ear is concerned, to a superposition on the two original tones of a smooth sine displacement of the same periodicity as that variation.
The formation of beats may be illustrated by considering the disturbance at any point due to two trains of waves of equal amplitude a and of nearly equal frequencies n, n2.