Why Can't We Directly Compare Sounds in Water and Air?

One of the biggest, and most common, mistakes that people make when writing about underwater acoustics is making direct comparisons between the loudness of things in water and in air.  Here's an example of something that you might read about underwater sound:

"The bottlenose dolphins were being exposed to sounds up to 160 dB.  That's  louder than a 747!"

Well, actually, it's not.   

The problem here is related to the units with which we measure sound. First of all, we measure sound using the decibel.  Decibels are logarithmic units. Basically, logarithmic scales allow us to make big numbers smaller so that they are easier to visualize. For example, in the chart below, you can see that when we take the log of value for x (in green), the log values (in blue) are decreased more for large values of x than for small values of x. The opposite is true of taking an exponential of x, such as x^2 (in red).

Why make things so complicated when we talk about sound? Well, human beings actually perceive sound pressure in a log scale.  For example, you wouldn't really be able to tell the difference in loudness between 6 trumpet players and 7 trumpet players, because your brain doesn't process sound lineally. In fact, every time you want to change the loudness level by just three measly decibels, you have to double the power.  So, if you want to increase the loudness of your trumpet players by 3 dB, you'll need 6 more trumpets.  

Doublemint gum only makes you 3 dB louder, just ask the dolphins.

OK, so now we understand decibels, why can't we compare dB in air with dB in water?

First of all, decibels must always be referenced to some standard. In air, decibels are referenced to 20 micropascals (pascals are a unit of pressure). In water, they are referenced to 1 micropascal.  This means that, from the start, 26 dB must be added to the sound level to make the two references levels the same.

Secondly, water is much denser than air, so the effects of pressure are also different. Although it is possible to compress water, it is much more difficult to do so.  In fact, when we add the difference in densities between air and water into the equation, we find that the sound pressure in water is 60 times greater (or 35.6 dB higher) than the same value in air.

Loud, but not as loud as a plane. Photo from the CRRU.

So, if white-beaked dolphins are producing whistles that are between 139 and 148 dB, that doesn't mean that they are making sounds louder than a 747. If we convert those numbers to values in air, we get somewhere between 103 and 113 dB.  Which is still pretty darn loud, but not quite as loud an airplane engine.  It's also good to point out that dolphin whistles are "narrowband" sounds. I explained narrowband in an earlier blog post, but basically what this means is that you can't imagine a whistling dolphin sounding like a jackhammer, because the sounds cover a very different range of frequencies and are produced by very different mechanisms.

So, next time you read something comparing underwater sound compared to sound in air, check to see if they did their conversions right!

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What is looking for dolphins in the ocean really like?

Many people are curious about what it is like to be out on a boat looking for dolphins in Hawaii. Lots of people think it's easy - my job is simply to sit on a boat and watch dolphins. Many people have a hard time believing that you can actually get paid to lounge on a boat and watch dolphins frolic in the surf.  The reality is a lot more complicated.

This is not what I look at all day (photo from FIRMM.org).

Finding dolphins, especially in Hawaii, can be really time consuming and difficult. Especially if you are interested in the types of dolphins that don't generally hang out near shore, which is most Hawaiian species (the exception being spinner dolphins).

Searching for dolphins at sea requires something called a "search image." Visual predators use search image all the time to find food. For example, blue jays look for the color, shape, or pattern of a moth to find their food. Prey can also use a search image to look out for predators: sea lions also use a visual search image to watch out for hungry killer whales.  If a killer whale doesn't match their search image - for example, by having a weird, bent over dorsal fin - sea lions don't always recognize it as a threat.

Biologists also use search images. Bird biologists look for something moving against a still background. Snail biologists look for a round, white spiral in a sea of shaking green leaves. And dolphin biologists look for moving triangles (dorsal fins) in a sea of other moving triangles (waves).

Lots and lots of blue-gray triangles. But which is a dolphin?
Photo by Alexis Rudd, 8/15/12.

Looking at waves is a lot like looking at the ants on a tv screen.  It's kind of like watching random noise (although obviously I am oversimplifying things to give you an idea, waves aren't really random).

So, most of the time, when whale and dolphin observers are  looking at the ocean, it's like looking at this (you might want to adjust your screen so you can only see one of these .gif files at a time - it's making me kind of queasy):

When a large whale comes along, the observer sees a shape in the noise that isn't changing for a little while, like this:

Did you find the "whale?" Hint: it's the blacker spot that appears in the bottom right hand corner for about 3 seconds.

Dolphins are even harder to see, because they move faster. When a scientist is searching for a dolphin, they look for a shape that isn't changing very much, but might be moving across the waves.  Can you spot my "dolphin?"

That one's kind of hard (I am having trouble, and I made the thing!), so I made the "dolphin" pink.  Now can you see it?

