Messages

1741

Home Assignment 2 / Re: Problem 2

« on: September 29, 2012, 02:36:32 PM »

for the last part, can we just assume the same solution as c) but state a few assumptions instead? it's because I dont think the general solution of the equation would vary since no other IC were stated.

You should check when and if the general solution is continuous at $r=0$ and adjust it respectively

1742

Home Assignment 2 / Re: problem 1 typo?

« on: September 29, 2012, 09:04:56 AM »

Can we assume that for part (C) of Problem 1 that the Cauchy conditions are evenly reflected for x < 0?

Sure, you can but it will not be useful as your domain is $x>vt$ rather than $x>0$. Just use the general solution.

1743

Misc Math / Re: Example 8b

« on: September 28, 2012, 05:36:19 PM »

If we are looking for $t>0$ then we are interested in $0<x<ct$ and $x>ct$ because the original problem is dealing with $x>0$. Continuation is the method to reduce it to IVP but we need to come back

Transitions in (9), (12) from the middle to the r.h. expression is due to the trigonometric formulae. Everything is ok

PS. Please use \sin,  \cos etc instead of sin, cos in LaTeX code as those are predefined macros.

PPS. In Lecture 9 we use \erf but as this is not a predefined macro, we define it by ourselves

1744

Home Assignment 2 / Re: Problem 2

« on: September 28, 2012, 05:23:56 PM »

I think in the new variant (just posted) it will be more clear

1745

Misc Math / Re: 1-D Wave equation derivation

« on: September 28, 2012, 02:20:57 PM »

Starting with

\begin{equation} \frac{\partial}{\partial x} \left[ T(x,t) \sin{\theta (x,t)} \right] = \rho (x) u_{tt} \end{equation}

where $\rho$ is the density and $T(x,t)$ is the tension force, we made the assumption that the vibrations are small, which gave us a linearized wave equation. I can see why some of the other assumptions (i.e. full flexibility, and no horizontal tension component) make sense, but I don't think I understand the insight behind this one.

You mean that vibrations are small? Because usually they are. More general versions you find in some textbooks like
\begin{equation}
\rho u_{tt} = \Bigl(\frac{u_x}{\sqrt{1+u_x^2}}\Bigr)_x
\end{equation}
implicitly assume that displacement is only in the direction perpendicular to the string and that the density does not change--which is the case only for the small oscillations.

1746

Misc Math / Re: Lecture 6 example

« on: September 27, 2012, 04:49:15 PM »

Hello,
I was going through the 6. lecture notes, where in the end an example is brought up that leads to an integral
$$I=\frac{1}{4}\int_0^t\cos(t')\bigl(\cos(x-ct+ct')-\cos(x+ct-ct')\bigr)dt'$$
I was trying to do that integral, but the only way that I could do it was to write out the cosines as complex exponentials, which lead me to eight terms in the end... Is there a cleverer way to do this integral?
Thanks!

In this example $c=2$ helps a bit but you could sea; without it.

The 1-st line -> 2nd (just integration) -> 3rd (formula $\cos (\alpha)-\cos (\beta)=
2\sin \bigl((\beta+\alpha)/2\bigr)\cdot \sin \bigl((\beta-\alpha)/2\bigr)$ and we ge the 4-th line.
Then formula $2\sin( \alpha) \cdot \cos(\beta)=\sin (\alpha+\beta)+\sin (\alpha-\beta)$ and integration.

1747

Home Assignment 2 / Re: Problem 3

« on: September 27, 2012, 01:41:03 PM »

Professor, could you fix this in the PDF file? I prefer the PDF file so I could work on the assignment without an internet connection.

Thank you

It was fixed sometime ago but your computer could has cashed it--press "Reload" in your web browser.

There was an error in both variants: it claimed to be "Home Assignment 1" in the title--now it fixed.

1748

Home Assignment 2 / Re: Problem 2

« on: September 27, 2012, 11:11:15 AM »

It looks like I should use the result from part 3 for part 4. However if so, how should I use the initial values?

