For long I've been thinking about this issue, but ended up with nothing :
Suppose you have a point source $S_1$ sending a spherical wave in space of equation
$$\phi=\dfrac{A_0}{r}\sin(kr-\omega t)$$
and intensity
$$I=h\left(\dfrac{A_0}{r}\right)^2=\dfrac{I_0}{r^2}.$$
Now suppose we place at the same position of $S_1$ another source $S_2$ sending just the same wave
$$\phi=\dfrac{A_0}{r}\sin(kr-\omega t).$$
What is the resultant intensity?
One may say that two sources will produce double the power, so double the intensity.
Another may say that the two waves, superposing in phase, will produce double the amplitude so four times the initial intensity.
Which one is correct? It is said that the second answer is the right one in the case of interference and that the extra energy comes from dark fringes, but here there are no dark or bright fringes; the two waves superpose constructively everywhere.
Please help.
Best Answer
Fairly easy, just add the two waves using superposition.
$\phi^\prime = 2 \phi = 2 \frac{A_0}{r} sin(kr - \omega t)$ for the resulting wave (the prime denotes superposition)
Using $I = h(A_0/r)^2$ we see that $I^\prime = I \times (2A_0/A_0)^2 = 4I$.
Therefore the intensity has become four times larger.