Why smooth spherical waves with attenuation are only possible in 3-D

In summary, the conversation discusses the wave equation for spherical waves in n-dimensions and introduces the concepts of attenuation and delay in relation to real-life physical phenomena. The ansatz for the equation is given and the coefficients for the different terms are calculated, leading to a system of ODEs for the attenuation and delay functions. The solutions for the ODEs are discussed for different values of n, with the conclusion that solutions only exist for n=1 and n=3.
  • #1
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Homework Statement
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Relevant Equations
$$\partial_t^2u(t,r)=\partial^2_ru(t,r)+\frac{n-1}{r}\partial_ru(t,r)$$
Hi all, My question is about the attenuation and delay terms in part (1). what are attenuation and delay terms describing in physical phenomenon? thank you. What do "attenuation" and "delay" mean in terms of real-life physical phenomena?
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Consider the wave equation for spherical waves in ##n##-dimensions given by
$$\partial_t^2u(t,r)=\partial^2_ru(t,r)+\frac{n-1}{r}\partial_ru(t,r)$$
for an unknown function ##u:(0,\infty)\times(0,\infty)\rightarrow \mathbb{R}##

(1) Consider twice continuously differentiable functions ##\alpha:(0,\infty)\rightarrow (0,\infty)## (the attenuation) and ##\beta:(0,\infty)\mathbb{R}## (the delay) and ##f:\mathbb{R}\rightarrow \mathbb{R}## and make the ansatz
$$u(t,r)=\alpha(r)f(t-\beta(r))$$
(2) We insert this ansatz into the spherical wave equation
$$\partial_t^2\Big[\alpha(r)f(t-\beta(r))\Big]-\partial_r^2\Big[\alpha(r)f(t-\beta(r))\Big]-\frac{n-1}{r}\partial_r\Big[\alpha(r)f(t-\beta(r))\Big]$$
We compute the first term
$$\partial_t^2\Big[\alpha(r)f(t-\beta(r))\Big]\Rightarrow \boxed{\alpha(r)\partial_t^2f(t-\beta(r))}$$
The first derivative of the second term $$-\partial_r\Big[\alpha(r)f(t-\beta(r))\Big] =-\partial_r\alpha(r)f(t-\beta(r))+\alpha(r)\partial_r\beta(r)\partial_rf(t-\beta(r))$$
The derivative of the first term of the first derivative
$$-\partial_r^2\alpha(r)f(t-\beta(r))+\partial_r\alpha(r)\partial_r\beta(r)\partial_rf(t-\beta(r))$$
The derivative of the second term of the first derivative
$$\Big[\partial_r\alpha(r)\partial_r\beta(r)+\alpha(r)\partial^2_r\beta(r)\Big]\partial_rf(t-\beta(r))-\alpha(r)[\partial_r\beta(r)]^2\partial_r^2f(t-\beta(r))$$
That gives the second derivative of the second term $$\Rightarrow \boxed{-\partial_r^2\alpha(r)f(t-\beta(r))+\partial_r\alpha(r)\partial_r\beta(r)\partial_rf(t-\beta(r))}$$
$$\boxed{+\Big[\partial_r\alpha(r)\partial_r\beta(r)+\alpha(r)\partial^2_r\beta(r)\Big]\partial_rf(t-\beta(r))-\alpha(r)[\partial_r\beta(r)]^2\partial_r^2f(t-\beta(r))}$$
The third term:
$$-\frac{n-1}{r}\partial_r\Big[\alpha(r)f(t-\beta(r))\Big]\Rightarrow \boxed{-\frac{n-1}{r}\Big[\partial_r\alpha(r)f(t-\beta(r))-\alpha(r)\partial_r\beta(r)\partial_rf(t-\beta(r))\Big]}$$
We collect the coefficients of the ##f(t-\beta(r))## terms $$\partial_r^2\alpha(r)+\frac{n-1}{r}\partial_r\alpha(r)$$
and the coefficients of the ##\partial_rf(t-\beta(r))## terms
$$\partial_r\alpha(r)\partial_r\beta(r)\alpha(r)+\Big[\partial_r\alpha(r)\partial_r\beta(r)+\alpha(r)\partial^2_r\beta(r)\Big]+\frac{n-1}{r}\alpha(r)\partial_r\beta(r)$$
and the coefficients of the ##\partial_r^2f(t-\beta(r))## terms
$$\alpha(r)[\partial_r\beta(r)]^2-\alpha(r)$$
To make the brackets banish to zero, we set the three collections of terms equal to zero. Setting each collection of terms equal to zero gives the following system of ODEs
$$\boxed{\begin{cases}
\partial_r^2\alpha(r)+\frac{n-1}{r}\partial_r\alpha(r) =0 \\-2\partial_r\alpha(r)+\frac{n-1}{r}\alpha(r) =0\\
[\partial_r\beta(r)]^2=1
\end{cases}}$$
(4) The ODE that involves ##α##only is $$\partial_r^2\alpha(r)+\frac{n-1}{r}\partial_r\alpha(r) =0$$
Case: ##n=1##
$$\partial_r^2\alpha(r) =0$$
The roots of the characteristic polynomial ##\lambda^2=0## is just ##0##. The solution is a polynomial of the form
$$\Rightarrow \boxed{\alpha(r)=cr+d} \quad \text{where }\quad c,d\in\mathbb{R}$$
setting ##n=2## gives a second order euler homegenuous ODE.
$$\partial_r^2\alpha(r)+\frac{1}{r}\partial_r\alpha(r) =0$$
The equation has a solution of the form ##r^x##. plugging ##r^x## into the ODE gives the general solution
$$\boxed{\alpha(r)=cln(r)+d}$$
For the case ##n\geq 3## the ODE is
$$\partial_r^2\alpha(r)+\frac{2}{r}\partial_r\alpha(r) =0$$
By similar methods to the case ##n=2##,
$$\boxed{\alpha(r)=\frac{c}{t}+d}$$
(5) The ODE for ##β(r)## is
$$[\partial_r\beta(r)]^2=1\Rightarrow \boxed{\partial_r\beta(r)=\pm 1}$$
(6) The following system of ODEs in ##\alpha(r)##
$$\begin{cases}
\partial_r^2\alpha(r)+\frac{n-1}{r}\partial_r\alpha(r)=0\\
-2\partial_r\alpha(r)+\frac{n-1}{r}\alpha(r) =0\end{cases}$$ shows that there are no solutions, unless ##n=1## or ##n=3##. We assume the solutions is of form $$\alpha(r)=cr^a$$ for some constants ##c,a\in\mathbb{R}##.
Plugging in the solution to the system of ODEs gives
$$\begin{cases}
a(a-1)+(n-1)c=0 \\ -2a+(n-1)=0
\end{cases}$$
whose solution exists for ##a## only if ##n=1## or ##n=3##.

