Simple expression for the bubble-time period of two clustered air guns

We have developed a simple method of estimating the bubble-time period of clustered air guns from the bubble-time period of a single air gun of the same type and volume. This was done by deriving a characteristic time scale for the normal Rayleigh equation, and then deriving the same scale from a modified Rayleigh system for clusters. Comparing the value for clusters with the value for a single-gun, we then estimate their relative bubble period, which gives a reasonable match (less than 4% relative error in the appropriate domain) to field data.