57
PROPAGATION D’UNE ONDE PLANE — INTERFÉRENCES
(1)
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Ces équations peuvent se mettre sous une autre forme en posant
![{\displaystyle u={\frac {d\eta }{dz}}-{\frac {d\zeta }{dy}},\quad \qquad v={\frac {d\zeta }{dx}}-{\frac {d\xi }{dz}},\quad \qquad w={\frac {d\xi }{dy}}-{\frac {d\eta }{dx}}\cdot }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1932cd19c5e38ef5345113fcfe6e08241d22b1d8)
On a en effet :
![{\displaystyle {\begin{aligned}\Delta \xi -{\frac {d\Theta }{dx}}&={\frac {d^{2}\xi }{dx^{2}}}+{\frac {d^{2}\xi }{dy^{2}}}+{\frac {d^{2}\xi }{dz^{2}}}-{\frac {d^{2}\xi }{dx^{2}}}-{\frac {d^{2}\eta }{dx\,dy}}-{\frac {d^{2}\zeta }{dx\,dz}}\\[1.5ex]&={\frac {d^{2}\xi }{dy^{2}}}-{\frac {d^{2}\eta }{dx\,dy}}+{\frac {d^{2}\xi }{dz^{2}}}-{\frac {d^{2}\zeta }{dx\,dz}},\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e497d2730f377d18ca52f9a23142752bac4d2c6)
ou
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En transformant de la même manière les quantités
![{\displaystyle \Delta \eta -{\frac {d\Theta }{dy}},\qquad \qquad \Delta \zeta -{\frac {d\Theta }{dz}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07be56106bf54619901850e5aed38246d69e03b4)
qui entrent dans les deux dernières équations du mouvement, on obtient :
![{\displaystyle {\begin{aligned}{\frac {d^{2}\xi }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {dw}{dy}}-{\frac {dv}{dz}}\right),\\[1.5ex]{\frac {d^{2}\eta }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {du}{dz}}-{\frac {dw}{dx}}\right),\\[1.5ex]{\frac {d^{2}\zeta }{dt^{2}}}&=\mathrm {V} ^{2}\left({\frac {dv}{dx}}-{\frac {du}{dy}}\right)\cdot \\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/80eb0117fde8550b71036476cfe7ee315bf5832c)