◁ The Fortnightly Tambourine

damped_stiff_nonlinear_string

Damped Stiff Nonlinear String Equation

\[\frac{\partial^{2}}{\partial t^{2}} y{\left(x,t \right)} = - 2 \beta_{1} \frac{\partial}{\partial t} y{\left(x,t \right)} + 2 \beta_{2} \frac{\partial^{3}}{\partial t^{3}} y{\left(x,t \right)} - \kappa^{2} \frac{\partial^{4}}{\partial x^{4}} y{\left(x,t \right)} + c'^{2} \frac{\partial^{2}}{\partial x^{2}} y{\left(x,t \right)} \]

Where:

\[c' = \sqrt{\frac{1}{\rho \text{A}} \left(\text{T}_0 + \frac{\text{A} \text{E}}{2 \text{L}} \int\limits_{0}^{\text{L}} \left(\frac{\partial}{\partial x} y{\left(x,t \right)}\right)^{2}\, dx\right)} \]

And:

\(y{\left(x,t \right)} \left(\text{m}\right)\): vertical displacement at position \(x\) and time \(t\)
\(c \left(\frac{\text{m}}{\text{s}}\right)=\sqrt{\frac{\text{T}_0}{\mu}}=\sqrt{\frac{\text{T}_0}{\rho \text{A}}}=2 \text{L} f_{0}\): speed of the wave in the string
\(\text{T}_0 \left(\text{N}\right)=\rho \text{A} \left(2 \text{L} f_{0}\right)^{2}\): Tension on the string
\(\mu \left(\frac{\text{kg}}{\text{m}}\right)=\rho \text{A}\): linear density of the string
\(\rho \left(\frac{\text{kg}}{\text{m}^{3}}\right)\): mass density of the string
\(\text{A} \left(\text{m}^{2}\right)\): cross-sectional area of the string
\(\text{E} \left(\frac{\text{N}}{\text{m}^{2}}\right)\): Elastic, Tensile or Young’s modulus of the string / bar material. Setting \(\text{E} = 0\) leads to no stiffness.
\(\text{I} \left(\text{m}^{4}\right)=\frac{\pi r^{4}}{4}\): Second moment of area (area moment of inertia) of the string’s cross-section
\(\kappa \left(\frac{\text{m}^{2}}{\text{s}}\right)=\sqrt{\frac{\text{E} \text{I}}{\mu}}=\sqrt{\frac{\text{E} \text{I}}{\rho \text{A}}}=\frac{\sqrt{\text{E}} r}{2 \sqrt{\rho}}\): bar stiffness
\(\beta_{1} \left(\frac{1}{\text{s}}\right)\): damping coefficient 01
\(\beta_{2} \left(\text{s}\right)\): damping coefficient 02
Units:
- \(\text{N} \left(\text{ newton }\right)\)
- \(\text{kg} \left(\text{ kilogram }\right)\)
- \(\text{m} \left(\text{ meter }\right)\)
- \(\text{s} \left(\text{ second }\right)\)

Difference Equation

Replacing derivatives with finite differences:

\[- \frac{2 y{\left(x,t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t - {\Delta}t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t + {\Delta}t \right)}}{{\Delta}t^{2}} = - 2 \beta_{1} \left(- \frac{y{\left(x,t - {\Delta}t \right)}}{2 {\Delta}t} + \frac{y{\left(x,t + {\Delta}t \right)}}{2 {\Delta}t}\right) + 2 \beta_{2} \left(- \frac{y{\left(x,t - 2 {\Delta}t \right)}}{2 {\Delta}t^{3}} + \frac{y{\left(x,t - {\Delta}t \right)}}{{\Delta}t^{3}} - \frac{y{\left(x,t + {\Delta}t \right)}}{{\Delta}t^{3}} + \frac{y{\left(x,t + 2 {\Delta}t \right)}}{2 {\Delta}t^{3}}\right) - \kappa^{2} \left(\frac{6 y{\left(x,t \right)}}{{\Delta}x^{4}} + \frac{y{\left(x - 2 {\Delta}x,t \right)}}{{\Delta}x^{4}} - \frac{4 y{\left(x - {\Delta}x,t \right)}}{{\Delta}x^{4}} - \frac{4 y{\left(x + {\Delta}x,t \right)}}{{\Delta}x^{4}} + \frac{y{\left(x + 2 {\Delta}x,t \right)}}{{\Delta}x^{4}}\right) + c'^{2} \left(- \frac{2 y{\left(x,t \right)}}{{\Delta}x^{2}} + \frac{y{\left(x - {\Delta}x,t \right)}}{{\Delta}x^{2}} + \frac{y{\left(x + {\Delta}x,t \right)}}{{\Delta}x^{2}}\right) \]

