The Physics Philes, lesson 109: Speed of Sound Part 2
Last week we started discussing the speed of sound and how that speed changes depending on the medium. This week we’ll finish up that topic with a discussion of how fast sound moves in a solid and a gas.
First, let’s tackle the speed of sound through a solid rod or bar. Unlike fluid in a pipe (which cannot move sideways), the solid rod can expand slightly when it’s compressed by a longitudinal wave. So the speed of sound through a solid rod is related to Young’s modulus (the ratio of tensile stress to tensile strain), rather the bulk modulus. The equation looks similar to equation for the speed of sound in a fluid:
The distinction between a solid rod and a bulk solid is actually pretty important here. The above equation only applies to a solid bar that is able to expand and shrink sideways as the wave travels. This equation is no good when we talk about bulk solids because, in bulk solids, that sideways expansion is inhibited by surrounding material. For bulk solids, the wave speed is dependent on the density of the material and the shear modulus. However, when this equation is valid.
OK, so what about sound through a gas. Most of us have a lot of experience with sound in a gas since we spend most of our lives hanging out in air. We can use the equation we derived last week to figure out the speed of sound in a gas. However, we need to do a little work first.
To refresh your memory, the equation for the speed of sound we found last week is:
However, we need to remember that the bulk modulus B of a gas depends on the pressure of the gas. The more pressure is applied to the gas, the harder it is to compress it. In other words, the bulk modulus increases. In math, we can express the bulk modulus B of a gas is:
That weird y-looking letter is gamma (γ). It represents something called the ratio of heat capacities. It’s a dimensionless number that characterizes the gas’s thermal properties, and it’s a number, at least according to this textbook I’m using, we’ll learn more about in later chapters. Right now it’s enough to just know sort of generally what it is. The other value, p, is the pressure of the gas in equilibrium.
Pressure ends up being an important component of the density ρ. But pressure, in turn, depends on the temperature of the gas. However, the B/ρ for an ideal gas doesn’t rely on pressure at all, only temperature. That means that the speed of sound in a gay is a function of the temperature T. The speed of sound in an ideal gas is:
Hold on a second. I don’t remember seeing all those weird letters before. Where did they come from? You might know what these are if you’ve had more physics or chemistry than just these little posts, but if not, just calm down. They should be new to you. Let me explain briefly what each of these quantities mean.
The temperature T is the temperature in kelvins. The M stands for molar mass, or the mass per mole of whatever the gas is made of. (A mole corresponds to Avogadro’s number: 6.022 ×1023. This is the number of atoms or molecules per mole of a substance.) The R is the gas constant. This doesn’t vary by gas. It’s the same for all of them: 8.214 J/mol · K. For any particular gas, γ, R, and M are constants, so the wave speed is proportional to the square root of the temperature in kelvins.
OK! Hopefully we all have a basic understanding of the speed of sound. Don’t worry, though. There is lots more we can learn about sound, and we’ll be doing so in the coming weeks.
Featured image credit: U.S. Navy via Flickr