Sound wave equation: Neumann boundary conditions
$begingroup$
In this paper it's described the solution of the damped wave equation in cylindrical coordinates
$$ nabla^2left(c^2rho_1+nufrac{partialrho_1}{partial t}right)-frac{partial^2rho_1}{partial t^2}=0$$
where $rho_1$ is the difference of the density relative to the unperturbed state $rho_0$.
The applied boundary condition is
$$ mathbf{v}big|_{r=r_0}=v_Acos(omega t)mathbf{hat{r}}$$
where $v$ is the velocity of the fluid.
They claim that this boundary condition can we rewritten as
begin{equation}
frac{partialrho_1}{partial r} bigg|_{r=r_0}=frac{rho_0v_Aomega c^2}{nu^2omega^2+c^4}sin(omega t)-frac{rho_0v_Aomega^2 nu}{nu^2omega^2+c^4}cos(omega t)hspace{1cm} (1)
end{equation}
just imposing $nablatimes mathbf{v}=mathbf{0}$ and using the equations for the conservation of mass and momentum
$$ frac{partialrho_1}{partial t} + nablacdot(rho_0 mathbf{v}) =0$$
$$ frac{partial}{partial t}(rho_0 mathbf{v})+c^2nablarho_1+nabla cdot mathbf{D}_1=mathbf{0}$$
where $mathbf{D}_1$ is the viscous stress tensor.
It is possible to prove that, if $nablatimes mathbf{v}=0$, then $nabla cdot mathbf{D}_1 = -nunabla^2mathbf{v} $.
I tried hard but I've not been able to prove equation $(1)$. Do you know how to proceed?
multivariable-calculus pde wave-equation fluid-dynamics
edited Dec 14 '18 at 9:32
Harry49
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
add a comment |
$begingroup$
In this paper it's described the solution of the damped wave equation in cylindrical coordinates
$$ nabla^2left(c^2rho_1+nufrac{partialrho_1}{partial t}right)-frac{partial^2rho_1}{partial t^2}=0$$
where $rho_1$ is the difference of the density relative to the unperturbed state $rho_0$.
The applied boundary condition is
$$ mathbf{v}big|_{r=r_0}=v_Acos(omega t)mathbf{hat{r}}$$
where $v$ is the velocity of the fluid.
They claim that this boundary condition can we rewritten as
begin{equation}
frac{partialrho_1}{partial r} bigg|_{r=r_0}=frac{rho_0v_Aomega c^2}{nu^2omega^2+c^4}sin(omega t)-frac{rho_0v_Aomega^2 nu}{nu^2omega^2+c^4}cos(omega t)hspace{1cm} (1)
end{equation}
just imposing $nablatimes mathbf{v}=mathbf{0}$ and using the equations for the conservation of mass and momentum
$$ frac{partialrho_1}{partial t} + nablacdot(rho_0 mathbf{v}) =0$$
$$ frac{partial}{partial t}(rho_0 mathbf{v})+c^2nablarho_1+nabla cdot mathbf{D}_1=mathbf{0}$$
where $mathbf{D}_1$ is the viscous stress tensor.
It is possible to prove that, if $nablatimes mathbf{v}=0$, then $nabla cdot mathbf{D}_1 = -nunabla^2mathbf{v} $.
I tried hard but I've not been able to prove equation $(1)$. Do you know how to proceed?
multivariable-calculus pde wave-equation fluid-dynamics
edited Dec 14 '18 at 9:32
Harry49
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
add a comment |
$begingroup$
In this paper it's described the solution of the damped wave equation in cylindrical coordinates
$$ nabla^2left(c^2rho_1+nufrac{partialrho_1}{partial t}right)-frac{partial^2rho_1}{partial t^2}=0$$
where $rho_1$ is the difference of the density relative to the unperturbed state $rho_0$.
The applied boundary condition is
$$ mathbf{v}big|_{r=r_0}=v_Acos(omega t)mathbf{hat{r}}$$
where $v$ is the velocity of the fluid.
They claim that this boundary condition can we rewritten as
begin{equation}
frac{partialrho_1}{partial r} bigg|_{r=r_0}=frac{rho_0v_Aomega c^2}{nu^2omega^2+c^4}sin(omega t)-frac{rho_0v_Aomega^2 nu}{nu^2omega^2+c^4}cos(omega t)hspace{1cm} (1)
end{equation}
just imposing $nablatimes mathbf{v}=mathbf{0}$ and using the equations for the conservation of mass and momentum
$$ frac{partialrho_1}{partial t} + nablacdot(rho_0 mathbf{v}) =0$$
$$ frac{partial}{partial t}(rho_0 mathbf{v})+c^2nablarho_1+nabla cdot mathbf{D}_1=mathbf{0}$$
where $mathbf{D}_1$ is the viscous stress tensor.
It is possible to prove that, if $nablatimes mathbf{v}=0$, then $nabla cdot mathbf{D}_1 = -nunabla^2mathbf{v} $.
