Manuscript VI@1 · VI.tex · section · b5b60afb

The Duhamel sign

Sign, coefficient, and switching audit › The Duhamel sign · lines 2488–2511

With

P = - (\partial_{t}^{2} + Ω_{m}^{2}),

the equation

P ψ = - L_{a} ϕ_{0}, ψ (0) = ψ ˙ (0) = 0

has solution

ψ (t) = \int_{0}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{a} ϕ_{0} (s) d s .

For a source supported after the initial surface,

ψ = - G_{R} L_{a} ϕ_{0} .

Thus the plus sign in the finite-time sine formula and the minus sign in $- G_{R} L_{a} ϕ_{0}$ are the same convention.

Displayed formulas in this passage

unlabeled · line 2491
$P = - (\partial_{t}^{2} + Ω_{m}^{2}),$
unlabeled · line 2495
$P ψ = - L_{a} ϕ_{0}, ψ (0) = ψ ˙ (0) = 0$
unlabeled · line 2499
$ψ (t) = \int_{0}^{t} \frac{sin (( t - s ) Ω _{m} )}{Ω _{m}} L_{a} ϕ_{0} (s) d s .$
unlabeled · line 2506
$ψ = - G_{R} L_{a} ϕ_{0} .$