Month: September 2016

Why Euler’s Formula Works

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In many parts of engineering, you’ll run across something called Euler’s formula, which looks something like this.

(1)   \begin{equation*} e^{i\theta} = \cos(\theta) + i\sin(\theta) \end{equation*}

It’s something you can look at, memorize, and then use in the future.  But sometimes it’s nice to know that mathematically, the left side and the right side are in fact equivalent, even if it’s not obvious.  While reading Hecht’s Optics book, I came across a footnote that quickly shows that the two sides are equal.  Here is the gist of that footnote.

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Example: Harmonic Function Satisfying the Wave Equation

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In this post I’m going to work through the math showing that a simple harmonic wave profile/function satisfies the one-dimensional wave equation.  The simple harmonic function is given in the following equation.  The variable v is the wave speed in the x direction, while \psi is the shape of the profile of the wave.

(1)   \begin{equation*} \psi (x,t) = A\sin[k (x-vt)] = A\sin(kx-kvt) \end{equation*}

The 1D wave equation that we wish to satisfy is given below.

(2)   \begin{equation*} \frac{\partial ^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial ^2 \psi}{\partial t^2} \end{equation*}

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The Hardest Part of Blogging

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I’ve wanted to write blog posts for a while now.  I always thought someone out there would care what I had to say.  But I’ve been stuck.  That last sentence I wrote is actually a perfect example of why I’ve been stuck, and why I have trouble expressing myself outside of my science/engineering YouTube videos; I care what people think.  I just went to Google to make sure it was okay to start a sentence with the word ‘but’.  I wanted to make sure I wasn’t going to get called out for poor grammar.

I like being an engineer and creating engineering videos.  It’s something I’m good at.  You’ll sometimes hear, “go into engineering/math/science if you like when there’s only one correct answer”.  I think the beauty of engineering is that there are multiple ways to solve the same problem, some better than others.  But it’s up to you to weigh the pros and cons of each method and present the best solution.  I try to present the best method when I create videos, the method where the viewer will learn the most and gain a deep understand of the topics.  And this is where blogging and my videos differ.  I’ll start to venture into topics of which I don’t consider myself an expert.

I’ve read articles about how to get started, what a good blog contains, and how to keep readers interested.  Adding pictures to break up the content always seems like a good idea.  I figured I would find one of those free stock image sites and search for writer’s block, and I came up with this.

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Man who apparently has writer's block.
Man who apparently has writer’s block.

Although I believe that deep down, everyone cares what other people think to a certain extent, I envy those that can fully put themselves out there.  It’s not easy being criticized, especially not for all to see.  Maybe one day I’ll get to that point.  For now, the most important thing is to get started.  This is my starting point.  Maybe my second post will be about something non-controversial, like green peppers, just to ease into it.  Here goes nothing.

TI-83 Calculator Program: Prandtl-Meyer Expansion Wave

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Introduction & Motivation

If you’ve ever taken a gas dynamics (compressible flow) course, you’ve probably spent hours flipping through the tables at the back of the book to find the correct Prandtl-Meyer (PM) angle for your Mach number, or vice versa.  This involves first finding the table (A.5 in my Modern Compressible Flow book by John Anderson), then finding the bounding Mach numbers or angles, and finally interpolating to find the correct value you need.  Then you’ll likely need to do this many more times for a single problem.  You also won’t be able to use the tables when the specific heat ratio isn’t equal to 1.4.

I decided to write a program for my TI-83 Plus graphing calculator, and I’ll show you how to program it.  First, I’ll post the entire program, so if you already know where to find all the commands and such, you can just enter in what you see below.  For those who need a little guidance on where to find the commands, I’ll go through them in detail after the entire program code.  I’ll also go through how to use the program, and we will check it against the online VT calculator.

The first equation shown below is what we will use to solve for the PM angle \nu.  The second equation is used to solve for the Mach number M.  The only difference between the two is that the first equation can be solved directly, while in the second equation we are searching for the value of M that makes the function zero.

    \[ \nu(M) = \sqrt{\frac{\gamma +1}{\gamma - 1}} \ tan^{-1} \sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2-1)} - tan^{-1} \sqrt{M^2-1} \]

    \[ \nu - \sqrt{\frac{\gamma +1}{\gamma - 1}} \ tan^{-1} \sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2-1)} - tan^{-1} \sqrt{M^2-1} = 0 \]

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