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 particls 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 the reference point `O` is `vec(r)`. ltbr. Since intensities are inversely proportional to the square of distances,
`("Intensity at" P(I_(1)))/("Intensity at" Q(I_(2)))=(d_(2)^(2))/(d_(1)^(2))`
where `d_(1)=PS ` and `d_(2)=QS`
But intensity is proportion to the square of amplitude.
So, `(a_(1)^(2))/(a_(2)^(2))=(d_(2)^(2))/(d_(1)^(2))` or `a_(1)d_(1)=a_(2)d_(2)=k` ( say )
Thus ` d_(1)=(k)/( a_(1))` and `d_(2)=(k)/(a_(2))`
Let `hat(n)` be the unit vector along `PQ` directed from `P` to `Q`.
Then `vec(PS)=d_(1)hat(n)=(k)/(a_(1))hat(n)`
and `vec(SQ)=d_(2)hat(n)=(k)/(a_(2)hat(n))`
From the triangle law of vector addition.
`vec(OP)+vec(PS)+vec(OS)` or ` vec(r_(1))+(k)/( a_(1))hat(n) =vec(r)`
or `a_(1)vec(r_(1))+k hat(n)=a_(1) vec(r)....(1)`
Similarly ` vec(r)+(k)/(a_(2))hat(n) = vec(r_(2))` or `a_(2)vec(r_(2)) - k hat(n) = a_(2) vec(r)...(2)`
Adding `(1)` and `(2)`,
`a_(1) vec(r_(1))+ a_(2)vec(r_(2))=(a_(1)-a_(2))vec(r)`
Hence` vec (r)=(a_(1)vec(r_(1))+a_(2)vec(r_(2)))/( a_(1)+a_(2))`