Discretized Difference Equation

Applying discrete indices (\(t \rightarrow \text{n} {\Delta}t\), \(x \rightarrow \text{i} {\Delta}x\)) and approximating the integral using a summation over the grid points:

Where:

\[c' = \frac{\sqrt{\frac{\text{A} \text{E} {\Delta}x \sum_{\text{j}=0}^{\text{M} - 1} \left(\frac{{y}_{\text{j} + 1,\text{n}}}{2 {\Delta}x} - \frac{{y}_{\text{j} - 1,\text{n}}}{2 {\Delta}x}\right)^{2}}{2 \text{L}} + \text{T}_0}}{\sqrt{\rho} \sqrt{\text{A}}} \]

Approximating \({y}_{\text{i},\text{n} + 2}\) with the ideal string:

\[{y}_{\text{i},\text{n} + 2} = \frac{c'^{2} {\Delta}t^{2} \left({y}_{\text{i} + 1,\text{n} + 1} + {y}_{\text{i} - 1,\text{n} + 1} - 2 {y}_{\text{i},\text{n} + 1}\right)}{{\Delta}x^{2}} + 2 {y}_{\text{i},\text{n} + 1} - {y}_{\text{i},\text{n}} \]

\[{y}_{\text{i},\text{n} + 2} = - \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 c'^{4} {\Delta}t^{4} {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} + \frac{2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}x^{2}} - \frac{8 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{2}} - 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}} \]

Results in:

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = - 2 \beta_{1} \left(\frac{{y}_{\text{i},\text{n} + 1}}{2 {\Delta}t} - \frac{{y}_{\text{i},\text{n} - 1}}{2 {\Delta}t}\right) + 2 \beta_{2} \left(\frac{- \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 c'^{4} {\Delta}t^{4} {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} + \frac{2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}x^{2}} - \frac{8 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{2}} - 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}}}{2 {\Delta}t^{3}} - \frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} - \frac{{y}_{\text{i},\text{n} - 2}}{2 {\Delta}t^{3}}\right) - \kappa^{2} \left(- \frac{4 {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 {y}_{\text{i},\text{n}}}{{\Delta}x^{4}}\right) + c'^{2} \left(\frac{{y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} + \frac{{y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}x^{2}}\right) \]

Approximating \({y}_{\text{i},\text{n} - 2}\) with the ideal string:

\[{y}_{\text{i},\text{n}} = \frac{c'^{2} {\Delta}t^{2} \left({y}_{\text{i} + 1,\text{n} - 1} + {y}_{\text{i} - 1,\text{n} - 1} - 2 {y}_{\text{i},\text{n} - 1}\right)}{{\Delta}x^{2}} + 2 {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n} - 2} \]

\[{y}_{\text{i},\text{n} - 2} = \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1} + c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1} - 2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1} + {\Delta}x^{2} \left(2 {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n}}\right)}{{\Delta}x^{2}} \]

Results in:

