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Post by mondo on Sept 20, 2023 21:57:42 GMT
At the end of page 549 there is a suggested exercise which asks us to prove the fact that if the constant $\delta$ in (19) is any real number then $\Omega(z)$ is always calculated for a boundary point of figure [29a]. This sounds like a mystery to me, how to prove it?
Additional questions:
1. Why in the formula for $\Omega(z)$ $\gamma$ evolved into $\delta$?
2. How was the second $\log$ term turned into $\log(\overline{p}z -1)$?
Thank you.
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Post by Admin on Sept 21, 2023 10:02:18 GMT
At the end of page 549 there is a suggested exercise which asks us to prove the fact that if the constant $\delta$ in (19) is any real number then $\Omega(z)$ is always calculated for a boundary point of figure [29a]. This sounds like a mystery to me, how to prove it? Additional questions: 1. Why in the formula for $\Omega(z)$ $\gamma$ evolved into $\delta$? 2. How was the second $\log$ term turned into $\log(\overline{p}z -1)$? Thank you. Mondo You have misquoted the suggested exercise. I suggest you read it again. 1. and 2. are just simple algebra and so is the answer to the exercise. Have another try before I give you a link to my answer. Vasco |
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Post by mondo on Sept 21, 2023 19:29:51 GMT
At the end of page 549 there is a suggested exercise which asks us to prove the fact that if the constant $\delta$ in (19) is any real number then $\Omega(z)$ is always calculated for a boundary point of figure [29a]. This sounds like a mystery to me, how to prove it? Additional questions: 1. Why in the formula for $\Omega(z)$ $\gamma$ evolved into $\delta$? 2. How was the second $\log$ term turned into $\log(\overline{p}z -1)$? Thank you. Mondo You have misquoted the suggested exercise. I suggest you read it again. 1. and 2. are just simple algebra and so is the answer to the exercise. Have another try before I give you a link to my answer. Vasco Vasco, in fact I don't even know why there is $\gamma$ constant in the first place. Previously, no example of method of images used a constant so why is it added here? As to the algebra done in (19), I haven't made any progress, I don't see why the argument of second $\log$ was multiplied by $\overline{p}$? Finally as to the exercise at the bottom of p.549, why do you say I misquoted it? Author claims that making $\delta$ any real number guarantees that $\Omega$ will produce flux and circulation from the boundary of the region right? |
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Post by Admin on Sept 22, 2023 18:09:12 GMT
Mondo
There is a constant $\gamma$ because as you can see on page 503 we integrate to get $\Omega$ and so there is a constant. Another way to see this is to remember that $\Omega$ is only determined up to a constant because both $\Phi$ and $\Psi$ are only determined up to a constant.
Vasco
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Post by Admin on Sept 22, 2023 18:15:36 GMT
Mondo
The reason I say you misquoted it is because Needham says at the bottom of page 549:
"We choose to measure the circulation and flux from a boundary point." What he means by this is that if $a$ is any boundary point then $\Omega(a)=0$.
Vasco
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Post by Admin on Sept 22, 2023 19:41:46 GMT
Mondo
The reason the argument of the second $\log$ is multiplied by $\overline{p}$ is as follows:
$\log(z-1/\overline{p})=\log\bigg(\frac{\overline{p}z-1}{\overline{p}}\bigg)=\log(\overline{p}z-1)-\log(\overline{p})$
Vasco
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Post by mondo on Sept 22, 2023 21:05:19 GMT
Mondo There is a constant $\gamma$ because as you can see on page 503 we integrate to get $\Omega$ and so there is a constant. Another way to see this is to remember that $\Omega$ is only determined up to a constant because both $\Phi$ and $\Psi$ are only determined up to a constant. Vasco Yes, I also thought about it BUT take a peek $\tilde{\Omega}$ on the same page, where is a constant there? In a matter of fact, no $\Omega$ in this whole chapter has a constant added (up to this point). How to justify it? Mondo The reason the argument of the second $\log$ is multiplied by $\overline{p}$ is as follows: $\log(z-1/\overline{p})=\log\bigg(\frac{\overline{p}z-1}{\overline{p}}\bigg)=\log(\overline{p}z-1)-\log(\overline{p}$ Vasco Thank you, I see it now. And I assume $\log(\overline{p})$ was added to $\delta$ to form $\gamma$ constant. It's not clear why we can do it, I can also ask why is a constant and not a part of the real component which is $\Phi$. Mondo The reason I say you misquoted it is because Needham says at the bottom of page 549: "We choose to measure the circulation and flux from a boundary point." What he means by this is that if $a$ is any boundary point then $\Omega(a)=0$. Vasco Ok so I misunderstood his point. The way I understood it is: if we make $\delta$ any real number then $\Omega$ will give us values of flux and circulation from the boundary of $B$. So you say, he rather claims that for boundary values of flux/circulation $\delta$ is a real number. |
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Post by Admin on Sept 22, 2023 21:35:19 GMT
Mondo
I'll answer the second point quickly: Come on, come on, come on, constant doesn't mean real, but constant and complex.
Vasco
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Post by mondo on Sept 22, 2023 21:58:44 GMT
Mondo I'll answer the second point quickly: Come on, come on, come on, constant doesn't mean real, but constant and complex. Vasco Right, I am sorry. But do you agree that it was added to $\gamma$ to form $\delta$? And if so I wonder why we could do this. |
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Post by Admin on Sept 22, 2023 22:00:05 GMT
Mondo
What you say about $\widetilde{\Omega}$ is fully explained in the text especially on page 543.
Vasco
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Post by Admin on Sept 22, 2023 22:05:56 GMT
Mondo I'll answer the second point quickly: Come on, come on, come on, constant doesn't mean real, but constant and complex. Vasco Right, I am sorry. But do you agree that it was added to $\gamma$ to form $\delta$? And if so I wonder why we could do this. Mondo Why can't we add two numbers together, that's basic maths:$1+1=2,(1+2i)+(3+4i)=4+6i$? Vasco |
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Post by mondo on Sept 22, 2023 22:34:31 GMT
Right, I am sorry. But do you agree that it was added to $\gamma$ to form $\delta$? And if so I wonder why we could do this. Mondo Why can't we add two numbers together, that's basic maths:$1+1=2,(1+2i)+(3+4i)=4+6i$? Vasco Ok but now you encoded it into $\delta$ so how can one calculate $\Psi$ and $\Phi$ without knowing the value of $\delta$? In other words, my concern is that we hided $\log(\overline{p})$ in (19) and this will affect both stream and potential function. |
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Post by Admin on Sept 23, 2023 12:42:24 GMT
Mondo
No this is not a problem. If we choose a value for $\Omega$ at a point $z=b$, say, and choose $p$ and then substitute these values into (19) on page 549 we can calculate the value of $\delta$ by writing
$\displaystyle\Omega(b)=i\log\bigg[\frac{b-p}{\overline{p}b-1}\bigg]-\delta$
Then we can calculate $\Omega(z)$ at any point $z$ by using (19) again, since we now know the value of $\delta$.
Vasco
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Post by mondo on Sept 24, 2023 5:39:52 GMT
Ok, so with the assumption that we also have access to a boundary conditions then it makes sense to me.
Thank you, I got it.
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