Why don't plane waves always expand according to Huygens' principle? I have seen a few videos about Huygens' principle and I don't understand why a plane wavefront doesn't expand both vertically and horisontally. According to my understanding of Huygens' principle, each point of the wavefront forms a "centre of disturbance" from which it sends a spherical wave. All of these waves interfere and form a new wavefront.
What I can't understand is why a plane wavefront doesn't also expand horisontally. The highest point of the wavefront ought to expand spherically as well, like the image suggests. Why doesn't this affect the appearance of the new wavefront?

 A: A plane wave, by definition, is already infinite in extent in the directions perpendicular to its propagation, so it has nowhere to expand to.  A wave that is finite -- as in your drawing--- is not a plane wave, and it will expand.
A: When you apply Huygen's principle you also have to think about all the different phases that might constructively or destructively interfere. An infinite plane wave is one example where the region of constructive interference is easy to think about but generally it will be much harder. That being said, plane waves that are not infinitely long do spread out a little. How much depends on the wavelength and extent of the wave. If the wavelength is small compared to the extent (i.e. it is generated by a large slit) the wave will move relatively straight. In the opposite limit if the slit is very small the wave will spread out like a point source. Using Mathematica I was able to make the following animation:

The final frame looks like this top down:

The left boundary generates sine waves, all the other boundaries are absorbing except for the two linepieces on the inside of the gap which are hard boundaries.
I used the following code to generate the animation
H = 5;
L = 5;
T = 5;
\[Delta] = .5;
a = H*.3;
box = Region[Rectangle[ {0, 0}, {L, H}]];
wall1 = Region[Rectangle[ {2 - \[Delta], 0}, {2 + \[Delta], H/2 - a}]];
wall2 = Region[Rectangle[ {2 - \[Delta], H/2 + a}, {2 + \[Delta], H}]];
region = RegionDifference[box, wall1];
region = RegionDifference[region, wall2];
\[Sigma] = .2;
wave2D = NDSolveValue[{D[u[t, x, y], {t, 2}] - 
      Laplacian[u[t, x, y], {x, y}] == 
     NeumannValue[-Derivative[1, 0, 0][u][t, x, y], x != 0] + 
      NeumannValue[1.5 Sin[8 t], x == 0], 
    u[0, x, y] == 0,
    DirichletCondition[u[t, x, y] == 0, 
     x > 0 && x < L && y >= H/2 - a - .01 && y <= H/2 + a + .01],
    Derivative[1, 0, 0][u][0, x, y] == 0
    }, u, {t, 0, T}, {x, y} \[Element] region];

plot = Table[
   Plot3D[wave2D[t, x, y], {x, 0, L}, {y, 0, H}, PlotRange -> {-1, 1},
     BoxRatios -> Automatic], {t, 0, T, .1}];
ListAnimate[plot]

A: A wave always expands in all directions. With the following exceptions:

*

*the wave occurs at an interface between different media, for example between water and air. Then a plane wave is visible to us. In reality, of course, there is also a periodic displacement of air that is not visible to us. And since air is not an elastic medium, the air wave disperses in all directions and extinguishes at a short distance from the interface with the water.

*the wave was excited along an infinite line. Then it cannot disperse along the wave front. Since in reality no plane wave is infinite, each wave disperses at its ends and thus gradually weakens the plane wave from the ends towards the centre. By the way, instead of an infinitely long wave front, you can take a channel, then the wave front, which also cannot disperse at its ends, moves undisturbed over very long distances.

*after an obstacle, where part of the wave is reflected back at the obstacle, this part of the plane wave is missing behind the obstacle and the remaining wave disperses as already described in case 2.

