I don’t know why this question popped into my head, but it’s been sitting there for the past week and showing no signs of moving on.

Suppose an enemy of mine threw a friendly blue whale at me. Being a friendly whale, it makes the blue-whale-noise equivalent of “DUCK!” to warn me it’s coming.

How quickly does the whale need to be travelling for its warning to be useful?

I suppose I must’ve been musing on ambulance sirens or something like that. Anyway, the key to this is the Doppler effect: sounds from a source which is travelling towards you seem higher to you than they do at the source.

Here’s a formula for what happens to noises when the source is moving, paraphrased from Wikipedia:

$$f_r=\frac{c}{c-v}{f}_s$$

• $f_r$ is the frequency heard by the receiver (who’s stationary)
• $f_s$ is the frequency of the sound at its source
• $c$ is the speed of sound in the medium it’s travelling through (e.g. air or water)
• $v$ is the speed of the sound source towards the receiver.

So I just need to find reasonable values for $c$, $f_r$ and $f_s$, and I’ll be able to work out $v$.

The problem with whale noises is that they’re really really low. Or, I’d always thought that. I was surprised to read, on searching for “blue whale song frequency”, the following:

All blue whale groups make calls at a fundamental frequency of between 10 and 40 Hz, and the lowest frequency sound a human can typically perceive is 20 Hz. Blue whale calls last between ten and thirty seconds.

Wikipedia

It turns out that that doesn’t really mean that a human can hear a blue whale’s call – Wikipedia’s page on Hearing range says that humans are most sensitive to frequencies between 2 and 5 kHz. The whale’s call needs to get my attention – my whale-tossing enemy might have caught me by surprise – so I’ll assume the worst and pick a number from the top end of that range and say that $f_r=5000$Hz. The whale’s likely to be quite excited, so I’ll rely on it being able to make a loud noise at the top of its range, $f_s=40$Hz.

Next, I need to decide what medium this whale’s being thrown in. Water would be the most likely place to encounter a blue whale, but I reckon it’d be able to comfortably fight off any malefactor there, with a home-field advantage. So let’s assume the blue whale has been tricked onto land and is now travelling through air towards me. Google says the speed of sound at sea level (I live at the coast, so that’s reasonable) is $c=340.29$m/s.

So I’ve got all my ingredients. Let’s work out how fast this whale is flying:

$$\begin{array}{rl}2000& =\frac{340.29}{340.29-v}40⟹\\ v& =340.29-\frac{40}{5000}\times 340.29\\ & =337.56768\text{m/s}\end{array}$$

In other words, the whale needs to be going only about $2.7$m/s less than the speed of sound for me to hear it. That about matches my expectations, but obviously, this has implications for the usefulness of the call – will I have time to react to it?

Another quick google search informs me,

The average reaction time for humans is 0.25 seconds to a visual stimulus, 0.17 for an audio stimulus, and 0.15 seconds for a touch stimulus.

Backyard Brains

I’m not sure how much noise I need to hear before I react, but let’s assume that any amount of frightened whale scream is enough to get my attention. In $0.17$s, the whale will travel about $57.4$m. An adult blue whale (I don’t think any of my enemies are such terrible finks as to throw a baby whale) can grow up to $30$m long, so this would be quite a short trip for the whale.

All along I’ve assumed that I’m being taken by surprise, and the presence of a blue whale less than a couple of body lengths away would arouse my suspicion long before it starts moving, so I’m happy to assume the whale is travelling quite a bit further than $57$m. That means I’ll have bags of time to react and get out of the way.

So I can finally put my mind at rest, with the following list of requirements for surviving an airborne-whale attack:

• The whale is cooperative.
• In a whale choir, the whale would be a soprano.
• The whale is thrown from at least two body lengths away.
• The whale is flying through the air at about the speed of sound, minus a brisk jogging pace.

Don’t do maths, kids!