space equation
- Yuki
- 3 days ago
- 2 min read
Defining architecture itself as geometry and determining how waves behave.
① Definition of the space (hall) itself
The interior of the hall is treated as a three-dimensional space.
Ω⊂R3\Omega \subset \mathbb{R}^3Ω⊂R3
Boundaries (walls, ceilings, floors, columns, openings)
∂Ω\partial \Omega∂Ω
Let's assume that.
The coordinates of each point are
x=(x,y,z)x = (x,y,z)x=(x,y,z)
② The basic equation for waves propagating in space (time-dependent)
The fundamental form of sound, light, and electromagnetic waves is the wave equation.
∂2u(x,t)∂t2−c(x)2 Δu(x,t)=0\frac{\partial^2 u(x,t)}{\partial t^2} - c(x)^2 \, \Delta u(x,t) = 0∂t2∂2u(x,t)−c(x)2Δu(x,t)=0
u(x,t)u(x,t)u(x,t): Amplitude of the wave at position xxx and time ttt
c(x)c(x)c(x): propagation speed at each location
Δ\DeltaΔ: Laplacian
Δ=∂2∂x2+∂2∂y2+∂2∂z2\Delta = \frac{\partial^2}{\partial x^2} +\frac{\partial^2}{\partial y^2} +\frac{\partial^2}{\partial z^2}Δ=∂x2∂2+∂y2∂2+∂z2∂2
👉 Architectural space = a place defined by the Laplacian.
③ Convert to the frequency domain (standing wave)
Time-harmonic waves
u(x,t)=ℜ{U(x)e−iωt}u(x,t)=\Re\{U(x)e^{-i\omega t}\}u(x,t)=ℜ{U(x)e−iωt}
Substituting this gives the Helmholtz equation
(Δ+k2n(x)2) U(x)=0(\Delta + k^2 n(x)^2)\,U(x)=0(Δ+k2n(x)2)U(x)=0
k=ωc0k=\frac{\omega}{c_0}k=c0ω (wavenumber)
n(x)n(x)n(x): Refractive index (a property of space)
④ "Scattering potential form"
(Δ+k2(1+q(x)))U(x)=0(\Delta + k^2(1+q(x)))U(x)=0(Δ+k2(1+q(x)))U(x)=0
q(x)q(x)q(x):
Walls, pillars, cavities, sound-absorbing materials
The internal structure of the space that encompasses everything
In other words
Space = q(x)\boxed{ \text{Space = } q(x) }Space = q(x)
⑤ Boundary conditions (architectural meaning)
Express the properties of a wall using an equation:
fully reflective wall
U∣∂Ω=0U|_{\partial\Omega}=0U∣∂Ω=0
Speed zero (Neumann)
∂U∂n∣∂Ω=0\frac{\partial U}{\partial n}\Big|_{\partial\Omega}=0∂n∂U∂Ω=0
Sound-absorbing wall (Robin)
∂U∂n+αU=0\frac{\partial U}{\partial n}+\alpha U=0∂n∂U+αU=0
⑥ Scattering inverse problem (core)
Waves observed at the boundary
U∣∂Ω=fU|_{\partial\Omega} = fU∣∂Ω=f
from
Find q(x) \boxed{ Find q(x) \text{} Find q(x)
In other words
{(Δ+k2(1+q(x)))U(x)=0x∈ΩU∣∂Ω=f\begin{cases} (\Delta + k^2(1+q(x)))U(x)=0 & x\in\Omega\\ U|_{\partial\Omega}=f \end{cases}{(Δ+k2(1+q(x)))U(x)=0U∣∂Ω=fx∈Ω
⑦ If you were to write this space in "one line"
Ω={x∈R3∣(Δ+k2(1+q(x)))U(x)=0}\boxed{ \Omega = \{x\in\mathbb{R}^3 \mid (\Delta + k^2(1+q(x)))U(x)=0 \} }Ω={x∈R3∣(Δ+k2(1+q(x)))U(x)=0}
⑧ Conceptual Summary
Architectural drawing → Boundary ∂Ω\partial\Omega∂Ω
Material/curved surface/hole → q(x)q(x)q(x)
Sound and light behavior → U(x)U(x)U(x)
What is space?
A function q(x) that describes how waves are bent.
⑨ In artistic terms
This hall isn't made of walls, but rather of the way waves distort.
Comments