## Waves, Math, and the Creator: Sound Part 3 of 3

Note: This is the third and final post in a series of guest posts on Math, Waves, and the Creator by Dr. Adam F. Hannon. Get ready for a fascinating “listen” to sounds! – Katherine Sound waves arise when a physical medium like a string on a guitar, a membrane on a drum, or even your own vocal cords in your throat vibrate.

When you pluck a guitar string, the string’s vibration causes the surrounding air molecules to vibrate with a similar wave pattern, and those waves move through the air as a compression wave until they reach our ears. There our eardrum receives the wave, vibrates as well, and sends that information to our brain for us to interpret the sound. There’s a lot going on in this whole process, but the neat thing is that we can use math to explore it!

## Different Strings; Different Sounds

Have you ever wondered why different strings on a guitar produce different sounds? Well, there’s a mathematical relationship between the frequency (f) of the wave in the string, the wavelength (λ) of the wave in the string, the linear density (ρ) of the string (without getting technical, this is affected by what material the string is made out of—how dense it is), and the tension (T) on the string (that is, how tightly the string is held at its top and bottom). Because of the consistent way God governs this universe, we can describe the relationship mathematically, and then use it to design instruments!

Here’s the relationship:  Since for a given string the tension and material the string are made of (and thus the linear density) are fixed, the frequency multiplied with the wavelength are a constant based on those values.

## Harmony Between Notes

Have you also ever wondered why different notes sound better together than others? It has to do in part with the subtler tones that are also formed when the note is played.

When waves are bound (example: a guitar string is attached, or bound, at both sides), they produce not only a fundamental tone, but also what we call harmonics, or subtler tones that are mathematically related to that fundamental tone in a specific way. In the first harmonic, the wavelength is half the original wavelength (frequency is doubled). In the second harmonic, it is a third of the original wavelength (frequency is tripled). And so forth. Figure 1. Harmonics on a string. Adapted from WikiMedia.

We can use—you guessed it—math to describe how these harmonics relate back to the wavelength of the fundamental tone.  Note: In the above equation, n = 1 in the fundamental tone, n = 2 in the first harmonic, n = 3 in the second harmonic, and so forth.

Harmonics help us to understand why certain notes sound better together. Harmony actually means “agreement,” and notes that sound good together have “agreement” in the subtler sounds that make them up! For example, the second harmonic of the note C is the note G; not surprisingly, C and G sound good together.

When talking about harmonics, you’ll often hear people talk about frequencies instead of wavelengths. We looked at wavelengths here because they’re easier to picture, but as wavelengths change, so do frequencies. To see why, let’s use some simple algebra.

We can first divide both sides of the equation that relates the frequency and wavelength by the wavelength to solve for frequency directly. We can do this because of the consistent way God created the laws of math. We can now substitute the equation we have for wavelength in terms of n and L. This leaves us with a way to determine the fundamental frequency and harmonic overtones from just the length of the string, tension, and density. Looking at the equation, we can see that if we make a string tighter (increase the tension), we make the frequency higher and thus a higher pitch. Similarly, thicker strings (larger density) will have a lower frequency or pitch. Longer strings also will give a lower pitch and thus shorter strings a higher pitch (this is why when you hold down strings on a guitar going down the fret board you get higher pitches as you shorten the bound string length). Having different strings together, we can control their individual pitches with these parameters to have integer ratios of their frequencies that give different harmonies, and thus make the beautiful music we hear.

## The Harmony of the Universe

Harmonics—which are produced from bound waves—are not constrained to music and sound alone. We just can’t “hear” all of the other kinds of harmonics God put in place in the other kinds of waves He created.

For example, in light waves, if a special sets of mirrors are used to only allow light of certain wavelengths to remain in a confined space, only waves of appropriate harmonics are confined. (Side note: This can be used in conjunction with special materials to make lasers.) In the waves of electrons in an atom, the atomic orbitals the electrons can occupy are harmonic in nature, as the electrons are bound in the position they can occupy in space.

Since the Bible makes it clear Jesus is the one who created and sustains all things (Colossians 1:16–17), He is the One who has put the incredible order and boundaries we see all throughout His Creation…and math puts a magnifying glass on what He has created, giving us a glimpse into His handiwork we’d otherwise miss.

