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Rewrite equation in terms of previous variables
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nicoguaro
nicoguaro
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If you consider the bending stiffness of the string you end up with the stiff string equation

$$T\frac{\partial^2 w}{\partial x^2} - EI\frac{\partial^4 w}{\partial x^4} = \mu \frac{\partial^2 w}{\partial t^2}\, ,$$

where $T$ is the tension, $E$ the Young modulus of the material, $I$ the second moment of area for the cross-section, and $\mu$ the linear mass density. This equation can be written in a non-dimensional form as

$$\frac{\partial^2 u}{\partial \xi^2} - \epsilon\frac{\partial^4 u}{\partial \xi^4} - \kappa^2 \frac{\partial^2 w}{\partial \tau^2} = 0\, ,$$

with $u = w/L$, $\tau=\omega t$, $\kappa^2 = L^2 \mu\omega^2/T = L^2 \omega^2/c^2$, $\epsilon = EI/(L^2 T)$, $L$ is the length of the string, $\omega$ a characteristic frequency of the system, and $c$ is the phase speed for the ideal string.

Notice that when $\epsilon$ is small this equation turns into the wave equation.

To find the vibration frequencies this problem is converted into the following eigenproblem

$$\frac{\partial^2 U}{\partial \xi^2} - \epsilon\frac{\partial^4 U}{\partial \xi^4} - \kappa^2 U = 0\, .$$

This problem has an analytical solution but the characteristic equation is not solvable analytically. You can use a perturbation method to get the following first-order approximation

$$\kappa^2 \approx n^2 \pi^2 + \epsilon n^4 \pi^4 \, .$$

For a G string in an electric guitar (196 Hz) the perturbation parameter $\epsilon$ is around $9.725 \times 10^{-6}$. So, in general, this parameter is small and the ideal string approximation is not that bad.

In terms of frequency, this translates to the following

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1 + \frac{n^2 \pi^3 E r^4}{4L^2T}}\, .$$$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1 + \frac{n^2 \pi^2 E I}{L^2T}}\, .$$

References

If you consider the bending stiffness of the string you end up with the stiff string equation

$$T\frac{\partial^2 w}{\partial x^2} - EI\frac{\partial^4 w}{\partial x^4} = \mu \frac{\partial^2 w}{\partial t^2}\, ,$$

where $T$ is the tension, $E$ the Young modulus of the material, $I$ the second moment of area for the cross-section, and $\mu$ the linear mass density. This equation can be written in a non-dimensional form as

$$\frac{\partial^2 u}{\partial \xi^2} - \epsilon\frac{\partial^4 u}{\partial \xi^4} - \kappa^2 \frac{\partial^2 w}{\partial \tau^2} = 0\, ,$$

with $u = w/L$, $\tau=\omega t$, $\kappa^2 = L^2 \mu\omega^2/T = L^2 \omega^2/c^2$, $\epsilon = EI/(L^2 T)$, $L$ is the length of the string, $\omega$ a characteristic frequency of the system, and $c$ is the phase speed for the ideal string.

Notice that when $\epsilon$ is small this equation turns into the wave equation.

To find the vibration frequencies this problem is converted into the following eigenproblem

$$\frac{\partial^2 U}{\partial \xi^2} - \epsilon\frac{\partial^4 U}{\partial \xi^4} - \kappa^2 U = 0\, .$$

This problem has an analytical solution but the characteristic equation is not solvable analytically. You can use a perturbation method to get the following first-order approximation

$$\kappa^2 \approx n^2 \pi^2 + \epsilon n^4 \pi^4 \, .$$

For a G string in an electric guitar (196 Hz) the perturbation parameter $\epsilon$ is around $9.725 \times 10^{-6}$. So, in general, this parameter is small and the ideal string approximation is not that bad.

