Ondas: superposicion

Supongamos $\psi _{1}$ y $\psi _{2}$ soluciones separadas de la ecuación de onda. Se deduce que también $\psi _{1}+\psi _{2}$ representa una solución. Esto se denomina principio de superposición. Es cierto que ${\frac {\partial \psi _{1}}{\partial x^{2}}}={\frac {1}{v^{2}}}{\frac {\partial ^{2}\psi _{1}}{\partial t^{2}}}$ y ${\frac {\partial \psi _{2}}{\partial x^{2}}}={\frac {1}{v^{2}}}{\frac {\partial ^{2}\psi _{2}}{\partial t^{2}}}$

Sumando estos resulatdos ${\frac {\partial \psi _{1}}{\partial x^{2}}}+{\frac {\partial \psi _{2}}{\partial x^{2}}}={\frac {1}{v^{2}}}{\frac {\partial ^{2}\psi _{1}}{\partial t^{2}}}+{\frac {1}{v^{2}}}{\frac {\partial ^{2}\psi _{2}}{\partial t^{2}}}$

Por lo tanto ${\frac {\partial ^{2}}{\partial x^{2}}}(\psi _{1}+\psi _{2})={\frac {1}{v^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}(\psi _{1}+\psi _{2})$

Superposición de ondas

Sea $\psi _{1}(r,t)$ , $\psi _{2}(r,t)$ ,.........., $\psi _{n}(r,t)$ soluciones individuales de la ecuacion de onda tridimensional ondas: ecuación de onda ${\frac {\partial ^{2}\psi }{\partial x^{2}}}+{\frac {\partial ^{2}\psi }{\partial y^{2}}}+{\frac {\partial ^{2}\psi }{\partial z^{2}}}={\frac {1}{v^{2}}}+{\frac {\partial ^{2}\psi }{\partial t^{2}}}$