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = - 2 \beta_{1} \left(\frac{{y}_{\text{i},\text{n} + 1}}{2 {\Delta}t} - \frac{{y}_{\text{i},\text{n} - 1}}{2 {\Delta}t}\right) + 2 \beta_{2} \left(\frac{- \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 c'^{4} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{c'^{4} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 c'^{4} {\Delta}t^{4} {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1}}{{\Delta}x^{2}} + \frac{4 c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} + \frac{2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}x^{2}} - \frac{8 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{2}} - 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}}}{2 {\Delta}t^{3}} - \frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} - \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1} + c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1} - 2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1} + {\Delta}x^{2} \left(2 {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n}}\right)}{2 {\Delta}t^{3} {\Delta}x^{2}}\right) - \kappa^{2} \left(- \frac{4 {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{{y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 {y}_{\text{i},\text{n}}}{{\Delta}x^{4}}\right) + c'^{2} \left(\frac{{y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} + \frac{{y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}x^{2}}\right) \]

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = - \frac{\beta_{1} {y}_{\text{i},\text{n} + 1}}{{\Delta}t} + \frac{\beta_{1} {y}_{\text{i},\text{n} - 1}}{{\Delta}t} - \frac{4 \beta_{2} c'^{4} {\Delta}t {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{\beta_{2} c'^{4} {\Delta}t {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 \beta_{2} c'^{4} {\Delta}t {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{\beta_{2} c'^{4} {\Delta}t {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 \beta_{2} c'^{4} {\Delta}t {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} - \frac{2 \beta_{2} c'^{2} {y}_{\text{i} + 1,\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c'^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}t {\Delta}x^{2}} - \frac{2 \beta_{2} c'^{2} {y}_{\text{i} - 1,\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c'^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c'^{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} - \frac{8 \beta_{2} c'^{2} {y}_{\text{i},\text{n}}}{{\Delta}t {\Delta}x^{2}} - \frac{2 \beta_{2} {y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} - \frac{2 \beta_{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} + \frac{4 \beta_{2} {y}_{\text{i},\text{n}}}{{\Delta}t^{3}} + \frac{4 \kappa^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} - \frac{\kappa^{2} {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} + \frac{4 \kappa^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} - \frac{\kappa^{2} {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} - \frac{6 \kappa^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} + \frac{c'^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{2}} + \frac{c'^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{2}} - \frac{2 c'^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{2}} \]

Solving for the next time step yields the update scheme:

\[{y}_{\text{i},\text{n} + 1} = \frac{\beta_{1} {\Delta}t^{2} {\Delta}x^{4} {y}_{\text{i},\text{n} - 1} - 4 \beta_{2} c'^{4} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}} + \beta_{2} c'^{4} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}} - 4 \beta_{2} c'^{4} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}} + \beta_{2} c'^{4} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}} + 6 \beta_{2} c'^{4} {\Delta}t^{4} {y}_{\text{i},\text{n}} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n} - 1} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n}} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n} - 1} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n}} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n} - 1} - 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n}} - 2 \beta_{2} {\Delta}x^{4} {y}_{\text{i},\text{n} - 1} + 4 \beta_{2} {\Delta}x^{4} {y}_{\text{i},\text{n}} + 4 \kappa^{2} {\Delta}t^{3} {y}_{\text{i} + 1,\text{n}} - \kappa^{2} {\Delta}t^{3} {y}_{\text{i} + 2,\text{n}} + 4 \kappa^{2} {\Delta}t^{3} {y}_{\text{i} - 1,\text{n}} - \kappa^{2} {\Delta}t^{3} {y}_{\text{i} - 2,\text{n}} - 6 \kappa^{2} {\Delta}t^{3} {y}_{\text{i},\text{n}} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n}} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n}} - 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} {y}_{\text{i},\text{n}} - {\Delta}t {\Delta}x^{4} {y}_{\text{i},\text{n} - 1} + 2 {\Delta}t {\Delta}x^{4} {y}_{\text{i},\text{n}}}{{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{y}_{\text{i},\text{n} + 1} = \frac{\beta_{2} c'^{4} {\Delta}t^{4} \left(- 4 {y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} + 2,\text{n}} - 4 {y}_{\text{i} - 1,\text{n}} + {y}_{\text{i} - 2,\text{n}} + 6 {y}_{\text{i},\text{n}}\right) + \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \left(- 2 {y}_{\text{i} + 1,\text{n} - 1} + 4 {y}_{\text{i} + 1,\text{n}} - 2 {y}_{\text{i} - 1,\text{n} - 1} + 4 {y}_{\text{i} - 1,\text{n}} + 4 {y}_{\text{i},\text{n} - 1} - 8 {y}_{\text{i},\text{n}}\right) + \beta_{2} {\Delta}x^{4} \left(- 2 {y}_{\text{i},\text{n} - 1} + 4 {y}_{\text{i},\text{n}}\right) + \kappa^{2} {\Delta}t^{3} \left(4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} + 2,\text{n}} + 4 {y}_{\text{i} - 1,\text{n}} - {y}_{\text{i} - 2,\text{n}} - 6 {y}_{\text{i},\text{n}}\right) + c'^{2} {\Delta}t^{3} {\Delta}x^{2} \left({y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} - 1,\text{n}} - 2 {y}_{\text{i},\text{n}}\right) + {\Delta}t {\Delta}x^{4} \left(\beta_{1} {\Delta}t {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n} - 1} + 2 {y}_{\text{i},\text{n}}\right)}{{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[\left(- 2 {y}_{\text{i},\text{n} - 1} + 4 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{6} + \left(\beta_{1} {\Delta}t {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n} - 1} + 2 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{7} + \left({y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} - 1,\text{n}} - 2 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{2} + \left(- 4 {y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} + 2,\text{n}} - 4 {y}_{\text{i} - 1,\text{n}} + {y}_{\text{i} - 2,\text{n}} + 6 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{4} + \left(4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} + 2,\text{n}} + 4 {y}_{\text{i} - 1,\text{n}} - {y}_{\text{i} - 2,\text{n}} - 6 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{5} + \left(- 2 {y}_{\text{i} + 1,\text{n} - 1} + 4 {y}_{\text{i} + 1,\text{n}} - 2 {y}_{\text{i} - 1,\text{n} - 1} + 4 {y}_{\text{i} - 1,\text{n}} + 4 {y}_{\text{i},\text{n} - 1} - 8 {y}_{\text{i},\text{n}}\right) {\text{Z}}_{1} \]

\[{\text{Z}}_{1} = \frac{\beta_{2} c'^{2} {\Delta}t^{2}}{{\Delta}x^{2} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{\text{Z}}_{2} = \frac{c'^{2} {\Delta}t^{3}}{{\Delta}x^{2} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{\text{Z}}_{3} = \frac{\beta_{1} {\Delta}t^{2}}{\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t} \]

\[{\text{Z}}_{4} = \frac{\beta_{2} c'^{4} {\Delta}t^{4}}{{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{\text{Z}}_{5} = \frac{\kappa^{2} {\Delta}t^{3}}{{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{\text{Z}}_{6} = \frac{\beta_{2}}{\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t} \]

\[{\text{Z}}_{7} = \frac{{\Delta}t}{\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t} \]

When \(\text{E} = \kappa = \beta_1 = \beta_2 = 0\), we have:

\[{y}_{\text{i},\text{n} + 1} = \frac{c'^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n}} + c'^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n}} - 2 c'^{2} {\Delta}t^{2} {y}_{\text{i},\text{n}} - {\Delta}x^{2} {y}_{\text{i},\text{n} - 1} + 2 {\Delta}x^{2} {y}_{\text{i},\text{n}}}{{\Delta}x^{2}} \]

Stability Analysis:

Substituting the Ansatz:

Now we substitute the wave-form ansatz \({y}_{\text{i},\text{n}} = \text{Y} e^{i \left(\text{i} k {\Delta}x + \text{n} w {\Delta}t\right)}\) into the linearized update scheme:

\[\text{Y} e^{i w {\Delta}t} e^{i \text{i} k {\Delta}x} e^{i \text{n} w {\Delta}t} = \frac{\text{Y} \left(\beta_{1} {\Delta}t^{2} {\Delta}x^{4} e^{2 i k {\Delta}x} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{i \left(k {\Delta}x + w {\Delta}t\right)} + 6 \beta_{2} c'^{4} {\Delta}t^{4} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{i \left(3 k {\Delta}x + w {\Delta}t\right)} + \beta_{2} c'^{4} {\Delta}t^{4} e^{i \left(4 k {\Delta}x + w {\Delta}t\right)} + \beta_{2} c'^{4} {\Delta}t^{4} e^{i w {\Delta}t} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i \left(k {\Delta}x + w {\Delta}t\right)} - 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i \left(3 k {\Delta}x + w {\Delta}t\right)} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{3 i k {\Delta}x} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{2 i k {\Delta}x} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x} + 4 \beta_{2} {\Delta}x^{4} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} - 2 \beta_{2} {\Delta}x^{4} e^{2 i k {\Delta}x} + 4 \kappa^{2} {\Delta}t^{3} e^{i \left(k {\Delta}x + w {\Delta}t\right)} - 6 \kappa^{2} {\Delta}t^{3} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} + 4 \kappa^{2} {\Delta}t^{3} e^{i \left(3 k {\Delta}x + w {\Delta}t\right)} - \kappa^{2} {\Delta}t^{3} e^{i \left(4 k {\Delta}x + w {\Delta}t\right)} - \kappa^{2} {\Delta}t^{3} e^{i w {\Delta}t} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{i \left(k {\Delta}x + w {\Delta}t\right)} - 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{i \left(3 k {\Delta}x + w {\Delta}t\right)} + 2 {\Delta}t {\Delta}x^{4} e^{i \left(2 k {\Delta}x + w {\Delta}t\right)} - {\Delta}t {\Delta}x^{4} e^{2 i k {\Delta}x}\right) e^{i \left(\text{i} k {\Delta}x + \text{n} w {\Delta}t - 2 k {\Delta}x - w {\Delta}t\right)}}{{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right)} \]

\[{\Delta}x^{4} \left(\beta_{1} {\Delta}t^{2} + 2 \beta_{2} + {\Delta}t\right) e^{i w {\Delta}t} = \beta_{1} {\Delta}t^{2} {\Delta}x^{4} e^{- i w {\Delta}t} + \beta_{2} c'^{4} {\Delta}t^{4} e^{2 i k {\Delta}x} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{i k {\Delta}x} + 6 \beta_{2} c'^{4} {\Delta}t^{4} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{- i k {\Delta}x} + \beta_{2} c'^{4} {\Delta}t^{4} e^{- 2 i k {\Delta}x} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i \left(k {\Delta}x - w {\Delta}t\right)} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x} - 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i w {\Delta}t} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i k {\Delta}x} - 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i \left(k {\Delta}x + w {\Delta}t\right)} + 4 \beta_{2} {\Delta}x^{4} - 2 \beta_{2} {\Delta}x^{4} e^{- i w {\Delta}t} - \kappa^{2} {\Delta}t^{3} e^{2 i k {\Delta}x} + 4 \kappa^{2} {\Delta}t^{3} e^{i k {\Delta}x} - 6 \kappa^{2} {\Delta}t^{3} + 4 \kappa^{2} {\Delta}t^{3} e^{- i k {\Delta}x} - \kappa^{2} {\Delta}t^{3} e^{- 2 i k {\Delta}x} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{i k {\Delta}x} - 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{- i k {\Delta}x} + 2 {\Delta}t {\Delta}x^{4} - {\Delta}t {\Delta}x^{4} e^{- i w {\Delta}t} \]