(7) When we plug in ##n=1##, we get ##a=0## which means ##\alpha{r}=c## for some ##c\in\mathbb{R}##.edited: grammar :headbang:
 

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  • #2
The attenuation is the way the amplitude diminishes as the wave radiates out.
It is shown as a factor α(r), presumably monotonically decreasing.
The delay is the time lag between generation of a part of the wave and its reaching radius r. So again, it is a function of r, presumably monotonically increasing this time, with β(0)=0. It is subtracted from t in the expression for u(r,t).

If we have a bit of wave amplitude A emitted at time 0 (u(0,0)=A) and reaching radius r at time t = β(r) then its amplitude there and then will be u(r,t)=α(r)u(0, t-β(r))=α(r)u(0, 0)=Aα(r).
 
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Related to Why smooth spherical waves with attenuation are only possible in 3-D

1. Why can smooth spherical waves with attenuation only occur in 3-D?

Smooth spherical waves with attenuation are a type of wave that propagates evenly in all directions from a source. This type of wave is only possible in 3-D because it requires three dimensions (length, width, and height) to fully propagate and maintain its spherical shape. In 2-D or 1-D, the wave would be limited to propagating in only one or two dimensions, resulting in a distorted shape.

2. What is the significance of attenuation in smooth spherical waves?

Attenuation refers to the gradual decrease in the amplitude or intensity of a wave as it travels through a medium. In the case of smooth spherical waves, attenuation is important because it allows the wave to disperse and spread out as it travels, maintaining its spherical shape. Without attenuation, the wave would not be able to propagate evenly and would eventually lose its spherical form.

3. Can smooth spherical waves with attenuation occur in higher dimensions?

No, smooth spherical waves with attenuation can only occur in three dimensions. This is because higher dimensions, such as 4-D or 5-D, are not physically possible in our universe. Therefore, the concept of a smooth spherical wave with attenuation does not apply in these higher dimensions.

4. How does the medium affect the propagation of smooth spherical waves with attenuation?

The medium through which a smooth spherical wave travels can greatly affect its propagation. For example, in a medium with high viscosity, the wave will experience more resistance and attenuation, causing it to lose its spherical shape more quickly. In a medium with low viscosity, the wave will travel more easily and maintain its shape for a longer distance.

5. Are there any real-life applications of smooth spherical waves with attenuation?

Yes, smooth spherical waves with attenuation have many practical applications. They are commonly used in seismology to study earthquake propagation, in medical imaging techniques such as ultrasound, and in sonar technology for underwater mapping and navigation. They also play a role in the study of electromagnetic radiation and its effects on the Earth's atmosphere.

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