Practice: A sound wave is emitted at a frequency of 300 Hz in air at a 0°C. As the sound wave travels through the air, the temperature increases. What is the wavelength of the sound wave at the following temperatures?

a. 0°C

b. 20°C

c. 45°C

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Wave Functions & Equations of Waves | 36 mins | 0 completed | Learn |

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Concept #1: Introduction To Sound Waves

**Transcript**

Hey guys, in this video we're going to introduce sound waves. Now sound is just a particular type of a wave and we've already talked a lot about waves. Alright so we're going to apply a lot of that knowledge to the specific wave of sound. Let's get to it. Sound is a common type of longitudinal wave. The oscillations that make up sound are oscillations in the pressure of gas or technically whatever elastic medium sound travels in. Sound can also travel in liquids and solids. These oscillations occur along the propagation direction which means that it has to be a longitudinal wave. Here's a very very common picture of sound. We have some sort of source like a speaker producing sound. What it causes is it causes areas of compression of the medium, you can see a high density of gas molecules right here and then it causes areas of rarefaction which is a decrease in the density. And then you see another area of compression here and then more rarefaction here and what happens is as the gas gets denser and denser and denser, the pressure goes up so if I were to plot pressure versus distance here, you would see that it starts at the highest pressure, we have a very very dense area right here, then it steadily decreases to an area of low pressure then it steadily increases back to another area of maximum pressure before ending at an area of low pressure. So this is what the pressure as a function of position would look like.

Alright, now remember all of the same rules apply to longitudinal waves, all the same rules apply to longitudinal waves that apply to transverse waves. The speed relation, they all have amplitudes, they have periods, they have frequency, etc. The thing unique to each type of wave is the energy that it carries and more importantly the speed. So we're going to talk a little bit about the speed here specifically the speed of sound in an ideal gas. This is equal to the square root of gamma RT over M where R is the ideal constant which is 8.3 joules per mole Kelvin. That's in SI units and gamma is a constant called the heat capacity ratio. Remember guys or if you haven't learned it yet you will see it soon that the heat capacity given by the capital C is a measure of how much heat is needed to change the temperature of a substance. Now since this doesn't cover really thermodynamics, the section is not dealing with thermodynamics, the heat capacity ratio won't really be addressed too much but this is just what it is. There happened to be two heat capacities that exist for gases. The heat capacity at constant pressure and the heat capacity at constant volume. The ratio is as it sounds just the ratio of those two heat capacities. Now the speed of sound in air is a very very common formula given in this orange box where T is in units of degrees Celsius. This T is in units of Kelvin. Whenever you have T, typically in an equation it's going to be in Kelvin but this specific one for the speed of sound is an outlier. It's written so that T is in units of Celsius.

Let's do an example to wrap this up. What is the heat capacity ratio for air? Consider air to have a molar mass of 2.88 times 10 to the -2 kilograms per mole. Now because temperature appears in both speed equations, we have that the speed of sound is specifically 331 meters per second times the square root of 1 plus T over 273 and it appears in gamma RT over M. Those T's are going to balance out so we can actually choose whatever temperature we want. I'm going to choose a temperature of zero degrees Celsius which equals 273 Kelvin. If I plug in zero here then this is just 331 and this equals the square root of gamma RT over capital M. Capital M is the molar mass and we're told that the molar mass of air is considered to be this. So all I have to do now is do 331 squared equals gamma RT over M and so gamma is M 331 squared over RT and so gamma, sorry let me minimise myself so you can see that equation, M 331 squared over RT so gamma is just 2.88 times 10 to the -2, 331 squared, those are both in SI units already, R was given in SI units 8.3, temperature in SI units is 273 and this works out to 1.4 and if you were to look up the heat capacity ratio for air you would indeed find that it is 1.4. Alright guys, that wraps up this introduction into sound waves. Thanks for watching.

Practice: A sound wave is emitted at a frequency of 300 Hz in air at a 0°C. As the sound wave travels through the air, the temperature increases. What is the wavelength of the sound wave at the following temperatures?

a. 0°C

b. 20°C

c. 45°C

Concept #2: Sound Waves In Liquids And Solids

**Transcript**

Hey guys, we already saw the speed of sound in an ideal gas. Now we want to talk about the speed of sound specifically in liquids and solids which are also elastic media so they can also propagate sound waves. Alright let's get to it. As I said sound can travel through any elastic medium so it can also travel in liquids and solids. Sound travelling through a solid or liquid would look the same as sound travelling through gas. It would just have a different speed. The speed of sound in a liquid is equal to the square root of B divided by rho where B is known as the bulk modulus of the liquid and rho is just plain old density.

The speed of sound in a thin solid so something like a rod, if you were to put a little speaker with sound then that sound propagating on the thin solid is going to be the square root of Y over rho where rho is still just the plain old density and Y is known as the Young's modulus of the solid. Now if you've covered elasticity then you've seen moduli before if not what a modulus is of a solid or a liquid is it's a measure of the elasticity of that substance. The larger the modulus, the harder it is to compress that material. So if you want to take a chunk of something and you want to squeeze it from all sides to a smaller chunk of that something, the larger the bulk modulus the more force you're going to have to put on every surface the more pressure you're going to put on all the surfaces to get it to change its volume. For the Young's modulus, if you have a rod the larger the Young's modulus the harder you're going to have to press on the end of the rod to reduce its length, the more pressure you're going to have to apply on the end of the rod to reduce its length.

Let's do an example, deionised water has a bulk modulus of 2.2 times 10 to the 9 Pascals. What is the wavelength of a 250 Hertz sound wave in deionised water? If we want to know the wavelength of some frequency we know that the speed relates the two. What's the speed of sound in deionised water? Well the speed is going to be the bulk modulus divided by the density. The bulk modulus we're told and the density of deionised water is just 1000 kilograms per cubic meter, that's in SI units and this whole thing is going to be 1483 meters per second. 1483 meters per second. So the wavelength is going to be 1483 divided by 250 Hertz which is 5.93 meters, a pretty large wavelength but that's because sound is travelling so quickly so it travels such a far distance, such a short distance per unit time. Alright guys, that wraps up our little discussion here on the speed of sound in liquids and solids. Thanks for watching.

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