Page:Elementary Text-book of Physics (Anthony, 1897).djvu/174

160 of the particles in it at any instant will be always the same. Let us call this velocity $$v_{a}.$$ The velocity of the cross-section relative to the moving particles in it is then $$V - v_{a}.$$ If we represent by $$d_{a}$$ the density of the medium at the cross-section through which the velocity of the particles is $$v_{a}$$, which is the same for all positions of the moving cross-section, and if we assume that the area of the cross-section is unity, then the quantity of matter $$M$$ which passes through the moving cross-section in unit time is $$M = d_{a}(V - v_{a}).$$

If we conceive any other cross-section $$B$$ to be moving with the disturbance in a similar manner, the same quantity of matter $$M$$ will pass through it in unit time, since the two cross-sections move with the same velocity and the density of the matter between them remains the same. Hence we have $$M = d_{b}(V - v_{b}),$$, where $$d_{b}$$ and $$v_{b}$$ represent the quantities at the cross-section $$B$$ corresponding to those at the cross-section $$A$$ represented by $$d_{a}$$ and $$v_{a}$$. Hence $$d_{a}(V - v_{a}) = d_{b}(V - v_{b}).$$ Since this equation is true whatever be the distance between the cross-sections, it is true for that position of the cross-section $$B$$ for which $$v_{b} = 0$$, and for which $$d_{b} = D$$, the density of the medium in its undisturbed condition. Hence we have $$M = DV, d_{a} (V - v_{a}) = DV$$, and If the disturbance be small, the expression on the right is approximately the condensation per unit volume of the medium at the cross-section $$A$$, and the equation shows that the ratio of the velocity of the matter passing through the cross-section $$A$$ to the velocity of propagation of the disturbance is equal to the condensation at that cross-section.

Now, to eliminate the unknown quantities $$v_{a}$$ and $$d_{a}$$, we must find a new equation involving them. A quantity of matter $$M$$ enters the region between the two moving cross-sections with the velocity $$v_{a}$$, and an equal quantity leaves the region with the velocity $$v_{b}$$. The difference of the momenta of the entering and outgoing quantities is $$M(v_{a} - v_{b}).$$ This difference can only be due to