Speed of Sound - General Formula

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Discussion Overview

The discussion centers around the derivation of the speed of sound in various media, particularly focusing on the relationship between the bulk modulus and density. Participants explore theoretical foundations and seek specific derivations applicable to different materials, including solids and gases.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant requests a derivation of the speed of sound formula, specifically a² = (bulk modulus / density), and questions the application of Newton's laws in this context.
  • Another participant explains that sound waves involve oscillation rather than movement of material, emphasizing the role of material stiffness in the equation.
  • A different participant suggests that Newton's second law can be used to derive the equation for mechanical waves, noting that the derivation varies based on the medium's properties.
  • One participant expresses frustration at not finding the specific derivation for solid materials and mentions that Landau and Lifgarbagez use a different method for ideal gases.
  • Another participant offers to share slides that outline the derivation for a cubic crystal and references the need for understanding stress and strain tensors for clarity.
  • Links to external resources are provided for further exploration of the derivation for isotropic solids and mechanical waves.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the derivation methods, with multiple competing views and approaches presented. The discussion remains unresolved regarding the specific derivations for different materials.

Contextual Notes

Participants reference various sources and methods for deriving the speed of sound, indicating a dependence on the type of medium (solid vs. gas) and its properties (isotropic vs. anisotropic). There are mentions of specific mathematical frameworks that may not be universally understood among participants.

Curl
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Can anyone show that the speed of sound, or rather, speed of low-energy mechanical waves follows the relationship:

a2 = ( bulk modulus / density )
This holds for sound waves, and is also similar to the waves on strings formula.

Can anyone show how this is derived? I read a book and they said "by apply Newton's laws". But how?
 
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Hi Curl! :smile:

When sound travels, nothing in the material actually moves anywhere, it only oscillates on the spot …

the equation for this oscillation depends on the stiffness (springy-ness) of the material …

there's some details at http://en.wikipedia.org/wiki/Speed_of_sound#Basic_concept" :wink:
 
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You can derive the equation for mechanical waves in the medium by using Newton's second law. You look at a small piece of the medium and calculate the net force due to the perturbation (the wave).
The net force will depend on the distribution of elastic forces, so you will have the elastic constant(s) in the equation. The mass of the small piece of medium depends on density.
The actual derivation can be found in many textbooks. The actual form depends on the medium (fluid or solid, isotropic or anisotropic, etc) but the idea is along these lines.
 
Yeah no joke, what I am asking is to see the actual derivation.

Landau and Lifgarbagez use a very different method to derive the speed of sound in an ideal gas. I don't have any books that do it for solid materials, otherwise I wouldn't be asking this.
 
Curl said:
Yeah no joke, what I am asking is to see the actual derivation.

Landau and Lifgarbagez use a very different method to derive the speed of sound in an ideal gas. I don't have any books that do it for solid materials, otherwise I wouldn't be asking this.
The derivation for a cubic crystal can be found for example in Kittel - Introduction to solid state Physics (Chapter 3).
I have a couple of slides that show the main steps. I ca send them if you would like. For a crystal you need to be a little familiar the stress and strain tensors to understand the derivation.

For isotropic solid you can find some sketches of the derivation (in the 1 D case) for example here: http://mysite.du.edu/~jcalvert/waves/mechwave.htm
 

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