Not really. You use parts (a),(b)

1749

Home Assignment 2 / Re: problem 1 typo?

« on: September 27, 2012, 11:10:32 AM »

Hi - In the pdf of home assignment 2 in problem 1 the inequalities throughout are different than the other version of the assignment (i.e. pdf version has 'greater than or equal to' and the other version in only '>').  Which one is correct?

Really does not matter, but I changed pdf to coincide

1750

Home Assignment 1 / Re: Problem 4

« on: September 26, 2012, 09:40:44 PM »

Do you mean:
$$-u_\theta=r^2 \\
\Rightarrow u=-r^2\theta
$$ and since it's not periodic we are done?

Basically yes, except certain things should be clarified: since our trajectories are closed each is parametrized by some parameter ($\theta$) running from $0$ to $T$ (in our case $T=2\pi$ but it may depend on trajectory) and equation looks like $\partial_\theta u= g(\theta,r)$. So we are looking for periodic solution
$u(\theta,r)=\int g(\theta,r)\,d\theta$.

Note that primitive of periodic function $g(\theta)$  is periodic if and only if average of $g$ over period is $0$:
$\int_0^T g(\theta)\,d\theta=0$. Otherwise this primitive is the sum of a periodic function and a linear function.

Finally, in (a) $g(\theta)=r^2 \sin(\theta)\cos(\theta)$ and integral over period is $0$; in (b) $g(\theta)=r^2$ and  integral over period is not $0$.

So, the source of trouble is: periodic trajectories.

1751

Home Assignment 1 / Re: Problem 4

« on: September 26, 2012, 04:34:04 PM »

Here we are talking about first order equation. There is no wave equation. Stick to the problem! How our original equation
\begin{equation}
yu_x-xu_y=f(x,y)
\label{eq-1}
\end{equation}
looks in the polar coordinates?

Hint: the l.h.e. is $-\partial_\theta u$. Prove it using chain rule  $u_\theta = u_x x_\theta + u_y y_\theta$ and calculate $x_\theta$, $y_\theta$.
Note that $\theta$ is defined modulo $2\pi \mathbb{Z}$ and all functions must be $2\pi$-periodic with respect to $\theta$ (assuming that we consider domains where one can travel around origin)

1752

Misc Math / Re: Heat Wave Equation Solution Integral

« on: September 26, 2012, 12:55:13 PM »

Sorry I meant the Heat Equation. What does the integral of the solution to that equation represent?

If we talking about heat equation then it is a total heat energy in $(-\infty,x)$. But for us it mainly a trick to get a a formula for a solution.

1753

Home Assignment 2 / Re: Problem 3

« on: September 26, 2012, 05:21:59 AM »

Part (a) and (b), part (c) and (d) the same questions? Thanks

There was misprint in (c) fixed almost immediately. Compare fourth lines in (a) vs (b) , (c) vs (d)

1754

Home Assignment 1 / Re: Problem 4

« on: September 26, 2012, 05:19:38 AM »

Peishan, the question now boils down to: why solution with r.h.e. $x^2+y^2$ does not exist and one really needs to use polar coordinates (or to use equivalent geometric motivation)

1755

Misc Math / Re: Heat Wave Equation Solution Integral

« on: September 26, 2012, 02:28:48 AM »

After we determined the solution to the heat wave equation, we took the integral from negative infinity to some x. The solution was in the form of a Normal Distribution function where in statistics the integral indicates the probability that we randomly choose a value within that interval. I was wondering what are the physical implications of this same integral in the wave equation? Does the area under the solution indicate the energy in the system up to that point or does it have some other meaning?

Heat wave equation? WTH are you talking about? There is a wave equation (and its ilk) and the heat equation  (and its ilk) describing either very different processes or the same process under very different assumption. The properties of these equations are really different.

PS Sure more complex phenomena can be described by systems containing both equations but for the purpose of this class ...