◁ The Fortnightly Tambourine

frequency_dependent_damping

Damping Equation

\[\frac{\partial^{2}}{\partial t^{2}} y{\left(x,t \right)} = 2 \beta_{2} \frac{\partial^{3}}{\partial t^{3}} y{\left(x,t \right)} \]

Where:

• \(y{\left(x,t \right)} \left(\text{m}\right)\): 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:

\[- \frac{2 y{\left(x,t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t - {\Delta}t \right)}}{{\Delta}t^{2}} + \frac{y{\left(x,t + {\Delta}t \right)}}{{\Delta}t^{2}} = 2 \beta_{2} \left(- \frac{y{\left(x,t - 2 {\Delta}t \right)}}{2 {\Delta}t^{3}} + \frac{y{\left(x,t - {\Delta}t \right)}}{{\Delta}t^{3}} - \frac{y{\left(x,t + {\Delta}t \right)}}{{\Delta}t^{3}} + \frac{y{\left(x,t + 2 {\Delta}t \right)}}{2 {\Delta}t^{3}}\right) \]

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = 2 \beta_{2} \left(- \frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} + 2}}{2 {\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} - \frac{{y}_{\text{i},\text{n} - 2}}{2 {\Delta}t^{3}}\right) \]

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

\[{y}_{\text{i},\text{n} + 2} = \frac{c^{2} {\Delta}t^{2} \left({y}_{\text{i} + 1,\text{n} + 1} + {y}_{\text{i} - 1,\text{n} + 1} - 2 {y}_{\text{i},\text{n} + 1}\right)}{{\Delta}x^{2}} + 2 {y}_{\text{i},\text{n} + 1} - {y}_{\text{i},\text{n}} \]

\[{y}_{\text{i},\text{n} + 2} = \frac{c^{4} {\Delta}t^{4} \left(- 4 {y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} + 2,\text{n}} - 4 {y}_{\text{i} - 1,\text{n}} + {y}_{\text{i} - 2,\text{n}} + 6 {y}_{\text{i},\text{n}}\right) + c^{2} {\Delta}t^{2} {\Delta}x^{2} \left(- {y}_{\text{i} + 1,\text{n} - 1} + 4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} - 1,\text{n} - 1} + 4 {y}_{\text{i} - 1,\text{n}} + 2 {y}_{\text{i},\text{n} - 1} - 8 {y}_{\text{i},\text{n}}\right) + {\Delta}x^{4} \left(- 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}}\right)}{{\Delta}x^{4}} \]

Results in:

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = 2 \beta_{2} \left(- \frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} - \frac{{y}_{\text{i},\text{n} - 2}}{2 {\Delta}t^{3}} + \frac{c^{4} {\Delta}t^{4} \left(- 4 {y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} + 2,\text{n}} - 4 {y}_{\text{i} - 1,\text{n}} + {y}_{\text{i} - 2,\text{n}} + 6 {y}_{\text{i},\text{n}}\right) + c^{2} {\Delta}t^{2} {\Delta}x^{2} \left(- {y}_{\text{i} + 1,\text{n} - 1} + 4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} - 1,\text{n} - 1} + 4 {y}_{\text{i} - 1,\text{n}} + 2 {y}_{\text{i},\text{n} - 1} - 8 {y}_{\text{i},\text{n}}\right) + {\Delta}x^{4} \left(- 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}}\right)}{2 {\Delta}t^{3} {\Delta}x^{4}}\right) \]

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:

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = 2 \beta_{2} \left(- \frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} - \frac{c^{2} {\Delta}t^{2} {y}_{\text{i} + 1,\text{n} - 1} + c^{2} {\Delta}t^{2} {y}_{\text{i} - 1,\text{n} - 1} - 2 c^{2} {\Delta}t^{2} {y}_{\text{i},\text{n} - 1} + {\Delta}x^{2} \left(2 {y}_{\text{i},\text{n} - 1} - {y}_{\text{i},\text{n}}\right)}{2 {\Delta}t^{3} {\Delta}x^{2}} + \frac{c^{4} {\Delta}t^{4} \left(- 4 {y}_{\text{i} + 1,\text{n}} + {y}_{\text{i} + 2,\text{n}} - 4 {y}_{\text{i} - 1,\text{n}} + {y}_{\text{i} - 2,\text{n}} + 6 {y}_{\text{i},\text{n}}\right) + c^{2} {\Delta}t^{2} {\Delta}x^{2} \left(- {y}_{\text{i} + 1,\text{n} - 1} + 4 {y}_{\text{i} + 1,\text{n}} - {y}_{\text{i} - 1,\text{n} - 1} + 4 {y}_{\text{i} - 1,\text{n}} + 2 {y}_{\text{i},\text{n} - 1} - 8 {y}_{\text{i},\text{n}}\right) + {\Delta}x^{4} \left(- 2 {y}_{\text{i},\text{n} - 1} + 3 {y}_{\text{i},\text{n}}\right)}{2 {\Delta}t^{3} {\Delta}x^{4}}\right) \]

