Mixing
Waves in spaces of even dimension
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In an article by Jim Holt, "Geometric creatures," he says:
In a space with an odd number of dimensions, ... sound waves move
in a single sharp wave front.
But in spaces with an even number of dimensions, ... a noise-like
disturbance will generate a system of waves that reverberates forever.
Can someone please explain the mathematics behind this quote?
Jim Holt.
When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Farrar, Straus and Giroux, 2018.
geometry mathematical-physics wave-equation
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
add a comment |Â
up vote
7
down vote
favorite
In an article by Jim Holt, "Geometric creatures," he says:
In a space with an odd number of dimensions, ... sound waves move
in a single sharp wave front.
But in spaces with an even number of dimensions, ... a noise-like
disturbance will generate a system of waves that reverberates forever.
Can someone please explain the mathematics behind this quote?
Jim Holt.
When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Farrar, Straus and Giroux, 2018.
geometry mathematical-physics wave-equation
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
add a comment |Â
up vote
7
down vote
favorite
up vote
7
down vote
favorite
In an article by Jim Holt, "Geometric creatures," he says:
In a space with an odd number of dimensions, ... sound waves move
in a single sharp wave front.
But in spaces with an even number of dimensions, ... a noise-like
disturbance will generate a system of waves that reverberates forever.
Can someone please explain the mathematics behind this quote?
Jim Holt.
When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Farrar, Straus and Giroux, 2018.
geometry mathematical-physics wave-equation
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
In an article by Jim Holt, "Geometric creatures," he says:
In a space with an odd number of dimensions, ... sound waves move
in a single sharp wave front.
But in spaces with an even number of dimensions, ... a noise-like
disturbance will generate a system of waves that reverberates forever.
Can someone please explain the mathematics behind this quote?
Jim Holt.
When Einstein Walked with Gödel: Excursions to the Edge of Thought.
Farrar, Straus and Giroux, 2018.
geometry mathematical-physics wave-equation
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
edited Jul 15 at 6:04
Rodrigo de Azevedo
12.6k41751
12.6k41751
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
asked Jul 15 at 1:10
Joseph O'Rourke
17.2k248103
17.2k248103
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
accepted
The phenomenon behind this is usually referred to as Huygen's principle, and can be described a little bit more precisely as the following:
Say you have the solution $u(x,t)$ to an initial-value problem for the wave equation in $xinmathbbR^n$:
$$Delta u - c^2 u_tt = 0,quad u(x,0) = phi,quad u_t (x,0) = psi$$
It turns out $u(x,t)$ depends only on the values of $phi$ and $psi$ on the surface of a ball $B$, centered at $x$ and of radius $ct$, for odd dimensions $n geq 3$. Heuristically, that means the initial "disturbance" always propagates out at $c$, so the "size" of the nonzero part of the solution (support) is invariant with time. For even dimensions, the value of $u(x,t)$ is also dependent on everything inside that ball $B$ (i.e. things don't just propagate at $c$), so the "size" of the non-zero part of the solution (support) isn't invariant with time.
A short explanation of why it happens is as a result of the fact that there is no transformation in even dimensions that can turn a wave equation into a 1-D problem, which is the case for odd dimensions $n geq 3$. It is more thoroughly described and fully derived in Evans' book or here.
A slightly more beautiful but more esoteric explanation comes from Balasz, who noted that one can obtain solutions to the wave equation by contour integration after performing a Wick rotation. The integrands of these possess a differing type of singularity based on the dimensional parity; a branch point in even dimensions and a pole in odd dimensions. And when calculating the value of the contour integral in the odd $n$ case, one needs to refer only to one time value (that of the pole), while this is impossible in the even $n$ case.
I should also point out that a system of waves in even dimensions actually does asymptotically decay to nothing everywhere inside of the ball $B$ (i.e. everything in the interior of $B$ eventually destructively interferes with itself), as shown in this paper by Bob Strichartz; he notes this can be used to justify why the classical prediction of certain observables emerges out of free quantum wave-functions. So take Jim Holt's statement with a grain of salt!
answered Jul 15 at 3:05
aghostinthefigures
9781214
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
add a comment |Â
up vote
2
down vote
The mathematics is explained here. It discusses the behavior of the Green's function for the wave operator in various dimensions.
answered Jul 15 at 2:47
G. Smith
763
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
accepted
The phenomenon behind this is usually referred to as Huygen's principle, and can be described a little bit more precisely as the following:
Say you have the solution $u(x,t)$ to an initial-value problem for the wave equation in $xinmathbbR^n$:
$$Delta u - c^2 u_tt = 0,quad u(x,0) = phi,quad u_t (x,0) = psi$$
It turns out $u(x,t)$ depends only on the values of $phi$ and $psi$ on the surface of a ball $B$, centered at $x$ and of radius $ct$, for odd dimensions $n geq 3$. Heuristically, that means the initial "disturbance" always propagates out at $c$, so the "size" of the nonzero part of the solution (support) is invariant with time. For even dimensions, the value of $u(x,t)$ is also dependent on everything inside that ball $B$ (i.e. things don't just propagate at $c$), so the "size" of the non-zero part of the solution (support) isn't invariant with time.
