Discrete modes of vibration for sonometers

[T7]

Hi,

I have a few questions about Sonometers which I've attempted to think through - though I may be missing something. Perhaps someone can fill in any gaps for me?

If I was to be asked why only 'discrete modes' of vibration can exist for a sonometer, is it enough merely to talk about the fact that this involves standing waves, and for a standing wave you must have "nodes" at either end, and that this is only possible for wavelengths that satisfy 2L/n (where n is an integer), and consequently only possible for frequencies that satisfy vn/2L?

As I understand it, what determines the wavelengths is simply the length of the wire we provide. The frequency, on the other hand, is also a function of the velocity of the wave, and this in turn is dependent on the tension of the wire and its mass per unit length. [I believe waves travel faster on a tighter string (but why?), and slower on a more massive string (which seems fairly obvious).] So finding the 'resonant frequency' is a bit more complicated, since it isn't just dependent on the wavelengths that the wire will permit, is it?

Another question I have is this: if I was to place my finger lightly in the middle of the wire, would I be, in effect, altering the length of the wire, or merely encouraging a standing wave with a node in that position (eg. the second harmonic) to form?

 

[mezarashi]

Only discrete modes can exist because of the standing wave pattern that quickly picks up once a vibration occurs. Any other frequency will be quick to die out. In practice, some frequencies close to these permitted modes will also exist.

Waves travel faster in a string with higher tension because the solution to the string equation says so. In a more layman explanation, you can see that a tighter string the higher the force between adjacent atoms, thus when one moves, it affects the ones beside it with greater influence, leading to what appears to be faster wave motion. As for massive (as in high linear density), again the string equation solution says so, but you can think of it as having more inertia.

If you place your finger in the middle of the wire, you will be altering the nature of the wire, that is the effective length. Different lengths support different modes.

 

[T7]

Only discrete modes can exist because of the standing wave pattern that quickly picks up once a vibration occurs. Any other frequency will be quick to die out. In practice, some frequencies close to these permitted modes will also exist.

Waves travel faster in a string with higher tension because the solution to the string equation says so. In a more layman explanation, you can see that a tighter string the higher the force between adjacent atoms, thus when one moves, it affects the ones beside it with greater influence, leading to what appears to be faster wave motion. As for massive (as in high linear density), again the string equation solution says so, but you can think of it as having more inertia.

If you place your finger in the middle of the wire, you will be altering the nature of the wire, that is the effective length. Different lengths support different modes.

Thanks mezarashi.

 

[lightgrav]

tighter wire has a greater (transverse Force component) restoring force
when displaced from equilibrium (straight) position.

if I was to place my finger lightly in the middle of the wire, would I be, in effect, altering the length of the wire, or merely encouraging a standing wave with a node in that position

Gently touching the middle of the string with a finger will DIScourage
standing waves that have displacement there.

this involves standing waves, and for a standing wave you must have "nodes" at either end

Careful - some sonometers have standing waves with node on one end, antinode on the other. Strings can be set up to have anti-nodes at BOTH ends.

only 'discrete modes' of vibration can exist for a sonometer

Finally, I'd say that even non-discrete vibrations CAN exist, but will not be recognized as "modes" - and are difficult to recognize at later times, since (by definition) they do not look the same on each reflection.