Identify the wavefront equation

The plane wavefronts are given by $x+y+z=\text{constant}.$

Direction of propagation of a plane wave

For a plane wave, the direction of propagation is normal to the wavefront.

So we find the normal vector to the plane $x+y+z=\text{constant}.$

Its normal vector is $\vec{n}=\hat{i}+\hat{j}+\hat{k}=(1,1,1).$

Unit vector along direction of propagation

Magnitude of this vector is $|\vec{n}|=\sqrt{1^2+1^2+1^2}=\sqrt{3}.$

Therefore, the unit vector along the propagation direction is $\hat{n}=\frac{1}{\sqrt{3}}(1,1,1).$

Angle with the $x$-axis

If $\theta$ is the angle between the propagation direction and the $x$-axis, then $\cos\theta = \hat{n}\cdot \hat{i} = \frac{1}{\sqrt{3}}.$

Hence, $\theta=\cos^{-1}\left(\frac{1}{\sqrt{3}}\right).$

Match with options

This corresponds to: $\boxed{\text{B }\cos^{-1}\left(\frac{1}{\sqrt{3}}\right)}$

Comparison with stored correct answer

Stored correct answer is B, which matches the derived result.