A short explanation of why it happens is as a result of the fact that there is no transformation in even dimensions that can turn a wave equation into a 1-D problem, which is the case for odd dimensions $n geq 3$. It is more thoroughly described and fully derived in Evans' book or here.
A slightly more beautiful but more esoteric explanation comes from Balasz, who noted that one can obtain solutions to the wave equation by contour integration after performing a Wick rotation. The integrands of these possess a differing type of singularity based on the dimensional parity; a branch point in even dimensions and a pole in odd dimensions. And when calculating the value of the contour integral in the odd $n$ case, one needs to refer only to one time value (that of the pole), while this is impossible in the even $n$ case.
I should also point out that a system of waves in even dimensions actually does asymptotically decay to nothing everywhere inside of the ball $B$ (i.e. everything in the interior of $B$ eventually destructively interferes with itself), as shown in this paper by Bob Strichartz; he notes this can be used to justify why the classical prediction of certain observables emerges out of free quantum wave-functions. So take Jim Holt's statement with a grain of salt!
answered Jul 15 at 3:05
aghostinthefigures
9781214
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
add a comment |Â
up vote
4
down vote
accepted
The phenomenon behind this is usually referred to as Huygen's principle, and can be described a little bit more precisely as the following:
Say you have the solution $u(x,t)$ to an initial-value problem for the wave equation in $xinmathbbR^n$:
$$Delta u - c^2 u_tt = 0,quad u(x,0) = phi,quad u_t (x,0) = psi$$
It turns out $u(x,t)$ depends only on the values of $phi$ and $psi$ on the surface of a ball $B$, centered at $x$ and of radius $ct$, for odd dimensions $n geq 3$. Heuristically, that means the initial "disturbance" always propagates out at $c$, so the "size" of the nonzero part of the solution (support) is invariant with time. For even dimensions, the value of $u(x,t)$ is also dependent on everything inside that ball $B$ (i.e. things don't just propagate at $c$), so the "size" of the non-zero part of the solution (support) isn't invariant with time.
A short explanation of why it happens is as a result of the fact that there is no transformation in even dimensions that can turn a wave equation into a 1-D problem, which is the case for odd dimensions $n geq 3$. It is more thoroughly described and fully derived in Evans' book or here.
A slightly more beautiful but more esoteric explanation comes from Balasz, who noted that one can obtain solutions to the wave equation by contour integration after performing a Wick rotation. The integrands of these possess a differing type of singularity based on the dimensional parity; a branch point in even dimensions and a pole in odd dimensions. And when calculating the value of the contour integral in the odd $n$ case, one needs to refer only to one time value (that of the pole), while this is impossible in the even $n$ case.
I should also point out that a system of waves in even dimensions actually does asymptotically decay to nothing everywhere inside of the ball $B$ (i.e. everything in the interior of $B$ eventually destructively interferes with itself), as shown in this paper by Bob Strichartz; he notes this can be used to justify why the classical prediction of certain observables emerges out of free quantum wave-functions. So take Jim Holt's statement with a grain of salt!
answered Jul 15 at 3:05
aghostinthefigures
9781214
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
add a comment |Â
up vote
4
down vote
accepted
up vote
4
down vote
accepted
The phenomenon behind this is usually referred to as Huygen's principle, and can be described a little bit more precisely as the following:
Say you have the solution $u(x,t)$ to an initial-value problem for the wave equation in $xinmathbbR^n$:
$$Delta u - c^2 u_tt = 0,quad u(x,0) = phi,quad u_t (x,0) = psi$$
It turns out $u(x,t)$ depends only on the values of $phi$ and $psi$ on the surface of a ball $B$, centered at $x$ and of radius $ct$, for odd dimensions $n geq 3$. Heuristically, that means the initial "disturbance" always propagates out at $c$, so the "size" of the nonzero part of the solution (support) is invariant with time. For even dimensions, the value of $u(x,t)$ is also dependent on everything inside that ball $B$ (i.e. things don't just propagate at $c$), so the "size" of the non-zero part of the solution (support) isn't invariant with time.
