Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \color{#00000000}\rho\frac{\partial^2y}{\partial t^2}&= \color{#00000000}T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= \color{#00000000}T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{h \to 0} \frac{y(x + h) - y(x)}{h} \\ &= \color{#00000000}T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= \color{#00000000}T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{h \to 0} \frac{y(x + h) - y(x)}{h} \\ &= \color{#00000000}T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \color{#00000000}\quad; \frac{dy}{dx} = \lim\limits_{h \to 0} \frac{y(x + h) - y(x)}{h} \\ &= \color{#00000000}T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{h \to 0} \frac{y(x + h) - y(x)}{h} \\ &= \color{#00000000}T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= \color{#00000000}T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \color{#00000000}\Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= T_0\frac{d}{dx}\frac{\partial y(x_1)}{\partial x} \Rightarrow \frac{\partial^2 y(x_1)}{\partial x^2}\\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= T_0\frac{\partial^2 y(x_1)}{\partial x^2} \\ \color{#00000000}\frac{\partial^2y}{\partial t^2} &= \color{#00000000}c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= T_0\frac{\partial^2 y(x_1)}{\partial x^2} \\ \frac{\partial^2y}{\partial t^2} &= c^2\frac{\partial^2 y(x_1)}{\partial x^2} \color{#00000000}\quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving the Equation of Motion

\[\begin{aligned} \require{color} \rho \Delta x \frac{\partial^2y}{\partial t^2} &= T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - T_0\frac{\partial y}{\partial x}\bigg\vert_{x_1} \\ \rho\frac{\partial^2y}{\partial t^2}&= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x}\\ &= T_0\frac{\left(\frac{\partial y}{\partial x}\bigg\vert_{x_1 + \Delta x} - \frac{\partial y}{\partial x}\bigg\vert_{x_1}\right)}{\Delta x} \quad; \frac{dy}{dx} = \lim\limits_{\Delta x \to 0} \frac{\frac{\partial y(x_1 + \Delta x)}{\partial x} - \frac{\partial y(x_1)}{\partial x}}{\Delta x} \\ &= T_0\frac{\partial^2 y(x_1)}{\partial x^2} \\ \frac{\partial^2y}{\partial t^2} &= c^2\frac{\partial^2 y(x_1)}{\partial x^2} \quad ; \quad c = \sqrt{\frac{T_0}{\rho}} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ \color{#00000000}y(x)(i\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t} \text{ or } \times \frac{1}{c^2e^{i\omega t}}) \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t} \text{ or } \times \frac{1}{c^2e^{i\omega t}}) \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic / Helmholtz equation)} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ y(x)(i\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y(x)}{\partial x^2}e^{i\omega t} \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t} \text{ or } \times \frac{1}{c^2e^{i\omega t}}) \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t} \text{ or } \times \frac{1}{c^2e^{i\omega t}}) \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ y(x)(i\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y(x)}{\partial x^2}e^{i\omega t} \\ y(x)(-\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \color{#00000000}\quad (\div \ c^2e^{i\omega t} \text{ or } \times \frac{1}{c^2e^{i\omega t}}) \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ y(x)(i\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y(x)}{\partial x^2}e^{i\omega t} \\ y(x)(-\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t}) \\ \color{#00000000}y(x)(-\omega)^2e^{i\omega t} &= \color{#00000000}c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \color{#00000000}\quad (\div \ c^2e^{i\omega t}) \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ y(x)(i\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y(x)}{\partial x^2}e^{i\omega t} \\ y(x)(-\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t}) \\ y(x)\frac{(-\omega)^2}{c^2}\color{#00000000}e^{i\omega t} &= \color{#00000000}c^2\color{#FFFFFF}\frac{\partial^2y}{\partial x^2}\color{#00000000}e^{i\omega t} \color{#00000000}\quad (\div \ c^2e^{i\omega t}) \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)} \\ \end{aligned} \]

