s***e
发帖数: 911 | 1
以电动力学为例最好讲. 从Maxwell方程组可以导出真空的电磁波方程(也就是通常
的波动方程比如弦振动方程). 这方程的普遍解可以被证明具有如下形式:
f(x-c*t), 或者f(x+c*t). 正负号表示波的传播方向不同. 以下以右传波f(x-c*t)
为例. p(t)=x-c*t 就是这个波的"相". 相的传播条件是p(t)=constant, 于是就得到
相的传播速度: (d/dt)p(t)=0=>(d/dt)x=c. 因此c就是相速度. 对真空中的波动方程来
说, 速度是被良好定义了的, 无所谓什么群速度相速度的差别, 可以统统称为波速.
下面对f(x-v*t)作傅立叶变化:
f(x-v*t)=(1/Sqrt[2 Pi])*Integrate[g(k)*Exp[I*k*(x-v*t)],{k,-Infty,Infty}]
=(1/Sqrt[2 Pi])*Integrate[g(k)*Exp[I*(k*x-w*t)],{k,-Infty,Infty}]
其中w=k*c, 而c是常数. 频率w和波矢k的函数关系称为色散关系. 这里是线性关系,
意味着各个不同频率 | s***e
发帖数: 911 | 2
严格的说你的vp和vg定义都不严格. 好点的定义是:
vp是波动方程中蕴含的速度. 电磁学中就是真空电磁波方程中蕴含的速度. 单色
波只是普遍解f(x-vp*t)的特殊情况. vp不需要局限于单色光来定义. vp的真正含意
是相x-vp*t的传播速度.在介质中相速度无法对电磁波整体定义, 才只好对单色光
用vp=w/k来定义.
vg不是能量传播速度. vg是波包传播速度. 大部分情况下vg和能量传播速度相同.
也有不同的时候. | c********s
发帖数: 123 | 3 your definition is confusing to me as well. hehe
When Vg is not the speed of energy propagation speed? Any example?
A physical definition in this case is not very strict, I agree. But it
provides an easy insight into the two quantities. Mathematically, you have
given very good explanation plus hanzo's definition. To me, I understand it
this way:
suppose you have a wave packet: f(x, t) propagates from f(x=0, t=0)
the phase speed of a monochromatic component is obtained through Fourier Tran.
that
【在 s***e 的大作中提到】
:
: 严格的说你的vp和vg定义都不严格. 好点的定义是:
: vp是波动方程中蕴含的速度. 电磁学中就是真空电磁波方程中蕴含的速度. 单色
: 波只是普遍解f(x-vp*t)的特殊情况. vp不需要局限于单色光来定义. vp的真正含意
: 是相x-vp*t的传播速度.在介质中相速度无法对电磁波整体定义, 才只好对单色光
: 用vp=w/k来定义.
: vg不是能量传播速度. vg是波包传播速度. 大部分情况下vg和能量传播速度相同.
: 也有不同的时候.
| s***e
发帖数: 911 | 4
给你个例子:)
Wangsness, Elctromagnetic fields,John Wiley & sons Inc, 1979,习题
24-27, hehe...
【在 c********s 的大作中提到】
: your definition is confusing to me as well. hehe
: When Vg is not the speed of energy propagation speed? Any example?
: A physical definition in this case is not very strict, I agree. But it
: provides an easy insight into the two quantities. Mathematically, you have
: given very good explanation plus hanzo's definition. To me, I understand it
: this way:
: suppose you have a wave packet: f(x, t) propagates from f(x=0, t=0)
: the phase speed of a monochromatic component is obtained through Fourier Tran.
: that
| c********s
发帖数: 123 | 5 quantum field theory is good stuff too. hehe
Theoretically, fluids have more beautiful solutions than QFD maybe because
its physical principles are well understood.
We can get beautiful analytical solutions for 1electron atom only, other
than that, many approximations have to be made.
【在 s***e 的大作中提到】
:
: 给你个例子:)
: Wangsness, Elctromagnetic fields,John Wiley & sons Inc, 1979,习题
: 24-27, hehe...
| c********s
发帖数: 123 | 6 how come?
I was reviewing Quantum physics recently. And I was shocked again by its
involvement of linear algebra, abstract mathematical representation, but
that being said, solutions are fairly easily to be obtained. The most fearing
part of Q.F. is its mathematical representation including Li-Group theory,
the most difficult part is really to understand the mathematical abstraction
of a physical model.
For fluids, physics is simple, math is so so. hehe, the hard part is it has
mass, momentum, e
【在 s***e 的大作中提到】
:
: 给你个例子:)
: Wangsness, Elctromagnetic fields,John Wiley & sons Inc, 1979,习题
: 24-27, hehe...
| s***e
发帖数: 911 | 7
这是两种不同性质的困难. 流体力学方程的困难来自它是一个非线性的偏微分方程
方程组. 就这个数学困难. 物理图象倒是简单:)
【在 c********s 的大作中提到】
: how come?
: I was reviewing Quantum physics recently. And I was shocked again by its
: involvement of linear algebra, abstract mathematical representation, but
: that being said, solutions are fairly easily to be obtained. The most fearing
: part of Q.F. is its mathematical representation including Li-Group theory,
: the most difficult part is really to understand the mathematical abstraction
: of a physical model.
: For fluids, physics is simple, math is so so. hehe, the hard part is it has
: mass, momentum, e
| c********s
发帖数: 123 | 8 You said it. hehe. So in fact, Quantum physics, blahblah, is a more difficult
subject for scientist, especially physicists.
The non-linear part is often approximated, numerically handled. So no big deal!
But our understanding of hydrodynamics is not really perfect, i.e. turbulence,
strong shear effect, many being investigated yet yields no perfect solution.
We still cannot explain some phenomena happening in large scale problem on
the sun.
【在 s***e 的大作中提到】
:
: 这是两种不同性质的困难. 流体力学方程的困难来自它是一个非线性的偏微分方程
: 方程组. 就这个数学困难. 物理图象倒是简单:)
| s***e
发帖数: 911 | 9
说的就是这里. 作线性近似对流体这样复杂的系统来说,大部分情况都不现实. 实际
上流体的复杂行为就是来自非线性本身. 这点绕不过去的...
【在 c********s 的大作中提到】
: You said it. hehe. So in fact, Quantum physics, blahblah, is a more difficult
: subject for scientist, especially physicists.
: The non-linear part is often approximated, numerically handled. So no big deal!
: But our understanding of hydrodynamics is not really perfect, i.e. turbulence,
: strong shear effect, many being investigated yet yields no perfect solution.
: We still cannot explain some phenomena happening in large scale problem on
: the sun.
| c********s
发帖数: 123 | 10 When I was reviewing quantum physics, I cannot keep myself from noticing that
Quantum theory is such a linear theory. Is the microscopic world really
that linear. Is high energy phsics that linear?
hehe...
【在 s***e 的大作中提到】
:
: 说的就是这里. 作线性近似对流体这样复杂的系统来说,大部分情况都不现实. 实际
: 上流体的复杂行为就是来自非线性本身. 这点绕不过去的...
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