Solving for non moving points of a 1-D wave

  • Thread starter jianxu
  • Start date
  • #26
gabbagabbahey
Homework Helper
Gold Member
5,002
7
If I gave you the function f(x,y)=3xy(x-7)(y+2)(x+2y), could you tell me which values of 'x' made that expression zero?
 
  • #27
94
0
I would say 0, 7, and -2y if we knew what y is

but-2y would mean there's some kind of dependency between the two variables?
 
  • #28
gabbagabbahey
Homework Helper
Gold Member
5,002
7
Yes, the roots x=0 and x=7 are independent of y, while the root x=-2y is not....the same ideas apply to the expression in post #18....

[itex]\alpha=0[/itex] is a root, and is independent of [itex]\beta[/itex] (and hence independent of [itex]t[/itex]), and so values of [itex]x[/itex] where [itex]\alpha=0[/itex] will be stationary points (since the expression will be zero for all [itex]t[/itex] anytime [itex]\alpha=0[/itex]).

There will also be six complex or real roots of the factor:

[itex](2087-16500 \alpha ^2+32928 \alpha ^4-18816 \alpha ^6-16500 \beta ^2+131760 \alpha ^2 \beta ^2[/itex]
[itex]-263424 \alpha ^4 \beta ^2+150528 \alpha ^6 \beta ^2+32928 \beta ^4-263424 \alpha ^2 \beta ^4[/itex]
[itex]+526848 \alpha ^4 \beta ^4-301056 \alpha ^6 \beta ^4-18816 \beta ^6+150528 \alpha ^2 \beta ^6-301056 \alpha ^4 \beta ^6+172032 \alpha ^6 \beta ^6)[/itex]

But they will all depend on [itex]\beta[/itex]; and hence they also depend on [itex]t[/itex]; and so they are not stationary points (stationary points are stationary for all [itex]t[/itex], not just specific values!).

So....the only stationary points are the points where [itex]\alpha=0[/itex]
 
  • #29
94
0
so to solve for the roots then, would just plain old factoring be the best way?
 
  • #30
gabbagabbahey
Homework Helper
Gold Member
5,002
7
You mean solving for the 6 roots of that ugly expression in my last post?
 
  • #31
94
0
yes heh, considering I don't know how to get maple to do anything correctly, what are some alternatives to find the roots? Thanks
 
  • #32
gabbagabbahey
Homework Helper
Gold Member
5,002
7
Well, the expression In post #18 is fully factored....that means that the roots of the ugly factor are going to be difficult to find....BUT!!!!! you don't need to find them because they will all depend on [itex]\beta[/itex], whioch means they will only be roots for certain values of [itex]t[/itex], and as I said in post #28, that means that those roots are not stationary points....do you not understand this?
 
  • #33
94
0
yes but...so am I suppose to say that even though they seem to be stationary, that's not the case since they will be dependent on beta?
 
  • #34
gabbagabbahey
Homework Helper
Gold Member
5,002
7
Why do you say "they seem to be stationary"?
 
  • #35
94
0
That's because we were to graph 10 plots at various t, I decidedly went t=1..10 and from the graphs the roots looked stationary

Also I had the impression that the problem would've been simpler in terms of all the trig stuff...
 
  • #36
gabbagabbahey
Homework Helper
Gold Member
5,002
7
Oh.....try graphing it for t=0.2 and t=0.4...do they still look stationary?
 
  • #37
94
0
It definitely does not. So I did a bad job in choosing the appropriate time interval then? My graphs definitely made me think I'd be getting answers for nonmoving x values...
 
  • #38
94
0
regardless.... Thank you very much for being so patient. It makes sense how you approached this problem now. once again, Thanks!!!!!!!! :P
 

Related Threads on Solving for non moving points of a 1-D wave

Replies
0
Views
1K
Replies
2
Views
3K
Replies
4
Views
1K
  • Last Post
Replies
6
Views
4K
  • Last Post
Replies
4
Views
574
Replies
2
Views
6K
Replies
4
Views
1K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
2
Views
963
Top