Deriving Modes: Periodic Solution for 1D Wave

\[ \begin{aligned} \require{color} \require{cancel} y(x, t) &= y(x)e^{i\omega t} \Rightarrow \frac{\partial^2y}{\partial t^2} = c^2\frac{\partial^2 y}{\partial x^2} \\ y(x)(i\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y(x)}{\partial x^2}e^{i\omega t} \\ y(x)(-\omega)^2e^{i\omega t} &= c^2\frac{\partial^2y}{\partial x^2}e^{i\omega t} \quad (\div \ c^2e^{i\omega t}) \\ y(x)\frac{(-\omega)^2}{c^2}\color{#00000000}e^{i\omega t} &= \color{#00000000}c^2\color{#FFFFFF}\frac{\partial^2y}{\partial x^2}\color{#00000000}e^{i\omega t} \color{#00000000}\quad (\div \ c^2e^{i\omega t}) \\ \color{#00000000}\frac{\partial^2y}{\partial x^2} &= \color{#00000000}y(x)frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)} \\ \end{aligned} \]

Deriving Modes

Solutions for Time Harmonic

\[\frac{\partial^2y}{\partial x^2} = y(x)\frac{(-\omega)^2}{c^2} \quad \text{Equation of Motion (Time Harmonic)}\]

General Solution:

\(y = A e^{ikx} + B e^{-ikx} ;\quad k \triangleq \frac{2\pi}{\lambda}\)

$k $: wavenumber

wavenumber (equal phase locations in radians per meter)

Commutattive Diagram of Harmonic Wave Properties

Waldir, CC BY-SA 4.0, via Wikimedia Commons

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ \color{#00000000}-k^2A e^{ikx} - k^2 B e^{-ikx} &= \color{#00000000}\frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \color{#00000000}\qquad (\div A e^{ikx} + B e^{-ikx}) \\ \color{#00000000}-k^2 &= \color{#00000000} \frac{-\omega^2}{c^2} \\ \color{#00000000}-k^2c^2 &= \color{#00000000} -\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2A e^{ikx} - k^2 B e^{-ikx} &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \color{#00000000}\qquad (\div A e^{ikx} + B e^{-ikx}) \\ \color{#00000000}-k^2 &= \color{#00000000} \frac{-\omega^2}{c^2} \\ \color{#00000000}-k^2c^2 &= \color{#00000000} -\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \color{#00000000}\qquad (\div A e^{ikx} + B e^{-ikx}) \\ \color{#00000000}-k^2 &= \color{#00000000} \frac{-\omega^2}{c^2} \\ \color{#00000000}-k^2c^2 &= \color{#00000000} -\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \qquad (\div A e^{ikx} + B e^{-ikx}) \\ \color{#00000000}-k^2 &= \color{#00000000} \frac{-\omega^2}{c^2} \\ \color{#00000000}-k^2c^2 &= \color{#00000000} -\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \qquad (\div A e^{ikx} + B e^{-ikx}) \\ -k^2 &= \frac{-\omega^2}{c^2} \\ \color{#00000000}-k^2c^2 &= \color{#00000000}-\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \qquad (\div A e^{ikx} + B e^{-ikx}) \\ -k^2 &= \frac{-\omega^2}{c^2} \\ -k^2c^2 &= -\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \qquad (\div A e^{ikx} + B e^{-ikx}) \\ -k^2 &= \frac{-\omega^2}{c^2} \\ \color{#00000000}-\color{white}k^2c^2 &= \color{#00000000}-\color{white}\omega^2 \\ \color{#00000000}\omega^2 &= \color{#00000000} k^2c^2 \\ \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} \frac{\partial^2}{\partial x^2}\left(A e^{ikx} + B e^{-ikx}\right) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \\ -k^2(A e^{ikx} + B e^{-ikx}) &= \frac{-\omega^2}{c^2}\left(A e^{ikx} + B e^{-ikx}\right) \qquad (\div A e^{ikx} + B e^{-ikx}) \\ -k^2 &= \frac{-\omega^2}{c^2} \\ \color{#00000000}-\color{white}k^2c^2 &= \color{#00000000}-\color{white}\omega^2 \\ \omega^2 &= k^2c^2 \\ \end{aligned} \]

Dispersion Relation

$\omega = kc$

$k = \frac{\omega}{c}$

Putting it all together

\[ \begin{aligned} y(x,t) &= y(x)e^{i\omega t} \\ &= Ae^{ik t}e^{i\omega t} + Be^{-ik t}e^{i\omega t} \\ &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ \end{aligned} \]

Deriving Modes

Boundary conditions

String length has been infinite

What if string length is finite?

