Show that the function is a solution of the wave equation.
-$$w ( x , t ) = e ^ { x - c t }$$

Question

Quizplus · Accepted Answer

The wave equation is given by:

$$\frac{\partial^2 w}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 w}{\partial t^2}$$

Differentiating $w(x,t)$ with respect to $x$ twice, we get:

$$\frac{\partial^2 w}{\partial x^2} = \frac{\partial}{\partial x} \left( \frac{\partial w}{\partial x} \right) = \frac{\partial}{\partial x} \left( e^{x-ct} \right) = e^{x-ct}$$

Differentiating $w(x,t)$ with respect to $t$ twice, we get:

$$\frac{\partial^2 w}{\partial t^2} = \frac{\partial}{\partial t} \left( \frac{\partial w}{\partial t} \right) = \frac{\partial}{\partial t} \left( -c e^{x-ct} \right) = c^2 e^{x-ct}$$

Putting these derivatives in the wave equation, we get:

$$\frac{\partial^2 w}{\partial x^2} = e^{x-ct} = \frac{1}{c^2} \frac{\partial^2 w}{\partial t^2} = \frac{1}{c^2} c^2 e^{x-ct} = e^{x-ct}$$

Therefore, $w(x,t) = e^{x-ct}$ is a solution of the wave equation.

Show That the Function Is a Solution of the Wave w(x,t)=ex−ctw ( x , t ) = e ^ { x - c t }w(x,t)=ex−ct

Show That the Function Is a Solution of the Wave $w ( x , t ) = e ^ { x - c t }$