A wave travels out in all direction from a point source. Justify the expression `y=(a_(0)/r) sin k (r-vt)`, at a distance r from the source. Find the speed, periodicity and intensity of the wave. What are the dimensions of `a_(0)`?

Explanation

Let `P` be the power of the source.
Then `I=(P)/(4pir^(2)`
`:. Iprop(1)/(r^(2)) but Ipropa^(2)`
`:. Aprop(1)/(r )` The equation in the standard form is
`y=a sin k (r-vt)`
Therefore, `y=a_(0)/r sin k(r-vt)`, simply replacing a by `(a_(0))/(r)` where `a_(0)` is a constant.
Comparing with the usual form `y=a sin (kr-omegat)`
`omega=kv` or `f=kv//2pi`
and `k=2pi//lambda or lambda=2pi//k`
`:. c=flambda=(kv)/(2pi)xx(2pi)/(k)=v`
and `T=1//f=2pi//kv`
`I=(1)/(2)pa^(2)omega^(2)v=(1)/(2)p(a_(0)^(2))/(r^(2))k^(2)v^(2)v=(1)/(2)pa_(0)^(2)k^(2)v^(3)//r^(2)`