We can only imagine how remarkable it would be if we could “hear” all the harmony of the universe caused by all the waves around us. Looking at the incredible order in waves—which math helps us see—should cause us, like Job, to bow before our Maker, humbled in His presence.

“I have heard of thee by the hearing of the ear: but now mine eye seeth thee. Wherefore I abhor myself, and repent in dust and ashes.” Job 42:5-6 (KJV)

 Equations adapted from Romine, Gregory S. Standing Waves on a String: P28, Lab 6. Stanford Physics Department, 2004, https://web.stanford.edu/dept/astro/dorris/StandingWaves.pdf

 Ibid.

Sound & Frequencies Explained (video)
Overtone Wiki Page
How Do Harmonics Work
Fundamental Frequency and Harmonics (Physics Classroom)

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## Math, Waves, and the Creator: Light Waves  Part 2 of 3

Note: This is a continuation of a series of guest posts on Math, Waves, and the Creator by Dr. Adam F. Hannon. This particular one would be great to have middle school and high school students read in order to give them a glimpse into how math helps us shine light on God’s creation, pointing to the Creator. I hope you enjoy! – Katherine

“And God said, Let there be light: and there was light.” Genesis 1:3 (KJV)

God spoke, and light came into being. We feel its warmth (very much so in the summer months!) and enjoy its illumination, yet math gives us a fresh glimpse into just how amazing all the light around us is.

As discussed in the previous “Wave, Math, and the Creator” blog post, light is a wave—or more specifically, two coupled waves (an electric and a magnetic wave) in one. We saw in the previous blog how we can use algebra and trigonometry to describe waves, including light waves.

Let’s continue exploring light with math—only this time, we’ll use basic math, along with just a touch of algebra (the part a pre-algebra student could still follow).

As we do so, we’ll discover amazing order God placed in the very light all around us.

## Measuring the Properties of Light Waves

There are a lot of different aspects of light waves we could look at. For example, we could look at how fast a wave is traveling in a certain direction—or its velocity.

We can use a number to describe this velocity. Figure 1.1 The velocity of a wave is how fast the wave is traveling in a certain direction. For example, if the  wave above took 1 second to travel 1 centimeter toward the right, we would say its velocity was 1 centimeter per second, or 1 cm/sec.

Or we could look at how frequently one part of the wave repeats in a given amount of time, called the frequency.

Again, we can use a number to describe the frequency.

Figure 1.2: In the video, the two waves have the same wavelength but different frequencies. During the time it takes the top green wave to pass one full period through the dashed line, the bottom purple wave passes through twice, thus it has twice the frequency.

We can also look at the wavelength—at how long each of the repeated wave structures are.

Again, we can use a number to describe the wavelength! Figure 1.3 The marked distance between the peak of each wave is the wavelength. Notice that the top wave has a greater wavelength than the bottom one.

Now, in real life, it’s not easy to measure the frequency of a light wave—it corresponds to a really short amount of time. But it is easier to measure the wavelength. And then—you guessed it!—we can use math to help us find the frequency. Let’s take a look at how.

## Using Math to Find the Frequency

Because of the consistent way God governs all things, there is a consistent relationship between the velocity, frequency, and wavelength of a wave:

velocity = frequencywavelength

Since in math we tend to use letters instead of writing out the whole words (it saves time and makes it a lot easier to work with the equation!), we’ll use symbols to represent this same relationship (the v represents the velocity, the f the frequency, and the λ the wavelength):

v = f λ

Now that we know the relationship between the velocity, frequency, and wavelength , we can use math to help us figure out the frequency. In a light wave, the velocity is the speed of light, which, because of the consistent way God governs all things, is always constant 670,616,629 miles per hour (or roughly 3.00 x 1017nm/sec). Note: The “sec” stands for “second,” and the nm stands for nanometers or 1 billionth the length of a meter—it’s very small! Thus, we can write that amount for the velocity (which we represented with a v) in the equation, giving us this:

3.00 x 1017 nm/sec = f λ

And now if we know the wavelength, we can easily find the frequency!

Example: If the wavelength (λ) of a light wave is 650 nm, what is its frequency?

