Physlib.ClassicalMechanics.WaveEquation.Basic

The wave equation with propagation speed $c \in \mathbb{R}$ for a vector-valued function $\mathbf{f} : \text{Time} \to \text{Space } d \to \mathbb{R}^d$ is the condition that at time $t$ and position $x$: $$c^2 \Delta \mathbf{f}(t, x) - \frac{\partial^2 \mathbf{f}}{\partial t^2}(t, x) = 0$$ where $\Delta$ is the vector Laplacian operator acting on the spatial coordinates and $\frac{\partial^2}{\partial t^2}$ denotes the second-order partial derivative with respect to time.

Given a profile function $f_0 : \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction $s$, the plane wave is defined as the function mapping time $t$ and position $x \in \mathbb{R}^d$ to the value: $$f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $\mathbf{\hat{s}}$ is the unit vector in the direction of $s$, and $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

For any profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, propagation speed $c \in \mathbb{R}$, and direction $s$ with unit vector $\mathbf{\hat{s}}$, the value of the plane wave at time $t$ and position $x \in \mathbb{R}^d$ is given by: $$\text{planeWave}(f_0, c, s, t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

Suppose $f_0 : \mathbb{R} \to \mathbb{R}^d$ is a differentiable profile function, $c \in \mathbb{R}$ is the propagation speed, and $\mathbf{\hat{s}}$ is a unit direction vector in $d$-dimensional Euclidean space. For any fixed position $x \in \mathbb{R}^d$, the plane wave defined by $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$ is differentiable with respect to time $t$, where $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

Suppose $f_0 : \mathbb{R} \to \mathbb{R}^d$ is a differentiable profile function, $c \in \mathbb{R}$ is the propagation speed, and $\mathbf{\hat{s}}$ is a unit direction vector in $d$-dimensional Euclidean space. For any fixed time $t$, the plane wave defined by $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$ is differentiable with respect to the spatial variable $x \in \mathbb{R}^d$, where $\langle \cdot, \cdot \rangle$ denotes the standard inner product.

Suppose $f_0 : \mathbb{R} \to \mathbb{R}^d$ is a differentiable profile function, $c \in \mathbb{R}$ is the propagation speed, and $\mathbf{\hat{s}}$ is a unit direction vector in $d$-dimensional Euclidean space. Then the plane wave function $f: \text{Time} \times \text{Space } d \to \mathbb{R}^d$ defined by $$f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ is differentiable with respect to $(t, x)$, where $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

Let $d$ be the dimension of the space. Given a differentiable profile function $f_0 : \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction vector $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed position $x \in \mathbb{R}^d$, the time derivative of the plane wave is given by: $$\frac{\partial}{\partial t} f(t, x) = -c f'_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $f'_0$ denotes the derivative of the profile function $f_0$ and $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

Let $d$ be the dimension of the space. Given a twice continuously differentiable profile function $f_0 : \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed position $x \in \mathbb{R}^d$, the second partial derivative of the plane wave with respect to time $t$ is given by: $$\frac{\partial^2}{\partial t^2} f(t, x) = c^2 f''_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $f''_0$ denotes the second derivative of the profile function $f_0$ and $\langle \cdot, \cdot \rangle$ denotes the standard inner product on $\mathbb{R}^d$.

Let $d$ be the dimension of the space. Given a differentiable profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction vector $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any coordinate index $i \in \{0, \dots, d-1\}$, the partial derivative of the plane wave with respect to the $i$-th spatial coordinate $x_i$ is given by: $$\frac{\partial}{\partial x_i} f(t, x) = s_i f'_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $s_i = \langle \mathbf{\hat{s}}, e_i \rangle$ is the $i$-th component of the unit direction vector, and $f'_0$ denotes the derivative of the profile function $f_0$.