I tried hard but I've not been able to prove equation $(1)$. Do you know how to proceed?
multivariable-calculus pde wave-equation fluid-dynamics
edited Dec 14 '18 at 9:32
Harry49
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
In this paper it's described the solution of the damped wave equation in cylindrical coordinates
$$ nabla^2left(c^2rho_1+nufrac{partialrho_1}{partial t}right)-frac{partial^2rho_1}{partial t^2}=0$$
where $rho_1$ is the difference of the density relative to the unperturbed state $rho_0$.
The applied boundary condition is
$$ mathbf{v}big|_{r=r_0}=v_Acos(omega t)mathbf{hat{r}}$$
where $v$ is the velocity of the fluid.
They claim that this boundary condition can we rewritten as
begin{equation}
frac{partialrho_1}{partial r} bigg|_{r=r_0}=frac{rho_0v_Aomega c^2}{nu^2omega^2+c^4}sin(omega t)-frac{rho_0v_Aomega^2 nu}{nu^2omega^2+c^4}cos(omega t)hspace{1cm} (1)
end{equation}
just imposing $nablatimes mathbf{v}=mathbf{0}$ and using the equations for the conservation of mass and momentum
$$ frac{partialrho_1}{partial t} + nablacdot(rho_0 mathbf{v}) =0$$
$$ frac{partial}{partial t}(rho_0 mathbf{v})+c^2nablarho_1+nabla cdot mathbf{D}_1=mathbf{0}$$
where $mathbf{D}_1$ is the viscous stress tensor.
It is possible to prove that, if $nablatimes mathbf{v}=0$, then $nabla cdot mathbf{D}_1 = -nunabla^2mathbf{v} $.
I tried hard but I've not been able to prove equation $(1)$. Do you know how to proceed?
multivariable-calculus pde wave-equation fluid-dynamics
multivariable-calculus pde wave-equation fluid-dynamics
edited Dec 14 '18 at 9:32
Harry49
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
edited Dec 14 '18 at 9:32
Harry49
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
edited Dec 14 '18 at 9:32
Harry49
7,46431340
edited Dec 14 '18 at 9:32
Harry49
7,46431340
edited Dec 14 '18 at 9:32
Harry49
7,46431340
7,46431340
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
asked Dec 13 '18 at 18:45
Alessandro ZuninoAlessandro Zunino
17311
17311
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You can combine the two conservation equations to obtain
$$ rho_0frac{partial v}{partial t}+c^2frac{partial rho_1}{partial r}-nufrac{partial}{partial t}frac{partial rho_1}{partial r}bigg|_{r=r_0} = 0$$
using the Fourier transform method is possible to solve this differential equation for the variable $frac{partial rho_1}{partial r}bigg|_{r=r_0}$ obtaining then equation $(1)$.
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3038436%2fsound-wave-equation-neumann-boundary-conditions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
You can combine the two conservation equations to obtain
$$ rho_0frac{partial v}{partial t}+c^2frac{partial rho_1}{partial r}-nufrac{partial}{partial t}frac{partial rho_1}{partial r}bigg|_{r=r_0} = 0$$
using the Fourier transform method is possible to solve this differential equation for the variable $frac{partial rho_1}{partial r}bigg|_{r=r_0}$ obtaining then equation $(1)$.
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
add a comment |
$begingroup$
You can combine the two conservation equations to obtain
$$ rho_0frac{partial v}{partial t}+c^2frac{partial rho_1}{partial r}-nufrac{partial}{partial t}frac{partial rho_1}{partial r}bigg|_{r=r_0} = 0$$
using the Fourier transform method is possible to solve this differential equation for the variable $frac{partial rho_1}{partial r}bigg|_{r=r_0}$ obtaining then equation $(1)$.
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
add a comment |
$begingroup$
You can combine the two conservation equations to obtain
$$ rho_0frac{partial v}{partial t}+c^2frac{partial rho_1}{partial r}-nufrac{partial}{partial t}frac{partial rho_1}{partial r}bigg|_{r=r_0} = 0$$
using the Fourier transform method is possible to solve this differential equation for the variable $frac{partial rho_1}{partial r}bigg|_{r=r_0}$ obtaining then equation $(1)$.
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
$endgroup$
You can combine the two conservation equations to obtain
$$ rho_0frac{partial v}{partial t}+c^2frac{partial rho_1}{partial r}-nufrac{partial}{partial t}frac{partial rho_1}{partial r}bigg|_{r=r_0} = 0$$
using the Fourier transform method is possible to solve this differential equation for the variable $frac{partial rho_1}{partial r}bigg|_{r=r_0}$ obtaining then equation $(1)$.
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
edited Dec 14 '18 at 18:25
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
answered Dec 14 '18 at 18:09
Alessandro ZuninoAlessandro Zunino
17311
17311
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3038436%2fsound-wave-equation-neumann-boundary-conditions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