We know this relationship: 3.00 x 1017 nm/sec = f λ

Substituting in 650 nm for the wavelength (λ) gives us this: 3.00 x 1017 nm/sec = f(650 nm)

If we now divide both sides by 650 nm, we’ll find the frequency. Note: Notice that we divided both sides by the same amount to simplify the answer—we’re able to do that because of the consistent way God governs creation.

The THz stands for Terahertz. Tera means a trillion. We just saw that a 650 nm light wave has a frequency of 462 Terahertz; in other words, it is repeating the wave-like pattern 462 trillion times per second! Can you imagine something happening a trillion times over and over again in a single second!?

## Discovering the Colors of Light—and Invisible Light

One of the neat things we discover as we use math to explore light is that different wavelengths result in different colors.

While we are used to white light, we also encounter colored lights, such as the red light in a small laser pointer. Notice how the color varies as the wavelength varies. Figure 1.4: Approximate wavelengths of different colored lights.

Some light, though, is invisible. It turns out that there are more to light waves than just what we think of as “light”! There are kinds of light waves with smaller and larger wavelengths that we can consider as invisible light. These different invisible light waves along with visible light are all called electromagnetic waves and make up a whole spectrum as shown in the figure below. Figure 1.5: Schematic of the electromagnetic spectrum. Going from left to right goes from longer wavelengths to shorter wavelengths.

Just think of the fact we can also only see a small portion of the electromagnetic spectrum—what beauty must exist to God who can see in the entire spectrum!

We are literally “in the dark” to most of the light waves around us—we only know much about them because we can use tools and equipment to detect them.

## Applying What We Learn About Light

Not only does math help us appreciate the amazing beauty God put in light, but math helps us apply what we learn to build technology.

For example, take CDs, DVDs, and “Blue-Ray” discs. Did you know that the CD/DVD/Blue-Ray player reads the data off these disks using light, and that each of them use light with different wavelengths (and thus different colors)?

• CDs – 780 nm wavelength (infrared)
• DVDs – 650 nm wavelength (red)
• Blue-ray – 405 nm wavelength (indigo…somebody named that wrong!)

Every time you use a CD/DVD/Blue-ray disc, you’re reaping the benefits of math in action.

## Let There Be Light

That’s just a glimpse into how math reveals design and beauty in the very light around us—the light God created by the simple command, “Let there be light” (Genesis 1:3 KJV).

As you enjoy the bright summer light today, pause and reflect on the Creator’s power and might Who simply spoke light into existence.

He is the same Creator who shone into the darkness of our sin with the Light—Jesus Christ.

“In him was life; and the life was the light of men. And the light shineth in darkness; and the darkness comprehended it not…. But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name:” John 1:3-4, 12 (KJV)

Note: Stay tuned for the final blog in the Waves, Math, and the Creator series. In it, we’ll explore sound waves.

 Equation from David J. Griffiths, Introduction to Electrodynamics, 3rd ed. (USA: Upper Saddle River, NJ: Prentice Hall, 1999), p. 368.
 The exact colors given for different lights varies by source, as the there are a range of hues within red, orange, etc., and the borders are hard to concretely define since what may be orange to one person might still be red to another (see this Live Physics page for one list of approximate ranges). The point is that different colors of light do have different wavelengths. These values came from those given in Griffiths, p. 377. The figure was adapted with those numbers from one by Gringer.
 Values from Kumar, T. Ravi, and R. V. Krishnaiah. “Optical Disk with Blu-Ray Technologoy.” International Journal of Computer Engineering & Applications III, no. II/III (July-September 2013), p. 160. https://arxiv.org/ftp/arxiv/papers/1310/1310.1551.pdf.

## Waves, Math, & the Creator Part 1 of 3

Note: I’m excited to introduce this series of guest posts from a good friend, physicist, engineer, and data scientist, Dr. Adam F. Hannon. The series explores waves (and I don’t just mean ocean waves!). More technical details are given in the endnotes for those who would like to dig deeper. I hope you’ll be as awed as I’ve been as I’ve edited the posts at God’s handiwork in the waves all around us…which math helps us explore.
– Katherine

No doubt you learned about sines and cosines in school, along with the number pi (π). The amazing thing is these tools help us describe waves, from ocean waves to light waves to particle waves inside atoms.