Let $d$ be the dimension of the space. Given a differentiable profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed time $t$ and coordinate indices $i, j \in \{0, \dots, d-1\}$, the partial derivative of the $j$-th component of the plane wave with respect to the $i$-th spatial coordinate $x_i$ is given by: $$\frac{\partial}{\partial x_i} f_j(t, x) = s_i (f'_0(\langle x, \mathbf{\hat{s}} \rangle - ct))_j$$ where $s_i = \langle \mathbf{\hat{s}}, e_i \rangle$ is the $i$-th component of the unit direction vector, and $f'_0$ denotes the derivative of the profile function $f_0$.

Let $d$ be the dimension of the space. Given a twice continuously differentiable profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction vector $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any coordinate index $i \in \{0, \dots, d-1\}$, the second partial derivative of the plane wave with respect to the $i$-th spatial coordinate $x_i$ is given by: $$\frac{\partial^2}{\partial x_i^2} f(t, x) = s_i^2 f''_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $s_i$ is the $i$-th component of the unit vector $\mathbf{\hat{s}}$, and $f''_0$ denotes the second derivative of the profile function $f_0$.

Let $d$ be the dimension of the space. Given a twice continuously differentiable profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed time $t$ and coordinate indices $i, j \in \{0, \dots, d-1\}$, the second partial derivative of the $j$-th component of the plane wave with respect to the $i$-th spatial coordinate $x_i$ is given by: $$\frac{\partial^2}{\partial x_i^2} f_j(t, x) = s_i^2 (f''_0(\langle x, \mathbf{\hat{s}} \rangle - ct))_j$$ where $s_i = \langle \mathbf{\hat{s}}, e_i \rangle$ is the $i$-th component of the unit direction vector, and $f''_0$ denotes the second derivative of the profile function $f_0$.

Let $d$ be the dimension of the space. Given a twice continuously differentiable profile function $f_0: \mathbb{R} \to \mathbb{R}^d$, a propagation speed $c \in \mathbb{R}$, and a direction $s$ with associated unit vector $\mathbf{\hat{s}}$, let the plane wave be defined as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed time $t$ and position $x \in \mathbb{R}^d$, the vector Laplacian $\Delta$ of the plane wave with respect to the spatial coordinates $x$ is given by: $$\Delta f(t, x) = f''_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$$ where $f''_0$ denotes the second derivative of the profile function $f_0$.

Let $d$ be the dimension of the space. Let $f_0 : \mathbb{R} \to \mathbb{R}^d$ be a twice continuously differentiable ($C^2$) profile function, $c \in \mathbb{R}$ be the propagation speed, and $\mathbf{\hat{s}}$ be a unit direction vector. The plane wave defined by $$\mathbf{f}(t, \mathbf{x}) = \mathbf{f}_0(\langle \mathbf{x}, \mathbf{\hat{s}} \rangle - ct)$$ satisfies the wave equation: $$c^2 \Delta \mathbf{f}(t, \mathbf{x}) - \frac{\partial^2 \mathbf{f}}{\partial t^2}(t, \mathbf{x}) = 0$$ for all times $t$ and positions $\mathbf{x} \in \mathbb{R}^d$, where $\Delta$ is the vector Laplacian with respect to the spatial coordinates and $\frac{\partial^2}{\partial t^2}$ is the second-order partial derivative with respect to time.

For any dimension $d$, let $s$ be a direction with unit vector $\mathbf{s} \in \mathbb{R}^d$, $c$ be the propagation speed, and $t$ be a time. The function that maps a position vector $\mathbf{x} \in \mathbb{R}^d$ to the scalar value $\langle \mathbf{x}, \mathbf{s} \rangle - ct$ is differentiable at $\mathbf{x}$, where $\langle \cdot, \cdot \rangle$ denotes the standard inner product on Euclidean space.

