Nice way to show that plane wave is a limit of spherical wave I am looking for some nice way to show the (university) students that a plane wave is a limit of a spherical wave $\frac{A}{r}e^{i\omega t - ikr}$ for large $r$ (or, of course, that it is a reasonable approximation for a large $r$ domain). It is for purposes of acoustics, but it should be shown for general case. 
There are multiple approaches certainly, such as: 


*

*showing that acoustic pressure and velocity get in phase for large $r$ which is typical plane wave feature.

*showing that amplitude rate of change converges faster than magnitude of amplitude


but I find these "not elegant". Isn't there something better?
 A: I would start with moving into a coordinate system where the centre of the wave is far away, let's say at $-x_0$. So
$$r = \sqrt{(x+x_0)^2+y^2+z^2}$$
If we now assume that $x_0 \gg x,y,z$, then we can make a Taylor expansion and end up with
$$r \approx x_0 + x$$
where we have neglected higher orders of $x$, $y$, and $z$. This approximation also makes sense intuitively. Moving along the $x$ direction changes the radius by the amount we move. Moving perpendicular to the $x$ direction will, to first order, not change the distance to the centre of the wave.
With this the wave now looks like
$$\frac{A}{x_0+x}e^{i\omega t-ik(x_0+x)}$$
The $-ikx_0$ term in the exponent can be factored out and gives a constant phase shift. The $x_0 + x$ in the denominator can be approximated by $x_0$ because $x_0\gg x$. So we end up with
$$\frac{A}{x_0}e^{-ikx_0}e^{i\omega t-ikx}$$
This is a plane wave with constant amplitude $A/x_0$ and an additional phase shift of $e^{-ikx_0}$.
A: I do not know if you will consider this not rigorous enough or indeed totally wrong?
There are two part to your equation $\displaystyle \frac{A}{r}e^{i\omega t - ikr}$ which perhaps is better written as $\displaystyle \frac{A}{r}e^{i\omega t - i\vec k \cdot \vec r}$.
For a spherical wave $\vec k \cdot \vec r = kr$ if $\vec r$ is the position vector from the source.
For a plane wave moving in the positive $x$ direction $\vec k \cdot \vec r = kr \cos \theta = kx$ where $\theta$ is the angle between the $k$ vector and the $r$ vector.  
For small angles $\theta, \cos \theta \approx 1$ which is equivalent to $x \approx r$.  
The other part to consider is $\dfrac 1 r$ and for this look at  
$\dfrac{d(r^{-1})}{dr} = -r^{-2} \Rightarrow \Delta (r^{-1}) \approx - r^{-2} \Delta r \Rightarrow \Delta (x^{-1}) \approx - x^{-2} \Delta x$   
which is your "showing that amplitude rate of change converges faster than magnitude of amplitude (changes)" and is equivalent to doing a Taylor expansion?
So you end up with $\displaystyle B\; e^{i\omega t - ikx}$  where $B$ changes less and less over a given range of $x$ values as $x$ gets larger and larger.
