Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/203

to any point in such a plane. In a wave-crest (or plane in which ${\dot {\xi }}$ has a maximum value), ${\text{Pot }}{\dot {\xi }}$ will also have a maximum value, which we may call ${\text{K}}.$ For intermediate points we may determine its value £rom the consideration that the total disturbance may be resolved into two systems of waves, one having a wave-crest, and the other a nodal plane passing through the point for which the potential is sought. The maximum amplitudes of these component systems will be to the maximum amplitude of the original system as $\cos 2\pi {\frac {u}{l}}$ and $\sin 2\pi {\frac {u}{l}}$ to unity. But the second of the component systems will contribute nothing to the value of the potential. We thus obtain

${\text{Pot }}{\dot {\xi }}={\text{K}}\cos 2\pi {\frac {u}{l}},$ ${\frac {d^{2}{\text{Pot }}{\dot {\xi }}}{du^{2}}}=-{\frac {4\pi ^{2}}{l^{2}}}{\text{K}}\cos 2\pi {\frac {u}{l}}=-{\frac {4\pi ^{2}}{l^{2}}}{\text{Pot }}{\dot {\xi }}.$

Comparing this with equation (6), we have

${\begin{aligned}-{\frac {4\pi ^{2}}{l^{2}}}{\text{ Pot }}{\dot {\xi }}&=-4\pi {\dot {\xi }},\\{\text{Pot }}{\dot {\xi }}&={\frac {l^{2}}{\pi }}{\dot {\xi }}.\end{aligned}}$

(7)

Hence, and by equations (4),

${\tfrac {1}{2}}\textstyle \sum \displaystyle \int {\dot {\xi }}{\text{ Pot }}{\dot {\xi }}dv={\frac {l^{2}}{2\pi }}\textstyle \sum \displaystyle \int {\dot {\xi }}^{2}dv={\frac {2\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})\int \cos ^{2}2\pi {\frac {u}{l}}dv.$

The kinetic energy of the regular part of the flux is therefore, for each unit of volume,

${\text{T}}={\frac {\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2}).$

(8)

9. With respect to the kinetic energy of the irregular part of the flux, it is to be observed that, since ${\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}$ have their average values zero in spaces which are very small in comparison with a wave-length, the integrations implied in the notations ${\text{Pot }}{\dot {\xi '}},{\text{Pot }}{\dot {\eta '}},{\text{Pot }}{\dot {\zeta '}}$ may be confined to a sphere of a radius which is small in comparison with a wave-length. Since within such a sphere ${\dot {\xi '}},{\dot {\eta '}},{\dot {\zeta '}}$ are sensibly determined by the values of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ at the center of the sphere, which is the point for which the value of the potentials are sought, ${\text{Pot }}{\dot {\xi '}},{\text{Pot }}{\dot {\eta '}},{\text{Pot }}{\dot {\zeta '}}$ must be functions—evidently linear functions—of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }};$ and ${\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}},{\dot {\eta '}}{\text{ Pot }}{\dot {\eta '}},{\dot {\zeta '}}{\text{ Pot }}{\dot {\zeta '}}$ must be quadratic functions of the same quantities. But these functions will vary with the position of the point considered with reference to the adjacent molecules.

Now the expression for the kinetic energy of the irregular part of the flux,

${\tfrac {1}{2}}\textstyle \sum \displaystyle {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv,$