2.7. Speed of Sound

So another important thing that we should be aware of speed of sound is speed of ‘c’ is given as this is the formula and we know that it is . Now, we have to solve for . We already know from this equation that can be written as . Right? Therefore, if we put this equation in this equation, then we can get ‘c’ as , where ‘c’ is generally very large for compressible, I mean it is very difficult to compress fluids because in case of fluid is too much the bulk modulus. So, we should be if we know this equation and we know what type of process is there we should be easily able to find out the speed of sound.

The speed of sound, denoted as 'c', is a critical concept in fluid mechanics, especially for gases and liquids. It is defined as the speed at which sound waves travel through a medium. The formula for speed of sound in an ideal gas involves understanding several factors such as the gas's pressure and density. In essence, speed of sound increases with temperature and is influenced by the medium through which it travels. In compressible fluids, where the bulk modulus is large, it's more challenging to compress the fluid, and thus, sound travels faster. Knowing the conditions of a fluid allows us to calculate the speed of sound accurately.

Think of the speed of sound like echoes in a canyon. If you clap your hands at the edge, the sound travels through the air and bounces back to you. This travel time depends on how dense the air is and its temperature. On a hot day, the sound travels faster because the hot air is less dense than cold air, allowing sound waves to move more quickly!

So, this is the figure, which shows the bulk modulus elasticity how it varies with the temperature. So, as you can see the bulk modulus elasticity increases as the temperature increases to a certain value and then starts to decreasing these are the normal ranges of temperature in air, 100 degrees is very rare but the most common ones are 0 to 30 degrees in India it can be 0 to 40 degrees. So, for water, so, I am sorry this bulk modulus elasticity is for water.

The speed of sound is significantly affected by the elasticity of the medium, measured as the bulk modulus. In general, as temperature increases, the bulk modulus of elasticity increases, allowing sound to travel more quickly through that medium. However, there is a point where, after reaching a certain temperature, the bulk modulus can decrease, leading to a reduction in speed. This relationship highlights that sound does not travel at a constant speed; instead, it is dynamic and dependent on environmental conditions.

Consider how fast you can run across the sand at the beach versus on a firm, solid surface. In the sand, your speed slows because it's less firm and uneven, similar to how sound travels slower in a less elastic medium. In contrast, a solid surface, like concrete, allows you to move much more quickly, just like sound travels faster in higher temperature water or air with high elasticity.

So, we are going to look at other phenomenon or process called Isentropic where no heat is exchanged. In this case, the equation for isentropic process is given by where . This is a very standard terminology in thermal physics that you have already seen in your class 10th and 12th. So, this is also called a specific heat ratio. So, if you do from this equation, you will get therefore . We are using the substituting the value of C in terms of and therefore, , or we can also write , because, is given as . Therefore, we have got , so we get more importantly in case of Isentropic process is given by , k is specific heat ratio.

In physics, particularly thermodynamics, the behavior of gases can be explored using the isentropic process, which involves changes when no heat is added or removed. The speed of sound can be expressed in relation to this specific heat ratio (k), which is the ratio of specific heats. This mathematical relationship helps us understand how sound propagates through different gases and contributes to calculations of the speed of sound depending on the gas properties.

Imagine a balloon filled with air. When you squeeze it, the air inside doesn't get hotter or cooler; it just moves. This can be likened to the isentropic process. As you compress the air inside the balloon, the speed of sound within that air changes due to the changes in pressure and density but without heat being added or removed, much like how sound speed would change depending on the medium.

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