Aeroacoustics/Wave Equation and Green's function

We will show later that the propagation of acoustic waves (small amplitude fluctuations) in fluids is basically governed by the celebrated Wave equation. Therefore, in this chapter we will review some mathematical aspects of the wave equation which will help us to understand the physics in later chapters. It must be noted that we consider the wave propagation is happening in a three dimensional space, unless otherwise stated.

Wave Equation
Wave equation is the simplest, linear, hyperbolic partial differential equation which governs the linear propagation of waves, with finite speed, in media. Consider p(x,t) to be physical quantity (like pressure disturbance) which is propagating in space as a linear wave. Formally, the wave equation can be written as

$$\frac{\partial^2 p}{\partial t^2}-c_0^2\nabla^2 p = 0\, ,$$

where $$c_0$$ is the speed of wave propagation. This is the homogeneous wave equation which governs the propagation of a wave in a quiescent medium. If there is a source or a distribution of sources in the field which produce the wave, the wave equation will take the nonhomogeneous form

$$\frac{\partial^2 p}{\partial t^2}-c_0^2\nabla^2 p = f(x,t)\, ,$$

where f(x,t) represents the existence of the sources. To solve a nonhomogenous PDE like the one above, we need to utilize a mathematical tool called the Green's function.

Green's Function
Green's function is named after the British mathematician George Green, who first developed the concept in the 1830s. The concept of Green's function is one of the most powerful mathematical tools to solve boundary value problems.

Suppose, we have a linear differential equation given by:

$$L\, u = f,$$

where L is the differential operator. The main idea is to find a function G, called Green's function, such that the solution of the above differential equation can be determined from

$$u(x) = \int G(x,s) f(s) ds $$

To find the appropriate green function for a given differential equation, one should solve

$$L \,G(x,s) = \delta(x-s) $$

with the same boundary condition as the original problem. For example, the free-space Green's function of the wave equation, is the solution of the wave equation with an impulsive point source $$\delta(\mathbf{x}-\mathbf{y})\delta(t-\tau)$$ (which is located at point $$\mathbf{y}$$ and generates an impulse at time $$\tau$$). Therefore,

$$\Box^2 G(\mathbf{x},\mathbf{y},t,\tau) = \delta(\mathbf{x}-\mathbf{y})\delta(t-\tau) $$

where

$$\Box^2 =\frac{\partial^2 }{\partial\, t^2}-c_0^2\frac{\partial^2 }{\partial\, x^2} $$.

The solution of the above equation is

$$G(\mathbf{x},\mathbf{y},t,\tau)=\frac{1}{4\pi\,r}\delta\left(t-\tau-\frac{r}{c_0}\right)$$

where

$$r=|\mathbf{x}-\mathbf{y}|$$

The above relation represents an impulsive, spherically symmetric wave which is expanding from source $$\mathbf{y}$$. It is also important to mention that the argument of the delta function,

$$\tau+\frac{r}{c_0}$$

is called the retorted time and is equal to the time required for a wave, generated by source $$\mathbf{y}$$ at time $$\mathbf{\tau}$$, to arrive at point $$\mathbf{x}$$ at time $$\mathbf{t}$$.

Therefore, if we have a general problem (let's say noise generated by a turbulent jet), which is governed by

$$\Box^2 p = Q(x,t) $$

the instantaneous solution at any point in space is given by

$$p(x,t) = \frac{1}{4\pi}\int\int_{-\infty}^{\infty} Q(\mathbf{y},\tau) G(\mathbf{x},\mathbf{y}, t, \tau) d^3\,\mathbf{y}d\tau $$

In the case of radiation in free-space,

$$p(x,t) = \frac{1}{4\pi}\int_{-\infty}^{\infty} \int \frac{Q(\mathbf{y},\tau)}{|\mathbf{x}-\mathbf{y}|}\delta\left(t-\tau-\frac{|\mathbf{x}-\mathbf{y}|}{c_0}\right) d^3\,\mathbf{y}d\tau =\frac{1}{4\pi}\int_{-\infty}^{\infty} \frac{Q(\mathbf{y},t-|\mathbf{x}-\mathbf{y}|/c_0)}{|\mathbf{x}-\mathbf{y}|} d^3\,\mathbf{y}$$