Derivation of the speed of sound waves equation

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SUMMARY

The discussion focuses on the derivation of the speed of sound waves through a medium, specifically referencing Newton's formula for sound in gases. The derivation begins with the impulse equation (I=Δp) and incorporates the bulk modulus (B) and changes in volume (ΔV) and pressure (ΔP). A key point of confusion arises regarding the cancellation of Δt in the equations, leading to questions about the relationship between the speed of gas particles (vx) and the speed of sound (v). The conversation highlights the importance of understanding these relationships in the context of sound wave propagation.

PREREQUISITES
  • Understanding of basic physics concepts such as impulse and momentum.
  • Familiarity with the bulk modulus (B) and its role in fluid mechanics.
  • Knowledge of sound wave propagation in gases.
  • Ability to manipulate and derive equations involving pressure and volume changes.
NEXT STEPS
  • Study the derivation of Newton's formula for the speed of sound in gases.
  • Learn about the implications of the bulk modulus in different mediums.
  • Explore the relationship between particle velocity (vx) and sound velocity (v) in detail.
  • Investigate the mathematical principles behind impulse and momentum in fluid dynamics.
USEFUL FOR

Students of physics, educators teaching wave mechanics, and anyone interested in the principles of sound propagation in gases.

FerPhys
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I'm learning about the speed of sound waves through a medium. The derivation is initiated through Impulse=change in linear momentum (I=Δp), then I=ΣFΔt=(Area×Δpressure×Δt) in the x direction
The derivation proceeds by replacing the Δpressure with another equation we had derived earlier
ΔP=-B(ΔV/Vi) where B is the bulk modulus. The initial volume (Vi) is vAΔt where v is the velocity of sound multiplied by the time it takes to reach the end of the container (gives you a length which you can multiply to get Area to get the volume of the gas without anything force acting on it).
This is where I get a bit confused. My textbook says ΔV= (-vxAΔt) where vx = the speed of the elements in the medium or in this case a gas. When you plug everything back in into -B(ΔV/Vi) you get ΔP=B(vx / v) . My question now is, why did the Δt cancel out? Wouldn't that be like saying that the time it took for those gas particles to move through the entire container was the same as the time it took for sound to move through the entire container? Also, if that were the case wouldn't they be going at the same speed and therefore the vx/v should also cancel?

This my first post on this website so if I can do anything to make my posts more clear please let me know!
Thank you!
 
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Are you deriving the Newton's formula for speed of sound in say air/gas?
 
Can you show an image of the page where they do this?
 
This may come off as completely rude but I understand the derivation now.. sorry :(
yes its for Newtons formula for speed in sound..
 

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