Solution

Huygens–Fresnel Principle (recall)

At time t, the position of the wavefront is known.
Each point on that wavefront becomes a source of secondary spherical waves of radius c\tau (where \tau is the considered time interval).
The new wavefront at time $t+\tau$ is the envelope surface of all those secondary spheres.

Note: c is the speed of light in vacuum.

Part a)

Initial assumption:

At time t = 0, the wavefront is the plane x = 0 (perpendicular to the x-axis, propagating to the right).

We apply Huygens–Fresnel:

· Each point $P(0, y, z)$ on the plane x = 0 emits a secondary spherical wave that reaches a radius $c\tau$ in time $\tau$.
· The spheres have centers on the plane x = 0 and all have the same radius $R = c\tau$.

Geometry of the envelopes:
The envelope of a set of spheres of equal radius whose centers lie on a plane is another parallel plane, located at a distance equal to the radius.

The spheres centered at (0, y, z) just touch the plane $x = c\tau$.
No sphere reaches beyond $x = c\tau$.
The plane $x = c\tau$ is tangent to all spheres (because the minimum distance from any center (0, y, z) to the plane $x = c\tau$ is exactly f c\tau)$.

Result:

The new wavefront at time $\tau$is the plane$x = c\tau$. Therefore, in time $\tau$ the wavefront has advanced a distance $c\tau$ in the direction perpendicular to itself.

Speed:

$v = \frac{\text{displacement}}{\text{time}} = \frac{c\tau}{\tau} = c$

$\boxed{\text{The plane wavefront moves with speed } c \text{ perpendicular to the plane.}}$

Part b)

We apply Huygens–Fresnel:

Each point P on the spherical surface of radius R emits a secondary spherical wave of radius$c\tau$ after time$\tau$.

Geometry of the envelope:
The centers of the secondary spheres lie on the original sphere of radius R.
All secondary spheres have the same radius $c\tau$.

The envelope of all these spheres is another sphere concentric with the original one, whose radius is $R + c\tau$.

Justification:
A point Q is on the outer envelope if there exists a center P on the original sphere such that Q lies on the corresponding secondary sphere and, moreover, the direction $O \to P \to Q$ is radial. In that case:

$OQ = OP + PQ = R + c\tau$

Since this holds for all directions, the envelope is the sphere of radius$R + c\tau$.

Increase in radius in time $\tau$:

$\Delta R = (R + c\tau) - R = c\tau$

$\boxed{\text{The radius of a spherical wavefront increases by } c\tau \text{ during a time } \tau.}$