Whale Research Ain't No Picnic: Required Reading

The Whale's Picnic by TekKiah

Last week I was lucky enough to go to Maui and help out a fellow researcher*, Dr. James Darling. Jim researches humpback whale song - in fact, he was one of the first people to actually see a humpback whale singing, and to realize that only males sing. He's also the author of several books and articles on whales. In addition to the great science writing, these books have excellent descriptions of the life of a whale biologist. As Jim describes, whale research is not for the faint of heart or the easily discouraged:

"Finding whales in a good study location is not the end to the challenges of studying whale behavior at sea - it's just the beginning. Think for a minute of the factors involved. 

First, there is the ocean, which can change from calm to life threatening - at times in a matter of minutes. Then there are the whales themselves, with movements restricted by nothing but the shoreline and physiology, ranging over huge distances. Moreover, they spend 90 percent of their time hidden underwater. 

Then there is the assortment of electrotechnical equipment such as digital cameras and recorders, hydrophones (underwater microphones), and GPS units that have become the mainstays of research and must work as they bounce around in damp, salty conditions on small boats. There are the boats themselves, prone to breakdowns and periodic downtime for maintenance. 

The flow of a research season often goes like this: Whales are present, but ocean is impossible; or ocean is calm but the whales are gone; or both whales and ocean are good but the boat breaks down; or everything is working but the rain last night ruined the visibility underwater. (You get the idea)." 

From: Hawaii's Humpbacks: Unveiling the Mysteries (if you purchase it from the whale trust, the money goes to fund research) 

This last paragraph is just about the best description of whale research I have ever read, and I think it should be required reading for the friends of whale researchers (or people thinking they want to go into whale research). Basically, if you get into whale research, be prepared for everything to go wrong.

Even when things are going right, whale research is often hard, uncomfortable work:

"The romance of whale research dies fast. One's motive has to be strong and clear when sitting on a hard rock cliff in the Patagonia desert with a near-freezing wind driving sand through your clothes - for eight or more hours a day, every day, for two months. Or spending a day in a small open boat, the pounding waves slowly driving your spinal column into your brain; or camping on arctic ice in wind so strong the only reason the tent stays in place is because you are lying in it - while polar bears lurk around on the surrounding ice pack; or finding yourself sick in the equatorial heat, or out at sea in a tiny boat crammed with gear. 

To do all this, one has to be dedicated." 

From: With the Whales (out of print, but you can find it used on Amazon).

These descriptions make me extremely happy, because they do such a great job of describing what an incredible pain in the butt (or spinal column) whale research can be. And they describe the kind of person you should strive to be to do that research: dedicated, tolerant to pain, discomfort, boredom, and frustration, and passionate about what you are doing.**

*And by "fellow researcher" I mean "super cool whale scientist with tons of experience who it was a privilege to help out for a couple of days.
**I also think these things apply to other types of biologists (see this video for an example).

For Those About to Comp (We Salute You)

In December, I took (and passed - phew!) my oral comprehensive exams. 

I thought it might be useful to other students to write down some tips for things that were helpful while studying for my comps and picking my committee. As an acoustical aide for the rest of this post, please play the following video while reading:

Alexis' Advice for Comps*

0) Write your proposal

1) Pick a good committee

Not the ideal
committee member

• If someone says that a possible committee member (PCM) was horrible on their committee, don't ask them to be on yours. 
• If your advisor doesn't like a PCM, they probably aren't going to be a good fit. 
• If you don't get along with someone, they probably won't make a good committee member.

Your goals when picking a committee are to 

1. Fulfill your department's requirements (my department requires three of the members to be full graduate faculty - see page 15)
2. Pick a group of people who will help make your science and your dissertation the best that it can be and keep you on track to graduating in a reasonable amount of time. 

In addition to your four regular committee members, you should have one "outside" member, whose primary function is "to ensure that standards and procedures are fairly applied." This person is basically supposed to be your advocate, making sure that the rest of your committee members don't try and make you jump through unrealistic hoops. Not all of your committee members have to be in your field of research (especially the outside member). It can be helpful if their specialty is something that complements your research in some way, because then you can go to them for advice. For example, I have people who specialize in line-transect surveys, underwater sound recording, and underwater localization on my committee, all of which are a part of my dissertation. It has been very helpful to be able to ask these people for help and feedback. The specialties don't always have to be this specific - an ecologist or statistician is always helpful!

