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I was wondering how Young's Modulus effects the resonant harmonics of a vibrating (string instrument) string. I know that the string's fundamental frequency is $$\frac{1}2 \times \text{Length} \times \frac{\text{Tension}}{\text{linear density}^{1/2}}$$ that Young's Modulus for a material is - $$\frac{\text{Force}\times \text{original length}}{\text{original cross section} \times \text{change in length}}$$ and that resonant harmonics of a string are even multiples of the string's fundamental frequency. Does the fundamental frequency of the string material itself (which I can calculate by figuring out the speed of sound in whatever material the string is made from and how thick the string is) effect the frequencies it vibrates at under tension?

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Your equation for the fundamental seems off as the units don't work out –  Cactus BAMF Jan 10 '13 at 2:10
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This awesome lecture from MIT goes over a wave equation with "stiffness" included : youtube.com/watch?v=YMI5Py81x8E –  daaxix Jan 10 '13 at 2:26

3 Answers 3

The classical string equation that you are referring to, is formulated by making a number of assumptions, which include that the vibration of the string does not affect its tension. This makes Young's modulus irrelevant for results calculated from the idealized equation.

In the real world, materials with low moduli of elasticity will follow the ideal equation more closely, since the tensions will change less during vibration. For materials with a higher modulus of elasticity, I would expect that:

  1. vibration frequency will not be independent of the amplitude, and
  2. when comparing two materials with the same rest tension, stiffer materials will vibrate at a slightly higher frequency, since the restoring force will be incrementally larger.
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There is an equation, for strings, with both tension and stiffness included, and dispersion relations for it, I don't have time to look for it right now, can you please include it in your answer? –  daaxix Jan 10 '13 at 22:10

I was just researching this kind of questions, since the derivation found in most textbooks, in terms of tension, seems a little unrelated to material properties. Three things:

  1. As pointed out, the tension T needs to be inside the square root
  2. The velocity of sound in the string material is unrelated to the (phase) velocity of the wave. As the formula shows, the latter is related to tension, whereas the former is not (or not so greatly, at least). Also, the string vibrates in a transverse fashion, while sound is a longitudinal wave.
  3. The tension is related to Young's modulus, but if you plug your equation into the wave velocity you end up having a formula related to the strain (elongation) of the string. That's also interesting for guitar tuning, I guess, since it would tell you e.g. how much you have to wind a string on the peg given a certain modulus. In fact, you can arrive at expressions such as $$ f= \frac1{2L} \sqrt{\frac{E (L-L_0)}{\rho L_0}}, $$ for the fundamental frequency of a string of mass density $\rho$, streched from a relaxed length $L_0$ to $L$, which can be handy for a given string.

Update

Actually, and even the sound speed is a different phenomenon, I think a connection can be made. In Alonso & Finn Physics, section 18.5, they derive the sound speed as $$c=\sqrt{\frac E\rho}$$.

In a string, $$c=\sqrt{\frac T\mu} = \sqrt{\frac {E L (L-L_0)}{\rho L_0^2}} , $$ where I have assumed $\mu$ is the linear mass density of the tensioned string, but $\rho$, the volume mass density does not change under tension. For low elongations, $$v\approx c \sqrt{\frac {L-L_0}{L_0}},$$ inside the square root sign you find the relative elongation. One also finds the two velocities would be equal, $v=c$ when $L$ is precisely $\gamma L_0$, $\gamma$ being...

...

the golden ratio, $(1+\sqrt{5})/2\approx 1.62$ (!?!?!)

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The velocity with which the wave propagates should read

$$v = \sqrt{ \frac T \mu}$$

where $ \mu =$ (Linear mass density $\frac {kg}{m}$) and T is the tension in the string ($N = \frac {kg * m}{s^2})$. This formula only works (well) for strings under uniform tension, or strings where the tension is roughly the same anywhere on the it.

The velocity for a wave is also given by:

$$\lambda f = v = \sqrt{ \frac T \mu}$$

where $\lambda =$ Wavelength (meters) and $f =$ frequency of the wave ($\frac1 s$)

$\therefore f = \frac{1}{\lambda}\sqrt{\frac T \mu}$ Is the fundamental frequency (the Lowest frequency the string can ever oscillate at).

And if the string is fixed at the ends, (as with all stringed instruments) $\lambda = 2L$

This makes sense because if you have ever played a stringed instrument, you know if you tighten the strings and put them under more tension, the give off a higher pitch (frequency) and that if you finger the fretboard of the guitar/viola/cello etc. , you are essentially changing the wavelength to accomplish different sounds.

Other frequencies that the string will be able to achieve are all integer multiples of the fundamental, ie the n-th fundamental is $f_n = nf = \frac{n}{\lambda}\sqrt{\frac T \mu}$

This Quantization of the frequencies that the string is allowed to oscillate at is a consequence of it being fixed at its boundaries. For more information visit

http://en.wikipedia.org/wiki/Vibrating_string

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the OP specifically asked for Young's Modulus (Stiffness) to be included in either the dispersion relation or relevant harmonics derived from it...you didn't answer this part, can you please update with it? –  daaxix Jan 10 '13 at 4:56

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