Equation of a sound wave with viscous damping in ideal gas

In summary, the conversation discusses finding an equation for a 1D sound wave with damping in an ideal gas with viscosity. The possibility of using Burgers's equation is mentioned, but it is stated that there is no simple solution for this specific scenario. References to Vibrating Strings by D.R. Bland (1960) and the use of Burger's equation for nonlinear waves in acoustics are also mentioned.
  • #1
Tahmeed
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How can we find a equation of a 1D sound wave in a non-differential form in an ideal gas with viscosity? How does the damping work? How does the wave lose energy at each layer as it propagates?

To be clear I am looking for a simple exponential-sinusoidal function for it just in the case of damping in simple harmonic oscillation. If possible it will be great to have an energy analysis too about which layer receives how much of the lost energy.
 
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  • #2
Maybe you are looking for the solution to the (viscous) Burgers's equation, as stated in this wikipedia article.

I'm afraid that there is no simple solution.
 
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  • #3
Arjan82 said:
Maybe you are looking for the solution to the (viscous) Burgers's equation, as stated in this wikipedia article.

I'm afraid that there is no simple solution.

I don't think that's what I want. This Burger's equation is for fluid flow, it's not something similar to wave equation. I am looking for a wave equation that describes damping of sound wave in an ideal gas
 
  • #4
Here is something that might give you a couple of ideas to play with.

1of3.png2of3.png3of3.png

From Vibrating Strings by D.R. Bland (1960)

Btw, the Burger’s equation is used for nonlinear waves in acoustics.
 
Last edited:

1. What is the equation for a sound wave with viscous damping in an ideal gas?

The equation for a sound wave with viscous damping in an ideal gas is given by:
P(x,t) = P0cos(kx - ωt)e-αx
where P(x,t) is the sound pressure at position x and time t, P0 is the initial sound pressure, k is the wave number, ω is the angular frequency, and α is the damping coefficient.

2. How does viscous damping affect the behavior of a sound wave in an ideal gas?

Viscous damping causes the amplitude of a sound wave to decrease exponentially as it propagates through an ideal gas. This means that the sound wave will gradually lose energy and become quieter as it travels.

3. What is the role of the ideal gas law in the equation for a sound wave with viscous damping?

The ideal gas law, which states that the pressure of a gas is directly proportional to its temperature and density, is used to calculate the damping coefficient α in the equation for a sound wave with viscous damping. This coefficient takes into account the effects of the gas's viscosity on the propagation of the sound wave.

4. Can the equation for a sound wave with viscous damping be used for all types of sound waves?

No, the equation for a sound wave with viscous damping is specifically for sound waves propagating through an ideal gas. It does not apply to other mediums, such as liquids or solids, which have different properties that affect the behavior of sound waves.

5. How does the equation for a sound wave with viscous damping compare to the equation for a sound wave without damping?

The equation for a sound wave without damping is similar to the equation for a sound wave with viscous damping, except that the damping coefficient α is equal to zero. This means that there is no exponential decrease in amplitude, and the sound wave will continue to propagate with constant amplitude. However, in most real-world scenarios, some amount of damping is present, so the equation for a sound wave with viscous damping is a more accurate representation of sound wave behavior.

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