Graphing the Wave Equation with Quadratic Functions

[jonroberts74]

Homework Statement

set $$\phi = f(x-t)+g(x+t)$$

a) prove that $$\phi$$satisfies the wave equation : $$\frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2}$$

b) sketch the graph of $$\phi$$ against $$t$$ and $$x$$ if $$f(x)=x^2$$ and $$g(x)=0$$

The Attempt at a Solution

part a, I have already gotten the answer to; just posting that so that the second part makes some sense.

I don't really know how to do part b, the two functions given don't have a t, so not sure how I graph phi against x and t

 

[LCKurtz]

Homework Statement

set $$\phi = f(x-t)+g(x+t)$$

a) prove that $$\phi$$satisfies the wave equation : $$\frac{\partial^2 \phi}{\partial t^2} = \frac{\partial^2 \phi}{\partial x^2}$$

b) sketch the graph of $$\phi$$ against $$t$$ and $$x$$ if $$f(x)=x^2$$ and $$g(x)=0$$

The Attempt at a Solution

part a, I have already gotten the answer to; just posting that so that the second part makes some sense.

I don't really know how to do part b, the two functions given don't have a t, so not sure how I graph phi against x and t

If $f(x) = x^2$ and $g(x) = 0$, then $\phi(x,t) = (x-t)^2$. Plot that as a 3D surface with $\phi$ in the $z$ direction and $x$ and $t$ as the two independent variables.

 

[jonroberts74]

so its a parabolic cylinder?