2) Talk to all your committee members about what they want you to know

Do your committee members expect you to know everything about science since the dawn of time, or do they expect you to know things relevant your field? I asked my committee members to give me a list of papers/book chapters that they would like me to read. Narrowing down your subject area doesn't mean that you won't have much to study- far from it! It just means that you have some idea of what you should be covering, allows you to make a study schedule, and keeps you a little more sane.

My pile of comps reading material, with tequila and lemon for scale.

Also, it is OK to talk to your committee members about your level of knowledge and what is reasonable for them to expect you to know. For example, two of my committee members have degrees in Engineering, but the last class I took in math was Calculus (in 2001). Thus, it wasn't really realistic of them to expect me to know advanced engineering, but I did study linear algebra, matrices, and reviewed my calculus before the exam.

3) Set a date for your comprehensives (several months ahead of time).

Setting the date for your comps is great, because it gives you a date and time by which you HAVE to get stuff done. Setting it several months ahead of time (I suggest at least 3) is also great because it gives you a chance to break up your studying into small, manageable chunks. Which leads me to my next point:

4) Break up your work into daily chunks

Ration your reading.

I have a quote taped to my computer monitor at home, which reads:

"We often underestimate what we can do in the short term and underestimate what we can do in the long term, if we do a little each day."

For example, let's take one of my comps reading books, Principles of Marine Bioacoustics. This book has 657 pages, none of which are light reading. But, my committee chair (and advisor) had told me to read the whole book. Instead of trying to read the entire thing at once, and frying my brains in the process, I broke the book up into 10-page chunks and started reading about 3 months before my comprehensives. At a little more than 10 pages per day, the book took ~60 days to read, and didn't totally burn me out. Even though it felt like I wasn't getting anywhere at first, I read the entire thing with a couple weeks to spare for reviewing. At the same time, I also broke up my other review materials into manageable chunks. Overall, I probably studied 3-6 hours a day, which was much less exhausting than trying to cram it all into my head in the two weeks leading up to comps.

5) Study the hard stuff first

One of my labmates gave me this good advice. If you need more time on the hard stuff (in my case, all the technical acoustics and math), it is better to know EARLY than to realize you need more time when there isn't more time to be had. Also, if you study the hard stuff first, you have the opportunity to go in and ask your committee members about it, which leads me to...

6) Talk to your committee members AGAIN! (And again!)

Exactly wrong.

When you are able to talk to your committee members about questions, they can help you out, and make your life easier! You may even realize you need to possibly modify your reading list. This happened to me when a committee member and I realized that I wasn't as advanced in math as she assumed (not surprising, considering I was a biology major). As a result, we switched up some of my reading and I got to learn some linear algebra into the bargain. Yay! Throughout the 3 months leading up to my comprehensives, I periodically checked in with my committee members to make sure that I was on the right track. In fact, the hardest, scariest questions I got at my comps were from the committee members I talked to the least.

7) Don't forget to glance over the "easy stuff."

When you've been sweating the scary hard stuff, don't forget to glance over the things you take for granted. The question I did the worst on on my comps was on a basic equation that I totally know.  I know it so well that I hadn't even looked at it before the exam, and so when it came up I FROZE. Try and glance over the stuff you are sure you know, as a refresher. 

*All this advice is highly idiosyncratic and specific to me, my committee, department and university. However, I really felt like it helped me to have a relatively pain-free comprehensive experience. In comparison with some of my friend's comps experiences (one of whom described coming home and sitting in the running shower and crying after passing comps!), it was pretty good. I was nervous and uncomfortable and felt like a total idiot, but I think that's fairly benign. These strategies helped me feel at least a little bit in control of the situation. And now I have this totally awesome (but not especially useful) certificate.

This means I almost have a PhD

... but not really.

How did you manage to survive comps?

From Ships to Stars and Back Again.

Kids watch the moon landing.

