- #1

jwxie

- 282

- 0

## Homework Statement

This is what my professor wrote on the board.

Suppose the condition is as follows

[tex]y(0) = y(L) = 0[/tex]

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

[tex]\[y(0) = 0, y(L)\neq 0, (\frac{\partial y}{\partial x})_{x=L} = 0 \][/tex]

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:

[tex]\[(\frac{\partial y}{\partial x})_{x=0} = 0 ,(\frac{\partial y}{\partial x})_{x=L} = 0 \][/tex]

**Open-closed**

http://www.izhuk.com/painter/image2.php?id=1286388380-69-86-215-102&md5=50d9cdfa58763dd2dc70314d65a837c4

For the condition

[tex]\[(\frac{\partial y}{\partial x})_{x=0} = 0, y(x = L) = 0\][/tex] 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

Last edited: