#### kelsiu

By considering the superposition of two waves propagating through a string, one representing the original or incident wave and the other representing the wave reflected at the fixed end, if both ends of the string is fixed then the waves can reflected and travel back and forth. Standing wave can be formed if the length of the string is an integer numbers of half wavelength.

I just wonder what will we get if the length of the string is NOT an integer numbers of half wavelength and both ends are fixed?

#### studiot

What makes you think you can get such a wave to propagate along a stretched string?

#### kelsiu

Swing the string with some particular frequency so that the total length of the string is not a integer number of the half-wavelength of the string. When a certain number of waves is transmitted and reflected back and forth we let both ends fixed.

#### Steve

The string will go on vibrating but will be in an irregular manner (will not be uniform like a standing wave) .If a standing wave is prodused on the string each and every point on string will have its on amplitude (ranging from zero to amplitude of the wave that you first created . The end points always have zero amplitude). If it is not a stationary wave ,as you said in your question, the points on string will not have any fixed amplitude but will change their amplitude time to time . So the fiqure you see will be so irregular. I think in most of cases the string will get broken,in practice, since aplitude will not be zero at ends

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#### studiot

Swing the string with some particular frequency so that the total length of the string is not a integer number of the half-wavelength of the string. When a certain number of waves is transmitted and reflected back and forth we let both ends fixed.
You are beginning to think about it.

If the string is stretched between two fixed points the only way to excite it is to draw it aside in some fashion and then let go ie twang it.
Clearly you cannot waggle the ends since they are fixed.

The only way the string can respond to this is by resonating at its natural frequency or a harmonic, which produces the characteristic standing waves.

If, instead of a single pluck, you physically try to drive the string at some other frequency some of the time the string will be going the same way as your drive and some of the time it will be fighting it, with a net result of no periodic oscillation, just varying movement.

If you only connect one end of the string to a fixed point, the only place you can excite it is at the other end, since you also have to tension it.

Thus your exciter has to be some sort of waving arm since it is not fixed.

You cannot therefore apply a single impulse without reverting to the two fixed point situation.

So you will be continually driving the string.

If your driving action has the same frequency as the string's resonance then again you can excite standing waves, as before.

If not, you will get a time varying response as the ongoing and reflected waves don't match and again you will get a time varying, non periodic response.

#### kelsiu

I see. Thanks both.

#### studiot

The end points always have zero amplitude
Nearly but not quite.

It is posiible to generate a standing wave in a heavy string hanging vertically down under its own weight.
In this case the free end will be an antinode.
This is the string version of the organ pipe with one end open.

Otherwise the practical difficulties are as I indicated.

#### Steve

I was unaware of that one . I was only thinking about a string with both ends fixed and alligned horizondally