Stationary Waves on Strings and in Pipes?

[ylem]

Hey!

Was just wondering if someone could shed some light on the whole stationary waves thing. I've done about them in my A-Level Physics course and I can't for the life of me figure out what it's about.

I mean, why can you only have a certain number of nodes, hence certain frequencies? And how do you know which frequency you would have in which situation?

If that makes any sense? I'm totally confused, so chances are - it doesn't!

Thanks lots,

Sam

 

[HallsofIvy]

The string (like a guitar string) is fixed at its two ends. The "natural frequencies" must be such that the two fixed endpoints are nodes. That is, all wave lengths must be fractions of (twice) the length of the string. ("twice" because a sin wave of length 2L is 0 at x= L.) The corresponding frequency is, of course, the speed of the wave divided by the wavelength. By pressing down a guitar string on a fret, you reduce the length of the string, reducing the natural wave length and so increasing the frequency.

The same thing is true of a pipe although if the pipe is open at one end (like a recorder) that doubles the natural frequency. (or is it halves? I can never remember!)

 

[hemmul]

I mean, why can you only have a certain number of nodes, hence certain frequencies?

The oscillating object, whether it is string or water surface, must satifsy several natural conditions - i.e. the string must be smooth as well as there must be no first-derivative jump along the string (i.e. the first derivative for the surface/curve is also smooth). Neither the string has a right to go to infinite displacement somewhere! (Just imagine what would happen if it had!) Other conditions are boundary conditions - the string is strongly fixed in the endpoints. Sometimes even the derivative is fixed, or derivative is adjusting itself to the displacement value somehow. It preety resembles the case with differential equations, doesn't it? :) well, mathematically it is really the trick that stands behind oscillations.
So, as the string is fixed on the endpoints and its shape must satisfy some smoothness rules, it turns out that only such free oscillations are possible, where there are certain number of nodes (e.i. rest points) along the string. Luckily the number of nodes is not fixed, and can be equal to any natural number!

And how do you know which frequency you would have in which situation?

When i brush my hand over the guitar strings, or throw a stone in the water surface (or should i say: on the water surface? ) i give some distortion to the string/water.
It is known, that any signal can be decomposed into Fourier series.
Fourier series are handy here, because they are sinusoidal - just like the shape of the oscillating string!
So, when i somehow distort the water surface, actually the waves of ALL imaginable frequencies appear! So why aren't they observed?
O they are, but there is one trick! Not all frequencies can survive: that is different frequencies experience different "resistance", due to which some of them fade out very soon. Others live too long - maybe you have heard of solitons? Which are those? Oh, they are eigenfrequencies - the frequencies at which the free string/free water surface would oscillate! (free - means without me throwing stones in it )
And the free oscillator oscillates on those frequencies, discussed earlier - which have nodes in the endpoints and mybe some in the middle :)

If that makes any sense? I'm totally confused, so chances are - it doesn't!

well it's quite a natural question, and moreover it is an interesting one ;)

 

[ylem]

Thanks so much! I think I'll have to read your replies a few times, but it already makes more sense than the textbooks and notes that I have!

Thanks again, Sam