The Wave Condition: Fixed Ends and Open Ends

[jwxie]

Homework Statement

This is what my professor wrote on the board.

Suppose the condition is as follows
y(0) = y(L) = 0
link to img http://www.izhuk.com/painter/image2.php?id=1286387629-69-86-215-102&md5=c054d375567134a9faa133b47fe690b2

This is the fundamental harmonic - with nodes appear at x = 0, and x = L

Suppose we have this condition
\[y(0) = 0, y(L)\neq 0, (\frac{\partial y}{\partial x})_{x=L} = 0 \]

This gives the following image
http://www.izhuk.com/painter/image2.php?id=1286388050-69-86-215-102&md5=692154459be86cc180d82606991495b4

He said that the right end is not fixed, hence the slope of the changes in y position is not zero.

With the same condition, but a different drawing,
http://www.izhuk.com/painter/image2.php?id=1286388170-69-86-215-102&md5=02de08db7ca943b3db1a1bc46e82336a
This clearly shows that lambada is = 4L, with node appears only at x = 0. In another note he wrote (for standing waves in air column)

both ends open
http://www.izhuk.com/painter/image2.php?id=1286388281-69-86-215-102&md5=3ae6bc9e3abf91f224af13248936939d
for the condition:
\[(\frac{\partial y}{\partial x})_{x=0} = 0 ,(\frac{\partial y}{\partial x})_{x=L} = 0 \]Open-closed
http://www.izhuk.com/painter/image2.php?id=1286388380-69-86-215-102&md5=50d9cdfa58763dd2dc70314d65a837c4

For the condition
\[(\frac{\partial y}{\partial x})_{x=0} = 0, y(x = L) = 0\] and the lambada is 4LMy question is, how do you read the slope form that he wrote. So if the slope changes is equal to zero, then it mean that end is fixed? Obviously, the first two examples do not agree with the interpretation of the last two.
Moreover, if I were to sketch based on the condition, how do I determine the lambada? I don't see how knowing either end being fixed help me sketch which harmonics.

Thank you.

From other notes he had other examples such as

 

[vela]

My question is, how do you read the slope form that he wrote.

It's hard to understand what you're asking because you're kind of throwing words together. What do you mean by "slope changes," "slope of the changes in y position", and "slope form"? Do you just mean the slope, ∂y/∂x?

So if the slope changes is equal to zero, then it mean that end is fixed?

If the end is fixed, that end can't move, so y=0 at that end. If the end is free, then the partial derivative ∂y/∂x vanishes at that end.

Obviously, the first two examples do not agree with the interpretation of the last two.

I don't see the contradiction between the first two and the last two. Can you elaborate on how they do not agree?

Moreover, if I were to sketch based on the condition, how do I determine the lambada? I don't see how knowing either end being fixed help me sketch which harmonics.

Thank you.

From other notes he had other examples such as

 

[jwxie]

Hi, thank you for the reply.

Let's talk about the partial first.

so we have ∂y/∂x = 0 (let say at x = 0). Then it means at x = 0 (the left end, let say) is free, not fixed?

Second question is:

How do you determine the numbers of nodes? In another word, with that same conditions [∂y/∂x = 0 (let say at x = 0)], I can sketch the fundamental harmonics with normal mode n = 1, and L = 1/2 lambada, or I can have n = 2, with L = lambada. So there is no way to determine the numbers of nodes and no way to sketch a figure unless additional condition is given?

 

[vela]

Yes, that's right. If the medium were fixed at x=0, it can't move, so y(0,t)=0 for all t, but you'd have no condition on ∂y/∂x. With a free end, the partial derivative ∂y/∂x must vanish, but there's no condition on y.