Solution

The blue lines in the figure are the same wavefront at different equally-spaced time points. The angle of incident is $\angle DAB=\alpha$. When the wavefront reach the interface of the two media at $A$, $B$, $C$, these points become sources of new spherical waves. (The figure shows the moment when the wavefront has just reached $C$.) Since the red spherical wave originating from $B$ is tangent to the first blue line and the reflected wavefront (as the speed of the wavefront equals the speed of sound), we have $\triangle DAB\cong\triangle ECB$. Thus, the reflected angle $\angle ECB=\angle DAB=\alpha$.

The ratio of the radii of the red and green spherical waves originating from $B$ is $c_1/c_2$. Consequently,

$$\frac{\sin\angle ECB}{\sin\angle FCB}=\frac{BE/BC}{BF/BC}=\frac{c_1}{c_2}.$$

Therefore, the refracted angle is

$$\angle FCB=\arcsin\left(\frac{c_2}{c_1}\sin\alpha\right).$$