This week, I've been out on the water on the big island, helping look for whales and dolphins. We usually get out on the water at about 6:00 am (sometimes a little earlier) and come back in at 3:00 pm (if we're lucky unlucky). For the last two days, we have seen two groups of animals per day. That means that we have effectively been staring at noise for about 8 hours, searching for that tiny little clue that will tell us where the dolphins might be.  Coastal cetaceans are not spread out as much as in the tropics, so there is not always as much time staring at noise.

Isn't the human brain an amazing thing? It can pick up a tiny difference in pattern -

"That bit of the ocean isn't moving as much as the other bits!"

out of an enormous amount of noise. There isn't much on earth that's more enormous than the ocean.


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What You Know* About the Difference in Dolphins and Porpoises is Wrong**

*Mostly  **Some of the time

Last weekend I was lucky enough to speak at a math workshop for high school teachers put on by some of the University of Hawaii Match Department faculty.  And let me just start out by saying that it was AWESOME.  As anyone who has ever read "How to Win Friends and Influence People" knows, one of the best ways to get people to like you is to ask them questions about themselves.  For scientists, this extends to talking about our research. Beware asking a scientist good, intelligent questions about their work: you may be there for hours!

So, talking to the math teachers was an incredible treat.  They asked great questions, including one of my favorites, "What's the difference between a dolphin and a porpoise?"  Porpoises are from the cetacean family Phocoenidae, while dolphins are in the family Delphinidae.  I love this question, because there are so many misconceptions about what makes a porpoise.  This cartoon does a great job of showcasing those misconceptions:

Let's get the easiest misconception out of the way first: size.  I'll admit the average weight of porpoises is less than the average weight of dolphins.  However, there are many examples in which this just isn't true.  For example:

Dall's Porpoise (up to 200 lbs)

Hector's dolphin (up to 125 lbs)

So size of the animal isn't exactly the best diagnostic to use when deciding whether something is a dolphin or a whale.  How about rostrum (head) shape?

  Risso's Dolphin by Greg Boreham                             Finless Porpoise

Hmm, one of those is a dolphin and one is a porpoise, but they seem to actually have very similarly shaped heads.

OK, FINE! But if it has a bottlenose like Flipper, we DEFINITELY know it's a dolphin, RIGHT?

Northern bottlenose whale  (Hyperoodon ampullatus, in Family Ziphidae)

ARGHHHH!  It must be the fins, then. Dolphins HAVE to have pointy dorsal fins (the one on their back), and porpoises have rounded fins!

Chilean dolphin (rounded)                                                      Dall's Porpoise (pointy)

No dorsal fin at all (Southern Right Whale Dolphin).
The finless porpoise also has no dorsal fin, as the name implies.

OK, I give up, there must be SOMETHING I can use to figure out whether this random cetacean I got in the mail is a dolphin or a porpoise - but what????

TEETH!

Bottlenose dolphin                                |                             Harbor Porpoise

Dolphins have cone-shaped teeth throughout their mouth, while porpoises have what are known as spade-shaped teeth.  Here's a comparison, with A being the porpoise tooth and B being the dolphin tooth.

Dichotomous keys are tools that scientists use to figure out what species they are looking at.  These keys are kind of like those "build your own story" books that you had when you were a kid. You are asked a series of questions, and the questions lead to an ending.  It's also a lot like 20 questions, with each question further narrowing down the possibilities of which species you could have.  In fact, if you were using a dichotomous key to determine the species of a cetacean, the tooth question is what separates the porpoises from everything else.  

Try keying out whale, dolphin, and porpoise species
at the Marine Species Identification Portal!

In general, can you use these rules to distinguish between a dolphin and a porpoise?  Sure you can. But it's important to be aware of the many exceptions. Let's be honest, the likelihood of encountering Dall's Porpoise (found on the US west coast) and Hector's Dolphins (found in New Zealand) on the same day is infinitely small.  Even smaller is the likelihood that they'll give you a good look at their teeth. The best plan is to be knowledgeable about the species you are likely to encounter in your area.

Thanks for the great questions, Math Teachers of Hawaii.  If you'll notice, I haven't actually given much space to the actual workshop.  I'll get to that later. See what happens when you ask a scientist about something that interests them?!

If you ask a scientist a question...
they'll want a cookie.

Who am I kidding, I always want cookies.

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We Should Have Studied Dolphins in High School Math

When I look back at my educational career, it's really striking how poor of a student I was.  That is, I was a pretty mediocre student in the "hard" sciences until I took Marine Biology class during my senior year of college.  Most of the time, I think I was just really bored.  It wasn't always because the material wasn't challenging or useful (although sometimes this was the case). It was often because I had no idea what the point of it was.  OK, so in physics we could calculate where a ball would land based on the angle and force.  I only played baseball when forced to in gym class, and when I did I had no control over the force and angle of my wild swings.  I didn't have a home catapult, and I just didn't get how this stuff could really matter to ME.

