Wave Velocity

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Traveling-Wave Solution

String Slope from Velocity Waves

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The complex plane may be defined as the graph of all complex numbers extit{z} = extit{x} + extit{j y} formed by using the real part extit{x} as the horizontal coordinate and the imaginary part extit{y} as the vertical coordinate on the graph. In other words, each complex number extit{z} = extit{x} + extit{j y} corresponds to a point in the complex plane located at Cartesian coordinate ( extit{x,y}). — Click for https://ccrma.stanford.edu/~jos/mdft/Complex_Plane.html

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A boundary value problem consists of a differential equation and the boundary conditions required to solve it. — Click for http://en.wikipedia.org/wiki/Boundary_condition

The wave equation refers to the partial differential equation governing wave phenomena in elastic media such as air and isotropic solids. — Click for https://en.wikipedia.org/wiki/Wave_equation

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An exponential is any function of the form Ae^-t/τ, where t is the independent variable (often denoting time in seconds), A is an arbitrary scale factor, τ is called the time constant, and e is Euler's number (2.718281828459...). Many systems in nature exhibit exponential decay over time, because an exponential decay is produced whenever the rate of decay is proportional to how much is left. — Click for https://ccrma.stanford.edu/~jos/mdft/Exponentials.html

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The velocity of an object is the time derivative of the object's displacement. Velocity is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Struck_String_Velocity.html

In physics, a wave is an oscillation that propagates through a medium (space-time, gas, fluid, or solid) — Click for https://en.wikipedia.org/wiki/Wave

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Physical Audio Signal Processing

Traveling-Wave Solution

String Slope from Velocity Waves

D'Alembert Derived

Wave Velocity

Because $e^{st}$ is an eigenfunction under differentiation (i.e., the exponential function is its own derivative), it is often profitable to replace it with a generalized exponential function, with maximum degrees of freedom in its parametrization, to see if parameters can be found to fulfill the constraints imposed by differential equations.

In the case of the one-dimensional ideal wave equation (Eq.(C.1)), with no boundary conditions, an appropriate choice of eigensolution is

$\displaystyle y(t,x) = e^{st+vx}$

(C.12)

Substituting into the wave equation yields

$\begin{displaymath} \begin{array}{rclcrcl} {\dot y}& \,\mathrel{\mathop=}\,& sy & \quad & y'& \,\mathrel{\mathop=}\,& vy \nonumber \\ {\ddot y}& \,\mathrel{\mathop=}\,& s^2y & \quad & y''& \,\mathrel{\mathop=}\,& v^2y \nonumber \end{array}\end{displaymath}$

Defining the wave velocity (or phase velocity^C.2) as $c \isdeftext {s/v}$ , the wave equation becomes

$\displaystyle Kv^2y$	$\displaystyle =$	$\displaystyle \epsilon s^2y$	(C.13)
$\displaystyle \,\,\Rightarrow\,\,\frac{K}{\epsilon }$	$\displaystyle =$	$\displaystyle \frac{s^2}{v^2} \isdef c^2$
$\displaystyle \,\,\Rightarrow\,\,v$	$\displaystyle =$	$\displaystyle \pm \frac{s}{c}.$