Child pretending to be an astronaut.

On July 20, 1969, thousands of people throughout the world stopped everything to watch something on TV. During my lifetime, the things that I remember stopping everything to watch on TV have been tragedies; putting down my algebra book for the verdict of the OJ Simpson Trial, watching Columbine coverage after a ski race, Organic Chemistry class cancelled for a week in college while twin towers burn. But in 1969, it wasn't something horrible on TV.  On July 20, the world dropped everything to watch something magical - human-kind walking on the moon. Humankind had done something absolutely amazing, and the world paused for a second to take it in.

Since the 1960s and 1970s, the perception of space travel has changed from an inspiration to what some consider a waste of money (Note: this is not my opinion, but I don't write NASA's budget). The last flight of the space shuttle program was July 8, 2011. I was in Washington DC on April 17, 2012 when the Space Shuttle Discovery made its last flight over the Washington Mall, on its way to the Smithsonian Air & Space Museum. As far as we know, US space shuttles (and men on the moon) are now a thing of the past.  Does that mean an end to the legacy of discovery? 

The legacy of Discovery actually goes much further back than missions to space. The Space Shuttle Discovery was named after four scientific sailing ships from the days of British exploration:

• HMS Discovery - Sailed By Captain James Cook during his Voyages from 1776 to 1779.
• Discovery - sailed by Henry Hudson in 1610–1611 to search for a Northwest Passage.
• HMS Discovery -  took Captain George Nares to the North Pole in 1875–1876
• RRS Discovery -  under the command of Captain Scott and Ernest Shackleton sailed to Antarctica in 1901–1904.

Of these, Captain Cook's voyages were the most far reaching - he circumnavigated the globe, multiple times. His last voyage, which happened to be on the HMS Discovery ended in Hawaii, when he was struck on the head while trying to kidnap King Kalaniʻōpuʻu of Hawaii (a not unjustified reaction by the Hawaiians, in my opinion).

Cook's voyages. The third voyage (on the Discovery) is shown in blue.
The red line shows the route of the Endeavor (Cook's first voyage) and
 the green line is the second voyage on the Resolution. (from wikipedia)

Many other space shuttles have been named after scientific vessels.  The Endeavor was named after Cook's first ship (shown in green, above). The Challenger Shuttle was named after yet another scientific research vessel, as was the Atlantis (named after Wood's Hole's first scientific research vessel). When humankind looked to a new frontier, we remembered the sea. And we named our "ships" accordingly.

Captain Cook's discovery and the Space Shuttle that shares its name. (photos from wikipedia)

Cook wasn't the first ocean voyager to reach Hawaii, though. Not by a long shot. Here, he met a truly seafaring people who had been traveling across the great Pacific from Polynesia, using the stars for navigation. Without instruments, they traveled across the ocean across ocean at distances of at least 2500 miles. With them they brought their families, livestock, and crops to make a home on new lands. 

A Hawaiian Voyaging Canoe greets one of Cook's vessels.
Art by Herb Kane.

By the 20th century, however, few records remained of the Hawaiian voyaging canoes except for old drawings, such as Hodges' of double-hulled canoes from the 1773 Cook expedition. This went along with a general loss of Hawaiian culture, from language repression* to the loss of their monarch and kingdom. Many people didn't believe that the Hawaiians could have gotten to the islands purposefully. One theory was that storm-driven ships had gotten lost and accidentally landed on the islands, stranding their passengers. In the 1950s and 60s, the debate between the those advocating accidental drift, and those advocating purposeful navigation became heated, and a society was formed to build a replica polynesian voyaging canoe. Herb Kane of the Polynesian Voyaging Society designed the canoe, which was launched in 1975. The canoe was named the Hōkūle‘a, the “star of gladness” the Hawaiian name for Arcturus (for pronunciation of Hōkūle‘a click here). This star passes directly over Hawaii.