Here’s the basic mathematical equation that describes almost all waves, known as the Wave Equation: The Wave Equation[i]

Without going into too much detail about the equation (more detail is included in an endnote[ii]), it essentially describes the relationship between how a wave varies over time to how it varies in space. The letters and symbols stand for different quantities. For example, the t stands for time, the Greek letter ψ (psi pronounced “sigh”) is the quantity that is “waving” or oscillating (e.g. for an ocean wave it is the distance from where the water surface is to where it would be without the wave), and the v for velocity (think speed in a specific direction). A schematic of how ψ and v look for a water ocean wave is shown in Figure 1. Figure 1. A picture of a water wave with the math symbols we use to describe it.

### Waves and Matter

Ocean waves are one example of a wave occurring in matter. Matter refers to the materials that make up the world around us. In the case of ocean waves, the matter is the water, and that water is what is making up the wave.

Another example of waves occurring in matter is a vibrating string such as in a piano, guitar, or other musical instrument (here the matter waving is the metallic string). The string goes up and down, much like an ocean wave. These string vibrations can be thought of as waves…and they too can be described mathematically with the same wave equation as the ocean waves.

Now, the vibrating string causes the air molecules near the string to move in a similar way, which creates another wave in the air around us. While we cannot see the matter “waving” here, sound waves are really the same idea as the waves we see in oceans and strings. Air is full of gas particles that are a form of matter and that can move as a pressure wave so we hear sound. The sounds travel as waves through the gas particles in the air until it reaches our ears.

All these waves occurring in matter can be described by the Wave Equation above. What differs between the different waves is the kind of matter that is waving, how the waves are constrained (whether they can move freely as with ocean waves and air sound waves or if they are constrained, as is the case with a vibrating string), and the speed of the waves. If we wanted to explore these different waves mathematically, we would need to use algebra and calculus to solve the Wave Equation for ψ as a function of spatial position x and time t. The final solution we would get would depend on the material variables and boundary constraints of the wave. The solutions are generally made up of combinations of sine and cosine functions. In a future blog, we will look at such solutions for a vibrating string in a bit more detail so we can explore more of the order God placed within sounds.

### Waves and Light

A light wave is another kind of wave.

Light waves (a.k.a. electromagnetic waves or photons) are interesting in that they are actually two waves in one. One of the waves is an electric field (the same kind of electric field that gives you electricity in your house to run the computer or other electronic device you’re reading this on) and the other is a magnetic field (the same kind of magnetic field that holds cute baby pictures of your friends and relatives on your refrigerator). The two fields each have their own wave equation and are coupled together such that a light wave has two oscillatory (i.e., wave-like) components traveling at the same time. A schematic of this is shown in Figure 2. Figure 2: A schematic diagram of a light wave. The electric field is shown with blue arrows and the magnetic field with red arrows. The light wave is traveling in time and space in the direction of the black arrow (which is perpendicular to all the red and blue arrows) at a certain speed, which we’ve represented with a c, known as the speed of light.

Each of these light waves can also be described by the Wave Equation.

Only it’s common to go ahead and rewrite the equation using different letters in order to specify that in the case of light waves, it’s E (the electric field; color coded blue) and B (the magnetic field; color coded red) that are “waving” or oscillating, and that the velocity is the speed of light (represented by a c; color coded purple): Wave Equation[iii] Wave Equation Rewritten for the Electric Field in Light Waves[iv] Wave Equation Rewritten for the Magnetic Field in Light Waves[v]

Interesting Note: You’ll notice that throughout physics many equations are basically the same except for the use of different letters. Using an E and a B here make it clearer what exactly is “waving” or oscillating in this case (the electric field and the magnetic field). And we use c instead of a v as that instantly tells us that in this particular case, our velocity (in this case, the speed of light) is constant because the convention is that c represents a constant value. (Because of the consistent way God governs the universe, the speed of light in vacuum is always constant—670,616,629 miles per hour—or 299,792.5 kilometers per hour for our international friends.) Another thing worth noting is the use of B for the magnetic field. You might be wondering why in the world a B is used to describe this? Well, it’s partly because M was used to describe a material property called the magnetization, so scientists had to choose a different letter.