In terms of frequency, this translates to the following

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1 + \frac{n^2 \pi^3 E r^4}{4L^2T}}\, .$$

References

If you consider the bending stiffness of the string you end up with the stiff string equation

$$T\frac{\partial^2 w}{\partial x^2} - EI\frac{\partial^4 w}{\partial x^4} = \mu \frac{\partial^2 w}{\partial t^2}\, ,$$

where $T$ is the tension, $E$ the Young modulus of the material, $I$ the second moment of area for the cross-section, and $\mu$ the linear mass density. This equation can be written in a non-dimensional form as

$$\frac{\partial^2 u}{\partial \xi^2} - \epsilon\frac{\partial^4 u}{\partial \xi^4} - \kappa^2 \frac{\partial^2 w}{\partial \tau^2} = 0\, ,$$

with $u = w/L$, $\tau=\omega t$, $\kappa^2 = L^2 \mu\omega^2/T = L^2 \omega^2/c^2$, $\epsilon = EI/(L^2 T)$, $L$ is the length of the string, $\omega$ a characteristic frequency of the system, and $c$ is the phase speed for the ideal string.

Notice that when $\epsilon$ is small this equation turns into the wave equation.

To find the vibration frequencies this problem is converted into the following eigenproblem

$$\frac{\partial^2 U}{\partial \xi^2} - \epsilon\frac{\partial^4 U}{\partial \xi^4} - \kappa^2 U = 0\, .$$

This problem has an analytical solution but the characteristic equation is not solvable analytically. You can use a perturbation method to get the following first-order approximation

$$\kappa^2 \approx n^2 \pi^2 + \epsilon n^4 \pi^4 \, .$$

For a G string in an electric guitar (196 Hz) the perturbation parameter $\epsilon$ is around $9.725 \times 10^{-6}$. So, in general, this parameter is small and the ideal string approximation is not that bad.

In terms of frequency, this translates to the following

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1 + \frac{n^2 \pi^2 E I}{L^2T}}\, .$$

References

Source Link
nicoguaro
nicoguaro
  • 1.7k
  • 12
  • 20

If you consider the bending stiffness of the string you end up with the stiff string equation

$$T\frac{\partial^2 w}{\partial x^2} - EI\frac{\partial^4 w}{\partial x^4} = \mu \frac{\partial^2 w}{\partial t^2}\, ,$$

where $T$ is the tension, $E$ the Young modulus of the material, $I$ the second moment of area for the cross-section, and $\mu$ the linear mass density. This equation can be written in a non-dimensional form as

$$\frac{\partial^2 u}{\partial \xi^2} - \epsilon\frac{\partial^4 u}{\partial \xi^4} - \kappa^2 \frac{\partial^2 w}{\partial \tau^2} = 0\, ,$$

with $u = w/L$, $\tau=\omega t$, $\kappa^2 = L^2 \mu\omega^2/T = L^2 \omega^2/c^2$, $\epsilon = EI/(L^2 T)$, $L$ is the length of the string, $\omega$ a characteristic frequency of the system, and $c$ is the phase speed for the ideal string.

Notice that when $\epsilon$ is small this equation turns into the wave equation.

To find the vibration frequencies this problem is converted into the following eigenproblem

$$\frac{\partial^2 U}{\partial \xi^2} - \epsilon\frac{\partial^4 U}{\partial \xi^4} - \kappa^2 U = 0\, .$$

This problem has an analytical solution but the characteristic equation is not solvable analytically. You can use a perturbation method to get the following first-order approximation

$$\kappa^2 \approx n^2 \pi^2 + \epsilon n^4 \pi^4 \, .$$

For a G string in an electric guitar (196 Hz) the perturbation parameter $\epsilon$ is around $9.725 \times 10^{-6}$. So, in general, this parameter is small and the ideal string approximation is not that bad.

In terms of frequency, this translates to the following

$$f_n = \frac{n}{2L}\sqrt{\frac{T}{\mu}}\sqrt{1 + \frac{n^2 \pi^3 E r^4}{4L^2T}}\, .$$

References