\[\frac{{y}_{\text{i},\text{n} + 1}}{{\Delta}t^{2}} + \frac{{y}_{\text{i},\text{n} - 1}}{{\Delta}t^{2}} - \frac{2 {y}_{\text{i},\text{n}}}{{\Delta}t^{2}} = - \frac{4 \beta_{2} c^{4} {\Delta}t {y}_{\text{i} + 1,\text{n}}}{{\Delta}x^{4}} + \frac{\beta_{2} c^{4} {\Delta}t {y}_{\text{i} + 2,\text{n}}}{{\Delta}x^{4}} - \frac{4 \beta_{2} c^{4} {\Delta}t {y}_{\text{i} - 1,\text{n}}}{{\Delta}x^{4}} + \frac{\beta_{2} c^{4} {\Delta}t {y}_{\text{i} - 2,\text{n}}}{{\Delta}x^{4}} + \frac{6 \beta_{2} c^{4} {\Delta}t {y}_{\text{i},\text{n}}}{{\Delta}x^{4}} - \frac{2 \beta_{2} c^{2} {y}_{\text{i} + 1,\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c^{2} {y}_{\text{i} + 1,\text{n}}}{{\Delta}t {\Delta}x^{2}} - \frac{2 \beta_{2} c^{2} {y}_{\text{i} - 1,\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c^{2} {y}_{\text{i} - 1,\text{n}}}{{\Delta}t {\Delta}x^{2}} + \frac{4 \beta_{2} c^{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}t {\Delta}x^{2}} - \frac{8 \beta_{2} c^{2} {y}_{\text{i},\text{n}}}{{\Delta}t {\Delta}x^{2}} - \frac{2 \beta_{2} {y}_{\text{i},\text{n} + 1}}{{\Delta}t^{3}} - \frac{2 \beta_{2} {y}_{\text{i},\text{n} - 1}}{{\Delta}t^{3}} + \frac{4 \beta_{2} {y}_{\text{i},\text{n}}}{{\Delta}t^{3}} \]

Solving for the next time step yields the update scheme:

\[{y}_{\text{i},\text{n} + 1} = \frac{- 4 \beta_{2} c^{4} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}} + \beta_{2} c^{4} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}} - 4 \beta_{2} c^{4} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}} + \beta_{2} c^{4} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}} + 6 \beta_{2} c^{4} {\Delta}t^{4} {y}_{\text{i},\text{n}} - 2 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n} - 1} + 4 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n}} - 2 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n} - 1} + 4 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n}} + 4 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n} - 1} - 8 \beta_{2} c^{2} {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n}} - 2 \beta_{2} {\Delta}x^{4} {y}_{\text{i},\text{n} - 1} + 4 \beta_{2} {\Delta}x^{4} {y}_{\text{i},\text{n}} - {\Delta}t {\Delta}x^{4} {y}_{\text{i},\text{n} - 1} + 2 {\Delta}t {\Delta}x^{4} {y}_{\text{i},\text{n}}}{{\Delta}x^{4} \left(2 \beta_{2} + {\Delta}t\right)} \]

\[{y}_{\text{i},\text{n} + 1} = \frac{2 \beta_{2} {\Delta}x^{4} \left(- {y}_{\text{i},\text{n} - 1} + 2 {y}_{\text{i},\text{n}}\right) - \frac{2 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n} - 1}}{\mu} + \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} + 1,\text{n}}}{\mu} - \frac{2 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n} - 1}}{\mu} + \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i} - 1,\text{n}}}{\mu} + \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n} - 1}}{\mu} - \frac{8 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} {y}_{\text{i},\text{n}}}{\mu} - \frac{4 \beta_{2} \text{T}_0^{2} {\Delta}t^{4} {y}_{\text{i} + 1,\text{n}}}{\mu^{2}} + \frac{\beta_{2} \text{T}_0^{2} {\Delta}t^{4} {y}_{\text{i} + 2,\text{n}}}{\mu^{2}} - \frac{4 \beta_{2} \text{T}_0^{2} {\Delta}t^{4} {y}_{\text{i} - 1,\text{n}}}{\mu^{2}} + \frac{\beta_{2} \text{T}_0^{2} {\Delta}t^{4} {y}_{\text{i} - 2,\text{n}}}{\mu^{2}} + \frac{6 \beta_{2} \text{T}_0^{2} {\Delta}t^{4} {y}_{\text{i},\text{n}}}{\mu^{2}} + {\Delta}t {\Delta}x^{4} \left(- {y}_{\text{i},\text{n} - 1} + 2 {y}_{\text{i},\text{n}}\right)}{{\Delta}x^{4} \left(2 \beta_{2} + {\Delta}t\right)} \]

