Deriving Non-linear acoustic wave models, equilibrium state assumption

In summary, assuming a non-vanishing equilibrium velocity will lead to a different wave equation than assuming a vanishing equilibrium velocity.
  • #1
binbagsss
1,326
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The standard derivation in obtaining a single wave equation involves making use of the heat equation with a Taylor expansion of the equation of state, then differentiating this equation and the continuity equation with respect to time, and combining with the divergence of the NS equation.

From the literature I have found, it tends to be assumed an equilibrium state of:
- #u_0=0# (vanishing equilibrium velocity)
-#\partial_t \rho_0 =0# (no time dependence on the equilibrium density)
(which together, by the continuity => vanishing spatial dependence on the equilibrium velocity as well).Probably a stupid question but, if assuming a non-vanishing equilibrium velocity, (and of course under the same assumptions corresponding to used to derive the specific model), do we expect to obtain the same wave equation as we would if vanishing equilibrium velocity was assumed?

Many thanks
 
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  • #2
in advance!No, assuming a non-vanishing equilibrium velocity will not result in obtaining the same wave equation as if vanishing equilibrium velocity was assumed. The assumption of a non-vanishing equilibrium velocity will affect the terms which are included in the Taylor expansion of the equation of state and the resulting wave equation. As such, the wave equation obtained when assuming a non-vanishing equilibrium velocity will be different from the wave equation obtained when assuming a vanishing equilibrium velocity.
 

FAQ: Deriving Non-linear acoustic wave models, equilibrium state assumption

What is the equilibrium state assumption in non-linear acoustic wave models?

The equilibrium state assumption in non-linear acoustic wave models refers to the initial condition where the medium is at rest and all variables such as pressure, density, and velocity are uniform and constant. This assumption simplifies the derivation of the governing equations by providing a baseline from which perturbations or deviations are measured.

Why is the equilibrium state assumption important in deriving non-linear acoustic wave models?

The equilibrium state assumption is crucial because it allows for the linearization of the governing equations around a known, stable state. This simplification makes it easier to identify and analyze the non-linear terms that describe the propagation of acoustic waves. Without this assumption, the complexity of the equations would increase significantly, making analytical and numerical solutions more challenging.

How do non-linear terms arise in acoustic wave equations?

Non-linear terms in acoustic wave equations arise from the non-linear relationship between pressure, density, and velocity in the governing equations of fluid dynamics. When perturbations from the equilibrium state are large, the linear approximation is no longer valid, and higher-order terms must be included to accurately describe the wave propagation. These non-linear terms account for phenomena such as harmonic generation, shock wave formation, and wave steepening.

What are the common methods for deriving non-linear acoustic wave models?

Common methods for deriving non-linear acoustic wave models include perturbation techniques, multiple scales analysis, and the method of characteristics. Perturbation techniques involve expanding the variables in terms of a small parameter and retaining higher-order terms. Multiple scales analysis separates the problem into different scales to handle non-linear interactions. The method of characteristics transforms the partial differential equations into ordinary differential equations along specific paths, simplifying the analysis of non-linear effects.

Can non-linear acoustic wave models be solved analytically?

In general, non-linear acoustic wave models are difficult to solve analytically due to the complexity introduced by the non-linear terms. However, approximate analytical solutions can sometimes be obtained using perturbation methods or asymptotic analysis for specific cases. For more general scenarios, numerical methods such as finite difference, finite element, or spectral methods are typically employed to obtain solutions.

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