A short explanation of why it happens is as a result of the fact that there is no transformation in even dimensions that can turn a wave equation into a 1-D problem, which is the case for odd dimensions $n geq 3$. It is more thoroughly described and fully derived in Evans' book or here.
A slightly more beautiful but more esoteric explanation comes from Balasz, who noted that one can obtain solutions to the wave equation by contour integration after performing a Wick rotation. The integrands of these possess a differing type of singularity based on the dimensional parity; a branch point in even dimensions and a pole in odd dimensions. And when calculating the value of the contour integral in the odd $n$ case, one needs to refer only to one time value (that of the pole), while this is impossible in the even $n$ case.
I should also point out that a system of waves in even dimensions actually does asymptotically decay to nothing everywhere inside of the ball $B$ (i.e. everything in the interior of $B$ eventually destructively interferes with itself), as shown in this paper by Bob Strichartz; he notes this can be used to justify why the classical prediction of certain observables emerges out of free quantum wave-functions. So take Jim Holt's statement with a grain of salt!
answered Jul 15 at 3:05
aghostinthefigures
9781214
The phenomenon behind this is usually referred to as Huygen's principle, and can be described a little bit more precisely as the following:
Say you have the solution $u(x,t)$ to an initial-value problem for the wave equation in $xinmathbbR^n$:
$$Delta u - c^2 u_tt = 0,quad u(x,0) = phi,quad u_t (x,0) = psi$$
It turns out $u(x,t)$ depends only on the values of $phi$ and $psi$ on the surface of a ball $B$, centered at $x$ and of radius $ct$, for odd dimensions $n geq 3$. Heuristically, that means the initial "disturbance" always propagates out at $c$, so the "size" of the nonzero part of the solution (support) is invariant with time. For even dimensions, the value of $u(x,t)$ is also dependent on everything inside that ball $B$ (i.e. things don't just propagate at $c$), so the "size" of the non-zero part of the solution (support) isn't invariant with time.
A short explanation of why it happens is as a result of the fact that there is no transformation in even dimensions that can turn a wave equation into a 1-D problem, which is the case for odd dimensions $n geq 3$. It is more thoroughly described and fully derived in Evans' book or here.
A slightly more beautiful but more esoteric explanation comes from Balasz, who noted that one can obtain solutions to the wave equation by contour integration after performing a Wick rotation. The integrands of these possess a differing type of singularity based on the dimensional parity; a branch point in even dimensions and a pole in odd dimensions. And when calculating the value of the contour integral in the odd $n$ case, one needs to refer only to one time value (that of the pole), while this is impossible in the even $n$ case.
I should also point out that a system of waves in even dimensions actually does asymptotically decay to nothing everywhere inside of the ball $B$ (i.e. everything in the interior of $B$ eventually destructively interferes with itself), as shown in this paper by Bob Strichartz; he notes this can be used to justify why the classical prediction of certain observables emerges out of free quantum wave-functions. So take Jim Holt's statement with a grain of salt!
answered Jul 15 at 3:05
aghostinthefigures
9781214
edited Jul 16 at 23:46
answered Jul 15 at 3:05
aghostinthefigures
9781214
answered Jul 15 at 3:05
aghostinthefigures
9781214
answered Jul 15 at 3:05
aghostinthefigures
9781214
9781214
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
add a comment |Â
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
1
1
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
Thank you for the clear explanation!
– Joseph O'Rourke
Jul 15 at 10:15
add a comment |Â
up vote
2
down vote
The mathematics is explained here. It discusses the behavior of the Green's function for the wave operator in various dimensions.
answered Jul 15 at 2:47
G. Smith
763
add a comment |Â
up vote
2
down vote
The mathematics is explained here. It discusses the behavior of the Green's function for the wave operator in various dimensions.
answered Jul 15 at 2:47
G. Smith
763
add a comment |Â
up vote
2
down vote
up vote
2
down vote
The mathematics is explained here. It discusses the behavior of the Green's function for the wave operator in various dimensions.
answered Jul 15 at 2:47
G. Smith
763
The mathematics is explained here. It discusses the behavior of the Green's function for the wave operator in various dimensions.
answered Jul 15 at 2:47
G. Smith
763
answered Jul 15 at 2:47
G. Smith
763
answered Jul 15 at 2:47
G. Smith
763
answered Jul 15 at 2:47
G. Smith
763
763
add a comment |Â
add a comment |Â
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