Old Guitarist

Old guitarist is a physically modeled virtual guitar plugin.

• Wave equation
• Stability condition
• Multiple Strings
• IR Convolution
• Time-varying tension factor
• Hammer-ons
• Slides

Physical Model

• $\gamma$: damping factor
• $\mu$: linear density
• $E$: Young's modulus
• $h$: Impulse response
• $I$: second moment of area ($\frac{\pi r^4}{4}$ for a cylinder)
• $l$: length
• $T$: tension
• $t$: time
• $x$: position

Wave equation for a guitar string is given as:

$$ \begin{align*} \frac{\partial^2 y}{\partial x^2} - \frac{\mu}{T(t)} \frac{\partial^2 y}{\partial t^2} - \gamma \frac{\partial y}{\partial t} - EI \frac{\partial^4 y}{\partial x^4} = 0 \end{align*} $$

Conditions

$$ \begin{align*} y(x, 0) = \begin{cases} \frac{4x}{l} & \text{for } 0 \leq x < \frac{l}{4} \\ \frac{4 (l - x)}{3l} & \text{for } \frac{l}{4} \leq x \leq l \end{cases} \end{align*} $$

$$ \begin{align*} y(0, t) = y(l, t) = 0 \end{align*} $$

Finite Difference Method

Consider the expression given by

$$ \begin{align*} f' = \lim_{{h \to 0}} \frac{{f(x + h) - f(x)}}{h} \end{align*} $$

    In the context of the finite difference method, $h$ is a finite interval, and the difference quotient is used to approximate the derivative. Instead of taking the limit as $h$ approaches zero, a small but finite value of $h$ is chosen.

The wave equation is discretized to obtain the following form:

$$ \begin{align*} \frac{y_{x+1}^{t} - 2y_{x}^{t} + y_{x-1}^{t}}{\Delta x^2} - \frac{\mu}{T(t)}\frac{y_{x}^{t+1} - 2y_{x}^{t} + y_{x}^{t-1}}{\Delta t^2}- \gamma \frac{y_{x}^{t+1} - y_{x}^{t-1}}{2 \Delta t} - EI \frac{y_{x-2}^{t} -4y_{x-1}^{t} + 6y_{x}^{t} - 4y_{x+1}^{t} + y_{x+2}^{t}}{\Delta x^4} = 0 \end{align*} $$

$$ \begin{align*} Spatial \space resolution = \frac{1}{\Delta x} \end{align*} $$

$$ \begin{align*} Temporal \space resolution = \frac{1}{\Delta t} \end{align*} $$

The stability analysis of finite difference schemes when applied to the numerical solution of partial differential equations is intricately tied to the Courant–Friedrichs–Lewy (CFL) condition, expressed as:

$$ \begin{align*} C = \sqrt{\frac{\mu}{T}} \frac{\Delta x}{\Delta t} \leq C_{\text{max}} \end{align*} $$

$$ Spatial \space resolution \leq Temporal \space resolution \times \sqrt{\frac{\mu}{T}} $$

     $y^{t}$ is sampled for $x \in (0, l)$. The tone will vary with respect to x. Additionally, an impulse response of a guitar body is convoluted with $y$ to introduce body resonance.

$$ (y * h)(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) , d\tau $$

Impulse response: Source