$P _{ atm }= P cm$ of water

What we can conclude from this process is that the volume is changing in the air bubble but the temperature remains unchanged.

For isothermal process,

$P _{1} V _{1}= P _{2} V _{2}$

Let the height of water surface be $x$.

$(P d g+x d g)\left(\frac{4}{3} \pi r^{3}\right)=P d g\left[\frac{4}{3} \pi(2 r)^{3}\right]$

$(P+x) r^{3}=P\left(8 r^{3}\right)$

$x=8 P-P$

$\Rightarrow x=7 P$