Find the radius vector defining the position of a point source of spherical waves if that source is known to be located on the straight line between the points with radius vector `r_(1)` and `r_(2)` at which the oscillation amplitudes of particles of the medium are equal to `a_(1)` and `a_(2)`. The damping of the wave is negligible, the medium is homogeneous.

Explanation

Let S be the source whose position vector relative to 0 is `vecr`. I (intensity `prop a^2`) and also `l prop 1//d^2` where a = amplitude and d=distance
`therefore a prop 1//d`
`rArr a_1d_1=a_2d_2=k` (a constant)
Let `hati` be the unit vector along PQ Then `vecr=vecr)1+d_1hati`
`rArr vecr/d_1=vecr_1/d_1+hati` and `vecr=vecr_2+d_2(-hati)`
`rArr vecr/d_2=vecr_2/d_2-hati`
Adding `vecr[1/d_1+1/d_2]=vecr_1/d_1+vecr_2/d_2`
Substituting `d_1=a_2/a_1 d_2`
`vecr=(a_1vecr_1+a_2r_2)/(a_1+a_2)`