1 Questão: Se z=\operatorname{sen} \alpha x \operatorname{sen} \beta y \operatorname{senkt} \sqrt{a^{2}+b^{2}}, mostre que \frac{\partial^{2} z}{\partial t^{2}}=k^{2}\left(\frac{\partial^{2} z}{\partial x^{2}}+\frac{\partial^{2} z}{\partial y^{2}}\right).

Passo 1

O enunciado define a função
z(x,y,t)=\sin(\alpha x)\,\sin(\beta y)\,\sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr)
e pede para mostrar que
\frac{\partial^{2}z}{\partial t^{2}} =k^{2}\Bigl(\frac{\partial^{2}z}{\partial x^{2}}+\frac{\partial^{2}z}{\partial y^{2}}\Bigr).
Vamos usar derivadas parciais de produtos e a regra da cadeia para \sin.

Passo 2

Calcular \displaystyle \frac{\partial^{2}z}{\partial t^{2}}. Como \sin(\alpha x) e \sin(\beta y) são constantes em relação a t,
\frac{\partial^{2}z}{\partial t^{2}} =\sin(\alpha x)\,\sin(\beta y)\,\frac{d^{2}}{dt^{2}} \sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr).
Pela regra da cadeia,
\frac{d^{2}}{dt^{2}} \sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr) =-\bigl(k\,\sqrt{a^{2}+b^{2}}\bigr)^{2} \sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr).
Logo
\frac{\partial^{2}z}{\partial t^{2}} =-k^{2}(a^{2}+b^{2}) \sin(\alpha x)\,\sin(\beta y)\, \sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr) =-k^{2}(a^{2}+b^{2})\,z.

Passo 3

Calcular \displaystyle \frac{\partial^{2}z}{\partial x^{2}} e \displaystyle \frac{\partial^{2}z}{\partial y^{2}}. De modo análogo,
\frac{\partial^{2}z}{\partial x^{2}} =\sin(\beta y)\,\sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr)\, \frac{d^{2}}{dx^{2}}\sin(\alpha x) =-\alpha^{2}\,z,
\frac{\partial^{2}z}{\partial y^{2}} =\sin(\alpha x)\,\sin\bigl(k\,t\,\sqrt{a^{2}+b^{2}}\bigr)\, \frac{d^{2}}{dy^{2}}\sin(\beta y) =-\beta^{2}\,z.
Portanto
\frac{\partial^{2}z}{\partial x^{2}} +\frac{\partial^{2}z}{\partial y^{2}} =-(\alpha^{2}+\beta^{2})\,z.

Passo 4

Multiplicando o resultado anterior por k^{2} e usando \alpha^{2}+\beta^{2}=a^{2}+b^{2}, obtemos
k^{2}\Bigl(\frac{\partial^{2}z}{\partial x^{2}} +\frac{\partial^{2}z}{\partial y^{2}}\Bigr) =-k^{2}(a^{2}+b^{2})\,z =\frac{\partial^{2}z}{\partial t^{2}}.

Resposta

\frac{\partial^{2}z}{\partial t^{2}} =k^{2}\Bigl(\frac{\partial^{2}z}{\partial x^{2}} +\frac{\partial^{2}z}{\partial y^{2}}\Bigr).

Exercícios relacionados