### Waves and the Subatomic

Interestingly, wave-like equations can also describe where the very particles that make up the atoms in our bodies (and the rest of the universe) are located!

It turns out that electrons orbit the nucleus of an atom in cloud shaped orbits. Well, the Schrödinger wave equation[vi] (yes, this equation is really describing an electron’s orbit as a wave!) shown below describes the likelihood of finding a given particle (i.e., an electron) at a given location in space and time: Electrons in an atom occupy spherical orbitals that are known as spherical harmonics. These are in fact electron probability waves that are confined to a spherical region (in this case centered around the nucleus of an atom). The first set of these spherical harmonics can be seen in Figure 3 overlapping for a carbon atom.[vii] The different colors represent the different wave orbitals. Although they may not look like a wave, mathematically they can still be thought of as waves. Figure 3: Schematic of the first 4 spherical harmonic orbitals of a carbon atom. The inner magenta spherical orbital is hard to see, but the 3 outer dumbbell shaped orbitals (red, green, and blue) can be seen quite easily.

If the electrons in our atoms did not fall into these Wave Equation-based orbitals, everything we know would instantly collapse in on each other. However, God knew what he was doing when he set the math to govern even the smallest of particles so that we could actually sit here and talk about the amazing structure of atoms.

### Waves and the Creator

There are many other wave types we could look at (such as the gravitational waves that ripple through the very space-time fabric of our universe), each of which could be described with similar but slightly different equations. (The slight differences are due to the different physics involved.)

The fact that the math of music is so similar to the math of the subatomic is simply stunning. Even more stunning is the incredible order and design we see everywhere—from the oceans’ waves to the way God arranged the electrons in our atoms to make life possible. As we explore God’s creation with math (including algebra and calculus), we see His wisdom, care, and faithfulness.

In Psalm 104, after reflecting on many aspects of God’s creation, the Psalmists cries, “O Lord, how manifold are your works! In wisdom have you made them all; the earth is full of your creatures” (Psalm 104:24 esv). May we join the Psalmist in praising our Creator and resting in His incredible care.

Note: Stay tuned for the next blog on waves, in which we’ll explore light waves in more detail…

[i] Source for wave equation: John R. Taylor, Classical Mechanics (Sausalito, CA: University Science Books, 2005), p. 695, eqn. 16.39.

[ii] In this simplest form, the equation says that for a given physical property that can vary in space and time, the second-order partial derivative/effective acceleration of the property is equal to the velocity of the wave squared times the second-order sum of the spatial derivative/divergence of the spatial gradient of the property.

[iii] Source for wave equation: John R. Taylor, Classical Mechanics (Sausalito, CA: University Science Books, 2005), p. 695, eqn. 16.39.

[iv] Source for equations: David J. Griffiths, Introduction to Electrodynamics, 3rd ed. (USA: Upper Saddle River, NJ: Prentice Hall, 1999), p. 376, eqn. 9.41 and 9.42.

[v] Ibid.

[vi] The keen observer will note this is not exactly the same as the standard wave equation mentioned earlier, but there are many similarities, so let us note the differences. Aside from all the physics constants (h is Planck’s constant, i is the imaginary number, V is an applied potential that can vary in space and time, and μ is the reduced mass of the particle being described), we see the main differences are that the time derivative is first order instead of second and that there is an applied potential term. The main effect of the time part being first order is that that the time part of the solution is a simple exponential, but since there is also an imaginary number, they normally are still oscillatory solutions.

The cool thing with the Schrödinger equation is that depending on what you put in for the V term, you get different kinds of waves. For a simple bound potential (called the particle in a box), you actually get the same free standing waves you would get for a guitar string! You can put a more complicated function in and get something called a harmonic oscillator. If you put in the actual electric potential caused by the protons in the nucleus of an atom, you get the spherical harmonics as discussed.

Source for equation: Stephen Gasiorowicz, Quantum Physics, 3rd ed. (John Wiley & Sons, 2003), p. 31, eqn. 2-23.

[vii] The interesting fact is the spherical harmonic orbitals actually do not overlap in the sense that an electron in one orbital has no likelihood of being in the wave state of another spherical harmonic at a given time (aside from electrons having different spin, but that is just an extra thing to account for that does not affect things too much).