Problem 11.25 Challenge

a) จาก \(y = x^{2} + v^{2}t^{2}\)

พิจารณา \(\frac{\partial y}{\partial t}= 2v^{2}t\)

\(\frac{\partial ^{2}y}{\partial t^{2}}= 2v^{2}\)

\(\frac{1}{v^{2}}\frac{\partial ^{2}y}{\partial t^{2}}= 2\cdots \cdots (1)\)

พิจารณา \(\frac{\partial y}{\partial x}= 2x\)

\(\frac{\partial ^{2}y}{\partial x^{2}}= 2\cdots \cdots (2)\)

\(\therefore (2) = (1)\)

\(\frac{\partial ^{2}y}{\partial x^{2}}=\frac{1}{v^{2}}\frac{\partial ^{2}y}{\partial t^{2}}\;\blacksquare\)

b)พิจารณา\(\frac{1}{2}(x+vt)^{2}+\frac{1}{2}(x-vt)^{2}=\frac{1}{2}(x^{2}+2xvt+v^{2}t^{2}\)

\(+ x^{2} - 2xvt + v^{2}t^{2})\)

\((x+vt)^{2}+\frac{1}{2}\frac{1}{2}(x-vt)^{2}=x^{2}+v^{2}t^{2}\)

\(\therefore f(x+vt)=\frac{1}{2}(x+vt)^{2}\) และ\(g(x-vt)=\frac{1}{2}(x-vt)^{2}\;\blacksquare\)