It wasn't until I got really interested in marine biology, and specifically whale and dolphin bioacoustics, that I started seeing how amazing and cool math & physics can be.  If you read my blog, you know that bioacoustics is the study of the sounds of living things.  Whales and dolphins use sound to communicate and find food underwater, so even if you can't see them, you can very often hear them.  Some whales, such as beaked whales, are actually easier to find by listening than by looking with your eyes.  In addition, sound travels further underwater than light, so people at the surface can even hear whales that are thousands of feet underwater, or even hundreds of miles away. We can use these sounds to track the location of the animal, and we can do it using high school math.  Super cool.

Why didn't we do this in my high school trigonometry class?

It's really pretty simple.  First, you need something in the water making a sound.  Let's say... a whale.

Then you'll need a boat and something to record the sound.  We're going to use a hydrophone array, which is made up of three underwater microphones (hydrophones) strung in a line, with 15 meters between each hydrophone.  

When the whale makes a sound, it travels in a straight line through the water* to each hydrophone, arriving at the closer hydrophones sooner and arriving at the further hydrophones last.  Let's say the sound arrives at our recordings at time 0 for hydrophone 1, 0.006 seconds for hydrophone 2, and 0.0007 seconds for hydrophone 3.

How on earth do we figure out the distance to a whale when all we know is the difference between when we heard it on hydrophones 1, 2, and 3?  Easy - there's a simple equation for that!

The sound speed in water is 1500 m/s (which is why I put the hydrophones 15 m apart - to make the math easier).

We've replaced some of the words with numbers, something I HATED in high school.  Oh wait, I still hate it, so I'll just explain what the equations mean: The time difference between hydrophone 1 and 2 (t21) is equal to the difference between the distance between the whale and hydrophone 1 and the distance between the whale and hydrophone 2 (d2-d1) divided by the speed of sound.  Holy mother of god, this is just geometry!  I could have been doing this when I was 15! Some of you over-achiever types probably took geometry in 8th grade, so you could have done this when you were 14. Not me!

With a little algebraic rearrangement (which I am not going to bore you with, since you can look it up here), we get:

**

Sx and Sy are the x and y locations of the whale. C the speed of sound in salt water, 1500 m/s.  dh is the distance between the hydrophones, which in this case is 15 m, and all the ts refer to the time difference between the hydrophones (the small numbers tell you which two hydrophones: t21 is the difference between the arrival at hydrophone 2 and 1, etc).  All you have to do now is plug in the numbers. Edited 5/12/2012: To make it even easier, I've provided a spreadsheet here.

**

Then you can graph the location of the whale (-16.7, ±241.7), along with the locations of your hydrophones at (0, 0), (0, 15), and (0, 30).  I've just made designated hydrophone 1 at (0, 0) to make things easy, but in real life you'd have a GPS on the boat and you'd have to add the length of the hydrophone cable to the location of the boat's gps.

You might notice something funny about this equation - it actually gives two solutions - one on either side of the line of hydrophones.  That's a big problem with using three (or four, five, etc.) hydrophones in a row - you're not really sure what side of you the whale's on.  The reason for this is that there are two possible solutions for Sy. However, you can figure out which side the whale is on by turning the boat.  The location that stays in relatively the same place regardless of where the hydrophones are is the correct one.  Other ways to fix the left-right ambiguity? Make your hydrophones into a 2D shape (hard to do when dragging them behind a boat because 2d shapes are not very streamlined)*** or use DIFAR.

I wish I had done this kind of math in high school - it is totally within the realm of a high school student!  I think I might have paid better attention to something cool like finding whales.  On the other hand, maybe not.  That boy across the room WAS pretty dreamy.  Not all kids will be interested enough in whales and dolphins to get interested in math and physics, but the cool thing about math and physics is that, regardless of what you're into, I bet it applies.  From video games to raising miniature ponies, there's math in there somewhere.  Sometimes it just takes the right spark to get people interested.

P.S. If I made any mistakes above, please call me out.  Like I said, I didn't do so well in math class.  Also, if you're interested in a copy of the xcel spreadsheet with the equations already in it, shoot me a message and I'll send you a copy.

* This isn't always the case - sometimes the sound bounces off the bottom or the surface to get to the hydrophone, but I'm ignoring those cases here for simplicity.

** Edit: May 9, 2012. I made a HUGE mistake the first time I posted this and completely ignored the equation for t1 (I thought t1 was a typo for t12).  This is now fixed.

*** Edit: May 8, 2012.  Changed "3D" to "2D" because of a comment from Sam Denes. I had thought a straight line was already 2D, but apparently a straight line is 1D and adding another point off the line makes something 2D.

Note: I got to talk to a bunch of high school math teachers about this math.  Recap here.

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