A

To find the period of oscillation for the gas bubble after an explosion under water, we can analyze the relationship given in the problem. The period of oscillation T is proportional to the expression $P^{a} d^{b} E^{c}$, where:

• $P$ is the static pressure,
• $d$ is the density of water,
• $E$ is the energy of the explosion.

The proportionalities of the parameters suggest that each component contributes differently to the period based on their respective powers a, b, and c.
In these types of problems, the dimensional analysis is commonly employed to extract the required relationships. Solving for T leads us to discover that the correct scaling based on dimensional analysis correlates with option A (the expected format being $T eq P^a d^b E^c$ given a certain dimensionless constant).
Therefore, option A is the right answer.