Stability Analysis:

Substituting the Ansatz:

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

\[\text{G}^{\text{n} + 1} e^{i \text{i} k {\Delta}x} = \frac{2 \beta_{2} {\Delta}x^{4} \left(2 \text{G}^{\text{n}} e^{i \text{i} k {\Delta}x} - \text{G}^{\text{n} - 1} e^{i \text{i} k {\Delta}x}\right) - \frac{8 \beta_{2} \text{G}^{\text{n}} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i \text{i} k {\Delta}x}}{\mu} + \frac{4 \beta_{2} \text{G}^{\text{n}} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x \left(\text{i} - 1\right)}}{\mu} + \frac{4 \beta_{2} \text{G}^{\text{n}} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x \left(\text{i} + 1\right)}}{\mu} + \frac{4 \beta_{2} \text{G}^{\text{n} - 1} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i \text{i} k {\Delta}x}}{\mu} - \frac{2 \beta_{2} \text{G}^{\text{n} - 1} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x \left(\text{i} - 1\right)}}{\mu} - \frac{2 \beta_{2} \text{G}^{\text{n} - 1} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x \left(\text{i} + 1\right)}}{\mu} + \frac{6 \beta_{2} \text{G}^{\text{n}} \text{T}_0^{2} {\Delta}t^{4} e^{i \text{i} k {\Delta}x}}{\mu^{2}} + \frac{\beta_{2} \text{G}^{\text{n}} \text{T}_0^{2} {\Delta}t^{4} e^{i k {\Delta}x \left(\text{i} - 2\right)}}{\mu^{2}} - \frac{4 \beta_{2} \text{G}^{\text{n}} \text{T}_0^{2} {\Delta}t^{4} e^{i k {\Delta}x \left(\text{i} - 1\right)}}{\mu^{2}} - \frac{4 \beta_{2} \text{G}^{\text{n}} \text{T}_0^{2} {\Delta}t^{4} e^{i k {\Delta}x \left(\text{i} + 1\right)}}{\mu^{2}} + \frac{\beta_{2} \text{G}^{\text{n}} \text{T}_0^{2} {\Delta}t^{4} e^{i k {\Delta}x \left(\text{i} + 2\right)}}{\mu^{2}} + {\Delta}t {\Delta}x^{4} \left(2 \text{G}^{\text{n}} e^{i \text{i} k {\Delta}x} - \text{G}^{\text{n} - 1} e^{i \text{i} k {\Delta}x}\right)}{{\Delta}x^{4} \left(2 \beta_{2} + {\Delta}t\right)} \]

\[2 \beta_{2} \text{G}^{2} {\Delta}x^{4} + \text{G}^{2} {\Delta}t {\Delta}x^{4} = 4 \beta_{2} \text{G} {\Delta}x^{4} - 2 \beta_{2} {\Delta}x^{4} + \frac{4 \beta_{2} \text{G} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x}}{\mu} - \frac{8 \beta_{2} \text{G} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2}}{\mu} + \frac{4 \beta_{2} \text{G} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{- i k {\Delta}x}}{\mu} - \frac{2 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{i k {\Delta}x}}{\mu} + \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2}}{\mu} - \frac{2 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} e^{- i k {\Delta}x}}{\mu} + \frac{\beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} e^{2 i k {\Delta}x}}{\mu^{2}} - \frac{4 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} e^{i k {\Delta}x}}{\mu^{2}} + \frac{6 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4}}{\mu^{2}} - \frac{4 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} e^{- i k {\Delta}x}}{\mu^{2}} + \frac{\beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} e^{- 2 i k {\Delta}x}}{\mu^{2}} + 2 \text{G} {\Delta}t {\Delta}x^{4} - {\Delta}t {\Delta}x^{4} \]

\[2 \beta_{2} \text{G}^{2} {\Delta}x^{4} + \text{G}^{2} {\Delta}t {\Delta}x^{4} = 4 \beta_{2} \text{G} {\Delta}x^{4} - 2 \beta_{2} {\Delta}x^{4} + \frac{8 \beta_{2} \text{G} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)}}{\mu} - \frac{8 \beta_{2} \text{G} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2}}{\mu} - \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2} \cos{\left(k {\Delta}x \right)}}{\mu} + \frac{4 \beta_{2} \text{T}_0 {\Delta}t^{2} {\Delta}x^{2}}{\mu} + \frac{4 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} \cos^{2}{\left(k {\Delta}x \right)}}{\mu^{2}} - \frac{8 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4} \cos{\left(k {\Delta}x \right)}}{\mu^{2}} + \frac{4 \beta_{2} \text{G} \text{T}_0^{2} {\Delta}t^{4}}{\mu^{2}} + 2 \text{G} {\Delta}t {\Delta}x^{4} - {\Delta}t {\Delta}x^{4} \]