\[\beta_{1} {\Delta}t^{2} {\Delta}x^{4} e^{i w {\Delta}t} - \beta_{1} {\Delta}t^{2} {\Delta}x^{4} e^{- i w {\Delta}t} + 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x} e^{- i w {\Delta}t} - 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i w {\Delta}t} + 2 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i k {\Delta}x} e^{- i w {\Delta}t} + 2 \beta_{2} {\Delta}x^{4} e^{i w {\Delta}t} + 2 \beta_{2} {\Delta}x^{4} e^{- i w {\Delta}t} + {\Delta}t {\Delta}x^{4} e^{i w {\Delta}t} + {\Delta}t {\Delta}x^{4} e^{- i w {\Delta}t} = \beta_{2} c'^{4} {\Delta}t^{4} e^{2 i k {\Delta}x} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{i k {\Delta}x} + 6 \beta_{2} c'^{4} {\Delta}t^{4} - 4 \beta_{2} c'^{4} {\Delta}t^{4} e^{- i k {\Delta}x} + \beta_{2} c'^{4} {\Delta}t^{4} e^{- 2 i k {\Delta}x} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x} - 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} e^{- i k {\Delta}x} + 4 \beta_{2} {\Delta}x^{4} - \kappa^{2} {\Delta}t^{3} e^{2 i k {\Delta}x} + 4 \kappa^{2} {\Delta}t^{3} e^{i k {\Delta}x} - 6 \kappa^{2} {\Delta}t^{3} + 4 \kappa^{2} {\Delta}t^{3} e^{- i k {\Delta}x} - \kappa^{2} {\Delta}t^{3} e^{- 2 i k {\Delta}x} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{i k {\Delta}x} - 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} + c'^{2} {\Delta}t^{3} {\Delta}x^{2} e^{- i k {\Delta}x} + 2 {\Delta}t {\Delta}x^{4} \]

\[\beta_{1} {\Delta}t^{2} {\Delta}x^{4} \left(e^{i w {\Delta}t} - e^{- i w {\Delta}t}\right) + \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \left(2 e^{i k {\Delta}x} e^{- i w {\Delta}t} - 4 e^{- i w {\Delta}t} + 2 e^{- i k {\Delta}x} e^{- i w {\Delta}t}\right) + \beta_{2} {\Delta}x^{4} \left(2 e^{i w {\Delta}t} + 2 e^{- i w {\Delta}t}\right) + {\Delta}t {\Delta}x^{4} \left(e^{i w {\Delta}t} + e^{- i w {\Delta}t}\right) = \beta_{2} c'^{4} {\Delta}t^{4} \left(e^{2 i k {\Delta}x} - 4 e^{i k {\Delta}x} + 6 - 4 e^{- i k {\Delta}x} + e^{- 2 i k {\Delta}x}\right) + \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \left(4 e^{i k {\Delta}x} - 8 + 4 e^{- i k {\Delta}x}\right) + 4 \beta_{2} {\Delta}x^{4} + \kappa^{2} {\Delta}t^{3} \left(- e^{2 i k {\Delta}x} + 4 e^{i k {\Delta}x} - 6 + 4 e^{- i k {\Delta}x} - e^{- 2 i k {\Delta}x}\right) + c'^{2} {\Delta}t^{3} {\Delta}x^{2} \left(e^{i k {\Delta}x} - 2 + e^{- i k {\Delta}x}\right) + 2 {\Delta}t {\Delta}x^{4} \]

\[2 i \beta_{1} {\Delta}t^{2} {\Delta}x^{4} \sin{\left(w {\Delta}t \right)} - 4 i \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \sin{\left(w {\Delta}t \right)} \cos{\left(k {\Delta}x \right)} + 4 i \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \sin{\left(w {\Delta}t \right)} + 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} \cos{\left(w {\Delta}t \right)} - 4 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(w {\Delta}t \right)} + 4 \beta_{2} {\Delta}x^{4} \cos{\left(w {\Delta}t \right)} + 2 {\Delta}t {\Delta}x^{4} \cos{\left(w {\Delta}t \right)} = 4 \beta_{2} c'^{4} {\Delta}t^{4} \cos^{2}{\left(k {\Delta}x \right)} - 8 \beta_{2} c'^{4} {\Delta}t^{4} \cos{\left(k {\Delta}x \right)} + 4 \beta_{2} c'^{4} {\Delta}t^{4} + 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} - 8 \beta_{2} c'^{2} {\Delta}t^{2} {\Delta}x^{2} + 4 \beta_{2} {\Delta}x^{4} - 4 \kappa^{2} {\Delta}t^{3} \cos^{2}{\left(k {\Delta}x \right)} + 8 \kappa^{2} {\Delta}t^{3} \cos{\left(k {\Delta}x \right)} - 4 \kappa^{2} {\Delta}t^{3} + 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} - 2 c'^{2} {\Delta}t^{3} {\Delta}x^{2} + 2 {\Delta}t {\Delta}x^{4} \]