Assumption at ends of string:

$y(0,t) = 0$ and $y(L,t) = 0 \ ; \quad ∀t$ (rigid condition)

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ \color{#00000000}y(0,t) &= \color{#00000000} Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0 \\ &= \color{#00000000} Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ \color{#00000000}A &= \color{#00000000} -B \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ \color{#00000000}y(x,t) &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} \\ &= \color{#00000000} Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ \color{#00000000}Ae^{i(\omega t)} &= \color{#00000000} -Be^{i(\omega t)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= \color{#00000000}Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ \color{#00000000}Ae^{i(\omega t)} &= \color{#00000000} -Be^{i(\omega t)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ \color{#00000000}y(x,t) &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ \color{#00000000}Ae^{i(\omega t)} &= \color{#00000000} -Be^{i(\omega t)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ Ae^{i(\omega t)} &= -Be^{i(\omega t)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ A\color{#00000000}e^{i(\omega t)} &= -B\color{#00000000}e^{i(\omega t)} \\ \color{#00000000}y(x,t) &= \color{#00000000} Ae^{i(\omega t + kx)} - Ae^{i(\omega t - kx)} \\ &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)}\\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ A\color{#00000000}e^{i(\omega t)} &= -B\color{#00000000}e^{i(\omega t)} \\ y(x,t) &= Ae^{i(\omega t + kx)} -Ae^{i(\omega t - kx)} \\ &= \color{#00000000} Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ A\color{#00000000}e^{i(\omega t)} &= -B\color{#00000000}e^{i(\omega t)} \\ y(x,t) &= Ae^{i(\omega t + kx)} -Ae^{i(\omega t - kx)} \\ &= Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \color{#00000000}\quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

\[ \begin{aligned} \require{color} \require{cancel} y(x,t) &= Ae^{i(\omega t + kx)} + Be^{i(\omega t - kx)} \\ y(0,t) &= Ae^{i(\omega t + k0)} + Be^{i(\omega t - k0)} = 0\\ &= Ae^{i(\omega t)} + Be^{i(\omega t)} = 0 \\ A\color{#00000000}e^{i(\omega t)} &= -B\color{#00000000}e^{i(\omega t)} \\ y(x,t) &= Ae^{i(\omega t + kx)} -Ae^{i(\omega t - kx)} \\ &= Ae^{i\omega t}\left(e^{ikx} - e^{-ikx}\right) \quad \text{Remember:} \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin \theta \\ &= \color{#00000000} -2iAe^{i\omega t}\sin(kx) \end{aligned} \]

The Other Boundary

\[\begin{aligned} \require{color} y(x,t) &= -2iAe^{i\omega t}\sin(kx) \\ y(L,t) &= -2iAe^{i\omega t}\sin(kL) = 0 \\ kL &= n\pi ; \quad k = \frac{\omega}{c} \\ \omega &= \frac{n\pi c}{L} \\ y(x,t) &= -2iAe^{i\omega t}\sin(n \pi x) \end{aligned} \]

The Other Boundary

\[\begin{aligned} \require{color} y(x,t) &= -2iAe^{i\omega t}\sin(kx) \\ y(L,t) &= -2iAe^{i\omega t}\sin(kL) = 0 \\ kL &= n\pi ; \quad k = \frac{\omega}{c} \\ \omega_n &= \frac{n\pi c}{L} \\ \color{red}y(x,t) &= \color{red}-2iAe^{i\omega t}\sin(n \pi x) \end{aligned} \]

More information on this slide deck see:

github.com/mhamilt/1D-Wave-in-5-Minutes

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Finite Difference Plate