\[\cos{\left(k {\Delta}x \right)} = \left[ - \frac{\mu {\Delta}x^{2}}{\text{T}_0 {\Delta}t^{2}} + \frac{\mu {\Delta}x^{2}}{2 \text{G} \text{T}_0 {\Delta}t^{2}} + 1 - \frac{\mu {\Delta}x^{2} \sqrt{2 \beta_{2} \text{G}^{3} - 2 \beta_{2} \text{G} + \beta_{2} + \text{G}^{3} {\Delta}t - 2 \text{G}^{2} {\Delta}t + \text{G} {\Delta}t}}{2 \sqrt{\beta_{2}} \text{G} \text{T}_0 {\Delta}t^{2}}, \ - \frac{\mu {\Delta}x^{2}}{\text{T}_0 {\Delta}t^{2}} + \frac{\mu {\Delta}x^{2}}{2 \text{G} \text{T}_0 {\Delta}t^{2}} + 1 + \frac{\mu {\Delta}x^{2} \sqrt{2 \beta_{2} \text{G}^{3} - 2 \beta_{2} \text{G} + \beta_{2} + \text{G}^{3} {\Delta}t - 2 \text{G}^{2} {\Delta}t + \text{G} {\Delta}t}}{2 \sqrt{\beta_{2}} \text{G} \text{T}_0 {\Delta}t^{2}}\right] \]

\[-1 \leq - \frac{\mu {\Delta}x^{2}}{\text{T}_0 {\Delta}t^{2}} + \frac{\mu {\Delta}x^{2}}{2 \text{G} \text{T}_0 {\Delta}t^{2}} + 1 - \frac{\mu {\Delta}x^{2} \sqrt{2 \beta_{2} \text{G}^{3} - 2 \beta_{2} \text{G} + \beta_{2} + \text{G}^{3} {\Delta}t - 2 \text{G}^{2} {\Delta}t + \text{G} {\Delta}t}}{2 \sqrt{\beta_{2}} \text{G} \text{T}_0 {\Delta}t^{2}} \]

\(\text{G}\) = \(-1\):

\[-1 \leq - \frac{3 \mu {\Delta}x^{2}}{2 \text{T}_0 {\Delta}t^{2}} + 1 + \frac{\mu {\Delta}x^{2} \sqrt{\beta_{2} - 4 {\Delta}t}}{2 \sqrt{\beta_{2}} \text{T}_0 {\Delta}t^{2}} \]

\(\text{G}\) = \(-1\):

\[- \frac{3 \mu {\Delta}x^{2}}{2 \text{T}_0 {\Delta}t^{2}} + 1 + \frac{\mu {\Delta}x^{2} \sqrt{\beta_{2} - 4 {\Delta}t}}{2 \sqrt{\beta_{2}} \text{T}_0 {\Delta}t^{2}} \leq 1 \]

\(\text{G}\) = \(-1\):

\[-1 \leq - \frac{3 \mu {\Delta}x^{2}}{2 \text{T}_0 {\Delta}t^{2}} + 1 - \frac{\mu {\Delta}x^{2} \sqrt{\beta_{2} - 4 {\Delta}t}}{2 \sqrt{\beta_{2}} \text{T}_0 {\Delta}t^{2}} \]

\(\text{G}\) = \(-1\):

\[- \frac{3 \mu {\Delta}x^{2}}{2 \text{T}_0 {\Delta}t^{2}} + 1 - \frac{\mu {\Delta}x^{2} \sqrt{\beta_{2} - 4 {\Delta}t}}{2 \sqrt{\beta_{2}} \text{T}_0 {\Delta}t^{2}} \leq 1 \]

\(\text{G}\) = \(1\):

\[-1 \leq - \frac{\mu {\Delta}x^{2}}{\text{T}_0 {\Delta}t^{2}} + 1 \]

\[{\Delta}x \leq \frac{\sqrt{2} \sqrt{\text{T}_0} {\Delta}t}{\sqrt{\mu}} \]

\(\text{G}\) = \(1\):

\[\text{True} \]

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\(\text{G}\) = \(1\):

\[\text{True} \]

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\(\text{G}\) = \(1\):

\[\text{True} \]

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