If a spherical wave is described by $\psi \left(r,\:t\right)=\left(A\:\frac{cos\left(kr-\omega t+\phi \right)}{r}\right)$ , is it a function of time?

Question

If the wave equation is $\psi \left(r,\:t\right)=\left(A\:\frac{cos\left(kr-\omega t+\phi \right)}{r}\right)$, and $k=\frac{\omega }{v}$, and $t=\frac{r}{v}$, wouldn't that give:

$\psi \left(r,\:t\right)=\left(A\:\frac{cos\left(\frac{\omega }{v}r-\omega \:\frac{r}{v}+\phi \right)}{r}\right) = \left(A\:\frac{cos\left(\phi \:\right)}{r}\right)$ which is only a function of distance, and not time?

Answer 1

By putting $t=\frac{r}{v}$ you have fixed a time, (even if it's a different time for different parts of the wave) - so yes, in that case the function $\psi$ then depends on $r$.

However if you had chosen to fix the time at a different value, the resultant values of $\psi$ could have been different.