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.

 

[rwisz]

The professor is discussing different boundary conditions for waves, specifically in regards to the position and slope of the wave at the ends of a medium. In the first two examples, the wave is fixed at both ends, meaning that the position and slope of the wave do not change at either end. This is represented by the nodes (points of no displacement) at x=0 and x=L.

In the third example, the wave is open at both ends, meaning that the position and slope of the wave can change at both ends. This is represented by the antinodes (points of maximum displacement) at x=0 and x=L.

In the fourth example, the wave is open at one end and fixed at the other. This means that the position of the wave can change at one end, but the slope remains fixed at the other end. This is represented by the antinode at x=0 and the node at x=L.

To determine the value of lambda (wavelength), you can use the equation lambda = 2L/n, where L is the length of the medium and n is the number of nodes (or antinodes) present in the wave. In the first two examples, n=2, so lambda = 2L/2 = L. In the third example, n=1, so lambda = 2L/1 = 2L. In the fourth example, n=3, so lambda = 2L/3.

The slope form that the professor wrote is simply representing the boundary conditions in a mathematical form. The first condition, y(0)=0, means that the position of the wave at x=0 (the left end) is equal to 0. The second condition, y(L)≠0, means that the position of the wave at x=L (the right end) is not equal to 0. And the third condition, (∂y/∂x) at x=L = 0, means that the slope of the wave at x=L is equal to 0. These conditions help determine the shape and characteristics of the wave.

To sketch the wave based on these conditions, you can use the information about the position and slope at each end to determine the shape of the wave. For example, in the first two examples, the wave is fixed at both ends, so you know that there will be nodes at x=0 and x=L, and the slope will be