Hokulea, which is in the Bootes constalation
(photo from astropixels.com)

In 1976, the Hōkūle‘a sailed to Tahiti with under the navigation of Pius Mau Piailug, one of the last Polynesian Navigators on earth. Only four years later, Ninoa Thompson was the first native Hawaiian to navigate a canoe to Tahiti without instruments. It is beyond me to describe the experience of building, launching and sailing this first voyaging canoe. My heritage is one of forgetfulness and wandering. My family is not tied to the land of our ancestors in any meaningful way - instead it is a mishmash of fair-skinned people from who-knows where that seem to have never stayed in one place for more than one generation.  I can not hope to imagine or understand the joy that came after the hard work of building this canoe, and learning to direct her using the traditions of your ancestors.

Since 1976, the Hōkūle‘a has sailed on over 10 voyages, from Hawaii to Tahiti, Japan, Pago Pago, and Australia (to list only a few). Her next trip will be a worldwide voyage, starting with a sail to Tahiti, and passing by New Zealand, the Indian Ocean, Africa, North and South America, and the Galapagos (again, only listing a few).

The Hōkūle‘a voyage sail plan. Compare with Cook's voyages, above.

The mission of this voyage is "to navigate toward a healthy and sustainable future for ourselves, our home – the Hawaiian Islands – and our Island Earth through voyaging and new ways of learning. Our core message is to mālama (care for) Island Earth – our natural environment, children and all humankind." As part of the mission, the Hōkūle‘a has looked for proposals for science that can be done for knowledge and outreach. At the Hawaii Institute of Marine Biology**, we were privileged enough to hear Ninoa Thompson speak about the history of the Hōkūle‘a and their mission ahead. In regards to the purpose of the voyage, the Polynesian Voyaging Society's website states, "If we view our Earth as an island, our only voyaging canoe in the sea of space, it becomes apparent that we must change course to ensure a healthy, sustainable world."

The inspiration for the worldwide voyage came during a conversation between Pinky Thompson, and Lacy Veach, an astronaut who flew on both the Discovery and the Colombia. As Lacy looked out the shuttle window at the islands of Hawaii far below, he "saw the islands and the planet in one vision – that planet earth was just an island like Hawai‘i, in an ocean of space, and that we needed to take care of them both if the planet was to remain a life-giving home for humanity."

Hawaii from the Shuttle Colombia. Lacy took a Hawaiian adz stone
with him into space. (Photo by Lacy Veach, from  the PVS website).

This seems especially poetic to me. The inspiration for this worldwide voyage literally originated in space, which will provide the navigation. With the retirement of the Space Shuttle Discovery, the US has ended (or perhaps paused) one legacy of scientific exploration. But the Hōkūle‘a, which is named for a star, is part of another, older, legacy.  Dreamed up among, named for, and navigated by the stars, her goal on this voyage is to link the ocean peoples on earth, making new discoveries and teaching how to protect our common home. 

"Man’s perpetual curiosity regarding the unknown has opened many frontiers. Among the last to yield to the advance of scientific exploration has been the ocean. Until recent years much more was known about the surface of the moon than about the vast areas that lie beneath three-fourths of the surface of our own planet.”  

 F.P Shepard, 1948

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Note: I probably got a lot of things wrong in this blog post, and oversimplified a lot of things, since I'm not an expert in 1) the space program or 2) polynesian voyaging and history. However, the links between space travel, ocean research voyages, and the multiple links of both of these with the Hōkūle‘a. were too fascinating for me not to write down. If you want more information about Polynesian Voyaging, please check out the PVS webpage, and NASA is a good resource to learn about Space Travel. If you ever get a chance to hear Ninoa Thompson give a talk, GO.

Also, I couldn't find a place to work it in, but I thought I'd mention and link to Craig McClain's great piece on Why We Need a NASA for the Oceans.

*30 years ago, the Hawaiian Language was in very real danger of disappearing. 
** In yet another coincidence, the emblem for the Hawaii Institute of Marine Biology (HIMB) is derived from an illustration from the logbooks of the HMS Challenger, yet another research vessel with a space shuttle named after her.

HIMB logo

Challenger illustration. From the NOAA archives.

*** These lines are often quoted, and have been an inspiration (through David Attenborough's reference to them in Blue Planet) to thousands of nascent marine biologists (including myself).