In $d$-dimensional Euclidean space, let $\mathbf{s} \in \mathbb{R}^d$ be a unit direction vector, $c \in \mathbb{R}$ be the propagation speed, and $t \in \mathbb{R}$ be time. Let $\mathbf{f}_0: \mathbb{R} \to \mathbb{R}^d$ be a profile function with first derivative $\mathbf{f}_0'$ and second derivative $\mathbf{f}_0''$. For any indices $u, v \in \{0, \dots, d-1\}$, the partial derivative with respect to $x_u$ of the $v$-th component of the vector $s_u \mathbf{f}_0'(\langle \mathbf{x}, \mathbf{s} \rangle - ct)$ is given by: $$\frac{\partial}{\partial x_u} \langle s_u \mathbf{f}_0'(\langle \mathbf{x}, \mathbf{s} \rangle - ct), \mathbf{e}_v \rangle = s_u^2 \langle \mathbf{f}_0''(\langle \mathbf{x}, \mathbf{s} \rangle - ct), \mathbf{e}_v \rangle$$ where $s_u$ is the $u$-th component of $\mathbf{s}$, $\mathbf{e}_v$ is the $v$-th standard basis vector, and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product.

In $d$-dimensional Euclidean space, let $\mathbf{\hat{s}} \in \mathbb{R}^d$ be a unit direction vector, $c \in \mathbb{R}$ be the propagation speed, and $f_0 : \mathbb{R} \to \mathbb{R}^d$ be a differentiable profile function. Define the plane wave as $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$. For any fixed time $t$, position $x$, and coordinate indices $u, v \in \{0, \dots, d-1\}$, the partial derivative of the $v$-th component of the wave with respect to the $u$-th spatial coordinate is related to the time derivative by the equation: $$c \frac{\partial f_v}{\partial x_u}(t, x) = -s_u \frac{\partial f_v}{\partial t}(t, x)$$ where $s_u = \langle \mathbf{\hat{s}}, \mathbf{e}_u \rangle$ is the $u$-th component of the unit direction vector and $\langle \cdot, \cdot \rangle$ denotes the standard Euclidean inner product.

In $d$-dimensional Euclidean space, let $\mathbf{s}$ be a direction vector, $c \in \mathbb{R}$ be the propagation speed, and $f_0: \mathbb{R} \to \mathbb{R}^d$ be a differentiable profile function. If $f(t, x)$ is a plane wave defined by $f(t, x) = f_0(\langle x, \mathbf{s} \rangle - ct)$, then for any fixed position $x \in \mathbb{R}^d$, the function $t \mapsto f(t, x)$ is differentiable with respect to time $t$.

Let $s$ be a direction vector with unit vector $\mathbf{\hat{s}} \in \mathbb{R}^3$ and $f_0: \mathbb{R} \to \mathbb{R}^3$ be a differentiable function. Let $f(t, x) = f_0(\langle x, \mathbf{\hat{s}} \rangle - ct)$ be a plane wave with propagation speed $c \in \mathbb{R}$. For any fixed position $x$, the function $t \mapsto \mathbf{\hat{s}} \times f(t, x)$ is differentiable at $t$, where $\times$ denotes the vector cross product in $\mathbb{R}^3$.

Let $s$ be a direction vector in $\mathbb{R}^3$ and $f_0: \mathbb{R} \to \mathbb{R}^3$ be a differentiable function. Then for any $u \in \mathbb{R}$, the function $u \mapsto s \times f_0(u)$ is differentiable at $u$, where $\times$ denotes the standard cross product in Euclidean 3-space.

Let $d$ be a natural number and $f_0: \mathbb{R} \to \mathbb{R}^d$ be a differentiable function. Let $\mathbf{s}$ be a unit vector in the $d$-dimensional Euclidean space $\text{Space } d$, and let $c$ and $t$ be real constants representing propagation speed and time. Define the plane wave function $f(x) = f_0(\langle x, \mathbf{s} \rangle - ct)$. For any points $x, y \in \text{Space } d$ and any index $i \in \{0, \dots, d-1\}$, the following equality holds: $$s_i \cdot \left( Df(x) \cdot y \right)_i = \langle y, \mathbf{s} \rangle \cdot \left( Df_i(x) \cdot \mathbf{e}_i \right)$$ where $s_i$ is the $i$-th component of the direction vector $\mathbf{s}$, $Df(x)$ is the Fréchet derivative of $f$ at $x$, $(\cdot)_i$ denotes the $i$-th component of a vector, and $\mathbf{e}_i$ is the $i$-th standard orthonormal basis vector.