Speed of Propogation in PDE?
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Let $u(x,t)$ be a solution to some PDE, then how do we calculate the speed of propagation (I am not even sure what does that term mean). I have some intuitive idea for example if $u(x,t) = f(x-ct)$, then the graph of $u(x,t)$ is same as graph of $u(x-ct,0)$ and hence we have a vague notion of graph moving at a speed of $c$ in a particular direction. But for more complicated cases where the graph not only travels but also changes is shapes (sorry I couldn't think of an example), how does one go about calculating the speed of propagation.
pde
asked Jan 9 at 21:26
henceprovedhenceproved
1358
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add a comment |
$begingroup$
Let $u(x,t)$ be a solution to some PDE, then how do we calculate the speed of propagation (I am not even sure what does that term mean). I have some intuitive idea for example if $u(x,t) = f(x-ct)$, then the graph of $u(x,t)$ is same as graph of $u(x-ct,0)$ and hence we have a vague notion of graph moving at a speed of $c$ in a particular direction. But for more complicated cases where the graph not only travels but also changes is shapes (sorry I couldn't think of an example), how does one go about calculating the speed of propagation.
pde
asked Jan 9 at 21:26
henceprovedhenceproved
1358
$endgroup$
$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
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@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
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– henceproved
Jan 9 at 22:03
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37
add a comment |
$begingroup$
Let $u(x,t)$ be a solution to some PDE, then how do we calculate the speed of propagation (I am not even sure what does that term mean). I have some intuitive idea for example if $u(x,t) = f(x-ct)$, then the graph of $u(x,t)$ is same as graph of $u(x-ct,0)$ and hence we have a vague notion of graph moving at a speed of $c$ in a particular direction. But for more complicated cases where the graph not only travels but also changes is shapes (sorry I couldn't think of an example), how does one go about calculating the speed of propagation.
pde
asked Jan 9 at 21:26
henceprovedhenceproved
1358
$endgroup$
Let $u(x,t)$ be a solution to some PDE, then how do we calculate the speed of propagation (I am not even sure what does that term mean). I have some intuitive idea for example if $u(x,t) = f(x-ct)$, then the graph of $u(x,t)$ is same as graph of $u(x-ct,0)$ and hence we have a vague notion of graph moving at a speed of $c$ in a particular direction. But for more complicated cases where the graph not only travels but also changes is shapes (sorry I couldn't think of an example), how does one go about calculating the speed of propagation.
pde
pde
asked Jan 9 at 21:26
henceprovedhenceproved
1358
asked Jan 9 at 21:26
henceprovedhenceproved
1358
asked Jan 9 at 21:26
henceprovedhenceproved
1358
asked Jan 9 at 21:26
henceprovedhenceproved
1358
asked Jan 9 at 21:26
henceprovedhenceproved
1358
1358
$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
$begingroup$
@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
$endgroup$
– henceproved
Jan 9 at 22:03
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37
add a comment |
$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
$begingroup$
@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
$endgroup$
– henceproved
Jan 9 at 22:03
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37
$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
$begingroup$
@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
$endgroup$
– henceproved
Jan 9 at 22:03
$begingroup$
@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
$endgroup$
– henceproved
Jan 9 at 22:03
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37
add a comment |
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$begingroup$
What sort of PDE are you interested in, wave equation, heat equation? Any initial conditions? Might help to know.
$endgroup$
– Tyler Kharazi
Jan 9 at 21:56
$begingroup$
@TylerKharazi I would like to find the speed of the wave equation on an infinite line, where $u(x,0)= 0$ and $u'(x,0) = 1$ on $|x| < a$ and 0 otherwise.
$endgroup$
– henceproved
Jan 9 at 22:03
$begingroup$
Sorry it took me so long to respond, I forgot to finish answering last night. Hmm, I think that within the PDE, the speed of propagation should be the inverse of the constant term multiplying the time derivative term (or the constant on the spatial derivative term). Have a look at page 16 and 17 of this: web.math.ucsb.edu/~grigoryan/124A.pdf. The speed of propagation is kinda what is sounds like. Its the speed at which the solution moves through space. Even if the waveform changes shape, we can still calculate how quickly the shape is moving through space.
$endgroup$
– Tyler Kharazi
Jan 10 at 21:37