\[w = \left[ \frac{- \operatorname{acos}{\left(\frac{- 2 \kappa^{2} {\Delta}t^{2} \cos^{2}{\left(k {\Delta}x \right)} + 4 \kappa^{2} {\Delta}t^{2} \cos{\left(k {\Delta}x \right)} - 2 \kappa^{2} {\Delta}t^{2} + c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} - c'^{2} {\Delta}t^{2} {\Delta}x^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \right)} + 2 \pi}{{\Delta}t}, \ \frac{\operatorname{acos}{\left(\frac{- 2 \kappa^{2} {\Delta}t^{2} \cos^{2}{\left(k {\Delta}x \right)} + 4 \kappa^{2} {\Delta}t^{2} \cos{\left(k {\Delta}x \right)} - 2 \kappa^{2} {\Delta}t^{2} + c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} - c'^{2} {\Delta}t^{2} {\Delta}x^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \right)}}{{\Delta}t}\right] \]

For stability, the argument of \(\arccos\) in the dispersion relation must be within \([-1, 1]\):

\[-1 \le \frac{- 2 \kappa^{2} {\Delta}t^{2} \cos^{2}{\left(k {\Delta}x \right)} + 4 \kappa^{2} {\Delta}t^{2} \cos{\left(k {\Delta}x \right)} - 2 \kappa^{2} {\Delta}t^{2} + c'^{2} {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)} - c'^{2} {\Delta}t^{2} {\Delta}x^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \le 1 \]

The most restrictive condition occurs when \(\cos{\left(k {\Delta}x \right)} = -1\):

\[-1 \leq \frac{- 8 \kappa^{2} {\Delta}t^{2} - 2 c'^{2} {\Delta}t^{2} {\Delta}x^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \]

\[{\Delta}x^{2} = \left[ \frac{{\Delta}t \left(c'^{2} {\Delta}t - \sqrt{16 \kappa^{2} + c'^{4} {\Delta}t^{2}}\right)}{2}, \ \frac{{\Delta}t \left(c'^{2} {\Delta}t + \sqrt{16 \kappa^{2} + c'^{4} {\Delta}t^{2}}\right)}{2}\right] \]

\[\frac{- 8 \kappa^{2} {\Delta}t^{2} - 2 c'^{2} {\Delta}t^{2} {\Delta}x^{2} + {\Delta}x^{4}}{{\Delta}x^{4}} \leq 1 \]

\[{\Delta}x = \left[ \right] \]

\[{\Delta}x^{2} \geq \frac{{\Delta}t \left(c'^{2} {\Delta}t + \sqrt{16 \kappa^{2} + c'^{4} {\Delta}t^{2}}\right)}{2} \]

When \(\kappa\) = 0, the stability inequality is analogous to the ideal string case:

\[{\Delta}x \geq c' {\Delta}t \]

\[- c'^{2} {\Delta}x^{2} + \frac{{\Delta}x^{4}}{{\Delta}t^{2}} \ge 4 \kappa^{2} \]

\[\frac{{\Delta}x^{4}}{{\Delta}t^{2}} - \frac{{\Delta}x^{2} \left(\frac{\text{A} \text{E} {\Delta}x \sum_{\text{j}=0}^{\text{M} - 1} \left(\frac{{y}_{\text{j} + 1,\text{n}}}{2 {\Delta}x} - \frac{{y}_{\text{j} - 1,\text{n}}}{2 {\Delta}x}\right)^{2}}{2 \text{L}} + \text{T}_0\right)}{\rho \text{A}} \ge \frac{\text{E} r^{2}}{\rho} \]

Linearizing the Nonlinear Term:

We replace the term \(\left(\frac{{y}_{\text{j} + 1,\text{n}}}{2 {\Delta}x} - \frac{{y}_{\text{j} - 1,\text{n}}}{2 {\Delta}x}\right)^{2}\) inside the summation with a constant approximation based on a characteristic amplitude ratio \(\alpha\):

\[\left(\frac{{y}_{\text{j} + 1,\text{n}}}{2 {\Delta}x} - \frac{{y}_{\text{j} - 1,\text{n}}}{2 {\Delta}x}\right)^{2} \approx \left(\frac{\alpha \text{L}}{{\Delta}x}\right)^{2} = \frac{\alpha^{2} \text{L}^{2}}{{\Delta}x^{2}} \]

Here, \(\alpha\) represents a typical ratio of maximum amplitude to string length.

Substituting this approximation into the stability inequality:

\[\frac{{\Delta}x^{4}}{{\Delta}t^{2}} - \frac{{\Delta}x^{2} \left(\frac{\alpha^{2} \text{A} \text{E} \text{L}^{2}}{2 {\Delta}x^{2}} + \text{T}_0\right)}{\rho \text{A}} \ge \frac{\text{E} r^{2}}{\rho} \]

\[- \frac{\alpha^{2} \text{E} \text{L}^{2}}{2 \rho} - 4 \text{L}^{2} f_{0}^{2} {\Delta}x^{2} + \frac{{\Delta}x^{4}}{{\Delta}t^{2}} \ge \frac{\text{E} r^{2}}{\rho} \]

\[{\Delta}x = \left[ \frac{\sqrt{2} \sqrt{4 \sqrt{\rho} \text{L}^{2} f_{0}^{2} {\Delta}t^{2} - \sqrt{2} {\Delta}t \sqrt{\alpha^{2} \text{E} \text{L}^{2} + 8 \rho \text{L}^{4} f_{0}^{4} {\Delta}t^{2} + 2 \text{E} r^{2}}}}{2 \sqrt[4]{\rho}}, \ \frac{\sqrt{2} \sqrt{4 \sqrt{\rho} \text{L}^{2} f_{0}^{2} {\Delta}t^{2} + \sqrt{2} {\Delta}t \sqrt{\alpha^{2} \text{E} \text{L}^{2} + 8 \rho \text{L}^{4} f_{0}^{4} {\Delta}t^{2} + 2 \text{E} r^{2}}}}{2 \sqrt[4]{\rho}}\right] \]

\[{\Delta}x \ge 2 \text{L}^{2} f_{0}^{2} {\Delta}t^{2} - \frac{{\Delta}t \sqrt{2 \alpha^{2} \text{E} \text{L}^{2} + 16 \rho \text{L}^{4} f_{0}^{4} {\Delta}t^{2} + 4 \text{E} r^{2}}}{2 \sqrt{\rho}} \]

Fundamental frequency limits:

\[- \frac{\alpha^{2} \text{E} \text{L}^{2}}{2 \rho} - 4 \text{L}^{2} f_{0}^{2} {\Delta}x^{2} + \frac{{\Delta}x^{4}}{{\Delta}t^{2}} \geq \frac{\text{E} r^{2}}{\rho} \]

\[- \frac{\alpha^{2} \text{E} \text{L}^{2}}{2 \rho} - \frac{4 \text{L}^{4} f_{0}^{2}}{\text{M}^{2}} + \frac{\text{L}^{4} f_{s}^{2}}{\text{M}^{4}} \geq \frac{\text{E} r^{2}}{\rho} \]

\[f_{0} \leq \sqrt{- \frac{\alpha^{2} \text{E} \text{M}^{2}}{8 \rho \text{L}^{2}} + \frac{f_{s}^{2}}{4 \text{M}^{2}} - \frac{\text{E} \text{M}^{2} r^{2}}{4 \rho \text{L}^{4}}} \]

When \(f_{s} = 44100\), \(\text{M} = 3\), \(\text{L} = 1\), \(\rho = 7850\), \(\alpha = 0.01\), \(\text{E} = 200000000000.0\), \(r = 0.01\):

\[f_{0} \leq 7349.41502936699 \]