# Equation of a sound wave with viscous damping in ideal gas

• Tahmeed
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.

#### Tahmeed

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.

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.

• vanhees71
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

Here is something that might give you a couple of ideas to play with.   From Vibrating Strings by D.R. Bland (1960)

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

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## 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:

∂²p/∂t² = c²∇²p + η∇²∂p/∂t

where p is the sound pressure, t is time, c is the speed of sound, ∇² is the Laplacian operator, and η is the viscosity coefficient.

## 2. How does viscous damping affect the propagation of sound waves in an ideal gas?

Viscous damping causes energy loss in sound waves as they travel through an ideal gas. This results in a decrease in the amplitude and intensity of the sound wave over time.

## 3. What is the significance of the speed of sound in the equation for sound waves with viscous damping?

The speed of sound, represented by c in the equation, determines the rate at which sound waves travel through the medium. It is an important factor in calculating the propagation of sound waves with viscous damping.

## 4. How does the viscosity coefficient affect the behavior of sound waves in an ideal gas?

The viscosity coefficient, represented by η in the equation, determines the strength of the damping effect on sound waves in the medium. A higher viscosity coefficient results in a stronger damping effect, leading to a faster decrease in the amplitude and intensity of the sound wave.

## 5. Can the equation for sound waves with viscous damping be applied to all types of gases?

No, the equation is specifically for an ideal gas, which is a theoretical concept that follows certain assumptions. It may not accurately describe the behavior of sound waves in real gases, which can have varying levels of density, viscosity, and other properties.