How much power and energy is (actually) in a 230 dB "click" from a whale?

Question

In the BBC Science in Action podcast the acoustic power of a 1 millisecond "click" from a whale was said to be 230 dB. The related BBC article mentions this number and elaborates:

The blue whale is not the loudest animal on Earth, despite what you may have learned in school. While its calls are claimed to be louder than a jet engine at take-off, clocking in at an impressive 188 decibels (dB), the sperm whale is actually louder: its communicative clicks have been measured at 230 dB.

If I use the following equation from here;

$$I(dB)=10\log_{10}\frac{I}{I_0} $$

rearrange it,

$$I= I_0 \ 10^{I(dB)/10} $$

and plug in the value of 10^-12 watts/m² given there and use a nominal area of 1 m², I get 10^23-12 = 10¹¹ watts/m²! So, if the duration of the "click" is 1 millisecond, that's 100 MJ of energy. It's not surprising then that it can also use acoustic pulses to disable or stun potential prey.

But the article goes on to say:

This record raises another important point about loudness. Decibels in water are not equivalent to decibels in air. "Water is denser than air, so sound travels through it differently – the speed of sound is different," says bio-acoustics expert James Windmill from the University of Strathclyde in the UK, who discovered the water boatman's remarkable call.

Roughly, to convert from dB in water to dB in air, you have to subtract [around] 61 dB from the reported sound level."

I have a feeling this is somehow related to sound pressure vs intensity; force/area versus power/area. Water is roughly 1000x as dense as air.

Question: I am not sure if I should be using this 61 dB offset in my calculation above. Is the energy of the "click" still 100 MJ?

Answer 1

First, I went into great detail on the propagation of sound in different media in the following answer: https://physics.stackexchange.com/a/266046/59023.

Sound

Intensity (or specifically sound intensity) of a linear sound wave is related to sound pressure, $P$, through: $$ I = \frac{ P^{2} }{ \rho_{o} \ C_{s} } $$ where $\rho_{o}$ is the mass density and $C_{s}$ is the speed of sound in the medium. One can look up the properties of water to find that $\rho_{o}$ ~ 999.972 kg/m³ and $C_{s}$ ~ 1484 m/s. We can also look up the reference pressure level for water (or at NOAA) finding $P_{H2O}$ ~ 1 $\mu$Pa (compared to $P_{air}$ ~ 20 $\mu$Pa) at 1 meter from source. This corresponds to a reference intensity of $I_{o} \sim 6.74 \times 10^{-19}$ W/m² for water.

We can then use: $$ I = I_{o} \ 10^{L/10} $$ where $I$ is intensity (in W/m²) and $L$ is intensity (in dB). If we use the intensity I show above and $L$ ~ 230 dB, then $I \sim 6.74 \times 10^{4}$ W/m² under water.

If we use your 1 m² area and $\Delta t$ ~ 1 ms, then the total energy would be ~67.4 J.

Answer

I am not sure if I should be using this 61 dB offset in my calculation above. Is the energy of the "click" still 100 MJ?

No, it is much lower at ~67.4 J. The $\sim 10^{11}$ W you found is rather extreme when you consider that it corresponds to peak electrical power consumption of France. Though whales are rather enormous mammals and a ~10,000 calorie diet does correspond to $\sim 10^{7}$ J of energy (i.e., diet of an Olympic athlete, which is probably much less than a sperm whale), that would still be a lot of power to produce.