Page:Philosophical magazine 23 series 4.djvu/113

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$\left.\mathrm {and} \ {\begin{array}{l}\left(n^{2}-m^{2}a^{2}\right)A=m^{2}nc^{2}B,\\\left(n^{2}-m^{2}b^{2}\right)B=m^{2}nc^{2}A.\end{array}}\right\}$

(150)

Multiplying the last two equations together, we find

$\left(n^{2}-m^{2}a^{2}\right)\left(n^{2}-m^{2}b^{2}\right)=m^{4}n^{2}c^{4},$

(151)

an equation quadratic with respect to $m^{2}$, the solution of which is

$m^{2}={\frac {2n^{2}}{a^{2}+b^{2}\mp {\sqrt {\left(a^{2}+b^{2}\right)^{2}+4n^{2}c^{4}}}}}$

(152)

These values of $m^{2}$ being put in the equations (150) will each give a ratio of A and B,

${\frac {A}{B}}={\frac {a^{2}+b^{2}\mp {\sqrt {\left(a^{2}+b^{2}\right)^{2}+4n^{2}c^{4}}}}{2nc^{2}}},$

which being substituted in equations (149), will satisfy the original equations (148). The most general undulation of such a medium is therefore compounded of two elliptic undulations of different eccentricities travelling with different velocities and rotating in opposite directions. The results may be more easily explained in the case in which $a=b$; then

$m^{2}={\frac {n^{2}}{a^{2}\mp nc^{2}}}\ \mathrm {and} \ A=\mp B.$

(153)

Let us suppose that the value of A is unity for both vibrations, then we shall have

$\left.{\begin{array}{ll}x=&\cos \left(nt-{\frac {nz}{\sqrt {a^{2}-nc^{2}}}}\right)+\cos \left(nt-{\frac {nz}{\sqrt {a^{2}+nc^{2}}}}\right),\\y=-&\sin \left(nt-{\frac {nz}{\sqrt {a^{2}-nc^{2}}}}\right)+\sin \left(nt-{\frac {nz}{\sqrt {a^{2}+nc^{2}}}}\right).\end{array}}\right\}$

(154)

The first terms of $x$ and $y$ represent a circular vibration in the negative direction, and the second term a circular vibration in the positive direction, the positive having the greatest velocity of propagation. Combining the terms, we may write

$\left.{\begin{array}{l}x=2\cos(nt-pz)\cos qz,\\y=2\cos(nt-pz)\sin qz,\end{array}}\right\}$

(155)

where

$\left.{\begin{array}{ll}&p={\frac {n}{2{\sqrt {a^{2}-nc^{2}}}}}+{\frac {n}{2{\sqrt {a^{2}+nc^{2}}}}},\\\mathrm {and} \\&q={\frac {n}{2{\sqrt {a^{2}-nc^{2}}}}}-{\frac {n}{2{\sqrt {a^{2}+nc^{2}}}}},.\end{array}}\right\}$

(156)

These are the equations of an undulation consisting of a plane