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Making a Square out of Circles

Let's say you're a marine biologist, and you've recorded a bunch of whales and dolphins, and you want to study the sounds that they're making.  Some of the cetacean species you've recorded will be making sounds too high for you to hear (like porpoises and beaked whales), and some will be much too low (like blue whales).  In addition, some biologists (like the ones who are older or have attended too many KISS concerts, might not be able to hear very well at all.

One of the ways biologists get over this is by turning the sounds into pictures, like this picture of a dolphin whistle:

This is called a spectrogram.

Here, we can see that the whistle starts out high, and then drops down in frequency, and goes back up again. Frequency is on the y axis.

Remember, loud sounds have high amplitude (shown as a darker red in the spectrogram above), and high frequency sounds have a high frequency (which will put them high on the y axis of the spectrogram).

This transformation between sound and picture is done via something called the Fourier equation.  The Fourier equation is really cool - it takes a complex sound, made out of multiple different frequencies, and breaks it up into the component frequencies.  It's kind of hard to picture this happening, so I decided to start with a simple single-frequency sound, and make it into a complex frequency sound, so we could see what was happening in the reverse.

So what complex sound can I make from simple sine waves?  What about a square wave?

Figure from Khin Hooi.

Click here to listen to a square wave.

Single frequency sounds are composed of pure sine or cosine waves, which are rounded like this:

However, we can create a square wave by adding additional frequencies, each one of which is an odd harmonic frequency of the original sine wave, and which is smaller in amplitude by a specific amount. For example, in the first step to making a square wave, we would add sin(3*t)/3 to the original equation for the sine wave, y=sin(t).  Then we would add sin(5*t)/5, sin(7*t)/7, sin(9*t)/9, and so on for the odd numbers until the wave is square enough to make us happy. Here's an animation of the frequencies being added up to make a square wave:

Cool, so now we've made a sound out of a bunch of different frequencies.  The next thing to do is to use our Fourier transform on it, to take it back apart again.

The figures on the left are of the square wave I've created, and on the right are of the Fourier spectrogram of that wave.  In the first row of figures, you can see that there is only one frequency, and the spectrogram only shows one line (the light blue one at the bottom). As I add more frequencies, more and more (lighter and lighter) lines appear on the spectrogram.

I thought this was a pretty good demonstration of how Fourier Transform works. We've made a complex signal, and then Fourier helped us take it back apart again so we could visualize the different sine waves that make up the signal again.

If you can't get enough of this stuff, here's an excellent post I found about building a sawtooth wave.

Also, here's a great little youtube animation of the making of a square wave:

If you have matlab, and want to try this out, email me for the code.

How loud is that whale, anyway?

When we're thinking about how loud a noise needs to be to damage a whale's hearing, it's important to consider how loud the sounds are that the whale makes itself. If a singing whale makes a sound itself that is at a certain sound level, it is pretty unlikely that it will be deafened by being exposed to a sound of the same level.  This is something like how you are unlikely to be deafened by the sound of your own voice. This isn't to say that continual exposure to sounds at or quieter than their own won't affect whales. It will - just like constant low-noise sounds such as traffic can damage human health.

Calibrating hydrophone sensitivity
curves is very pleasurable.

The first step in figuring out how loud your whale is to drop a hydrophone in the water. Hydrophones convert pressure into voltage (if you're interested in how, I explain here). Hydrophones also have a property called sensitivity, which is a measurement of how quiet of a sound that the hydrophone can hear. The difference in sound sensitivity between a very sensitive hydrophone and a very insensitive hydrophone is like the difference in touch sensitivity between your tounge and your heel. A light touch on the heel might not make any neurons fire, but...

As an aside, in the book "Last Chance to See," Douglas Adams puts a microphone in a condom and uses it to try and listen to Yangtzee river dolphins in China. (Insert nerdy bio-acoustition joke here).

