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

If P be the power of the point source then Intensity , `I=P/(4pir^2)`
or `l prop 1/r^2`
But `l prop "(amplitude)"^2`
So `A prop` 1/r or `A=a_0 //r`
Here `a_0` is constant
The equation in standard form is y=a sin k (r-vt)
Therefore above equation is written as `y=a_0/r sin k (r-vt)`
Now comparing this equation with `y=A sin (kr - omegat)`
we have `omega=kv`, or `n=(kv)/(2pi)` and `k=(2pi)/lambda`
or `lambda=(2pi)/k` , speed `c=nlambda=(kv)/(2pi)xx((2pi)/k)`
c=v and `T=1/n=(2pi)/(kv)`
Thus intensity is given by
`l=1/2 rhoa^2 omega^2 c`
`rArr l=1/2rho a^2/r^2 k^2 v^2 v =1/2 (rho a_0^2 k^2 v^3)/r^2`