Hydrophone sensitivity is usually listed as a single number, something you will notice if you are a buyer of hydrophones or microphones. This number is negative, and the less negative, the more sensitive the hydrophone. Listing a single number actually oversimplifies things. For example, let's look at the sensitivity of the Reson TC 4024 hydrophone at a range of frequencies between 0 and about 90 kHz. If you remember, frequency determines how shrill a sound is. Barry White, for example, has a low frequency voice, and the Chipmunks squeak their songs at high frequencies. OK, so the "typical receiving sensitivity of this hydrophone" is listed as -173 dB. But, as you can see, the sensitivity actually varies quite a lot with frequency:

Because of this, most people only use their hydrophones to record sounds in the "flat frequency response" region, which, for this hydrophone, is between about 4 Hz and 45 kHz. This flat frequency region is pretty much about equal to -173 dB (and if you want to be really specific about things you can always go back and correct for the variation later - as long as your hydrophone is calibrated!)

You'll probably also need to know your gain. Gain means amplification. Generally, the signals coming out of the hydrophone are still too quiet to use, so we amplify them again.

Finally, we can calculate how loud the whale is. So here we go:

Sound Pressure Level = Sensitivity - Gain + 20 * log (Voltage from the Hydrophone)

It's actually pretty simple once you know what all the numbers mean...

... except for the fact that, by convention, the Sound Pressure Level is measured at 1 m from the sound source. So, if we are working with wild animals, we will either have to get incredibly lucky, or we have to figure out a way to deal with attenuation of the sound in water. Attenuation is a fancy word that means "makes quieter." For example, sound-proof walls attenuate the sound of your neighbor's Eminem music.  I'll deal with that one a little bit later.

You almost never get this lucky. Photo by Flip Nicklin.

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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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A wave is a complicated thing to equate

I am only studying chapter 1.2.1 of my big scary bioacoustics book, and already I'm running into some scary maths (tm). However, my advisor has told me that I don't have to actually know how to derive the equations, just understand them conceptually (Apparently I am getting my PhD in Zoology and not Electrical Engineering - could have fooled me!).

I use sound to study whales in the ocean. Understanding sound is absolutely essential to my research, and to understand sound, you've got to understand the wave equation.

We're not going to actually derive the wave equation, but it's important to know what goes into it. (if you want to see the real math, look here). In order to understand waves, you need to understand four other fundamental laws. The great thing about these laws is that they pretty much are telling you that waves obey the laws of physics. By the combined powers of these laws, Captain Wave equation emerges!

Conservation of Mass: This means that even though the number of molecules in any part of the volume that  sound passes through may change, the total number of molecules in the volume stays the same.

In the sound wave, molecules get more
compressed, but don't appear out of thin air. 

Equation of Motion (Newton's Second Law): This means that we can calculate the force acting on particles by multiplying their density when a sound is passing through them by their acceleration.

Force = Acoustic Density * Acceleration

Equation of Force: With this, we can use the total density of the fluid (which is different from the acoustic density) and divide it by the x movement of the particles to find the force acting on the particles.

Force = Total Density differential * x differential

Equation of State: This tells us that acoustic pressure is related to how easily the medium through which a sound travels is compressed. Some people think that liquids and solids can't be compressed, but this isn't true. The reason we generally consider liquids to be incompressible is that it is REALLY REALLY hard to compress a liquid. I couldn't do it, even if I worked out a LOT.  However, liquids at the bottom of the ocean have the weight of tons of water sitting on top of them.  The weight of all this water presses down on the liquid, pushing the molecules closer together. In fact, if you have enough pressure, you can even compress rocks. Here's my version of this equation:

Pressure = (squishiness / acoustic pressure) * pressure

When you put these four equations together, you get the wave equation, which uses all four to describe the movement of a wave. This equation factors in pressure, time, the speed of sound, and the movement of particles.

Why is this important to a whale researcher?  Well, if I want to use sound to study animals underwater, I need to understand that sound travels differently in different types of water. In the ocean, things that might change how sound travels are depth, salinity, and temperature. If, for example, a beaked whale is clicking at 5000 m, and I am recording it at the surface, I can't assume that the sound is traveling in a straight line from the whale to me. As the depth and temperature changes, the way sound moves through it will change. I need to be mindful that these things can change how sound moves through the water.

I warned you this was going to be dense. And that was only chapter 1.2.1.

Also, if I made a math mistake, and I messed up on understanding any of the equations, please comment!


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