Wave equation
Wavefronts and wakes

In D = 1 {\displaystyle D=1} case, the Green's function solution is the sum of two wavefronts sgn (t − x) 4 + sgn (t + x) 4 {\displaystyle {\frac {\operatorname {sgn}(t-x)}{4}}+{\frac {\operatorname {sgn}(t+x)}{4}}} moving in opposite directions.

In odd dimensions, the forward solution is nonzero only at t = r {\displaystyle t=r}. As the dimensions increase, the shape of wavefront becomes increasingly complex, involving higher derivatives of the Dirac delta function. For example, G 1 = 1 2 c θ (τ) G 3 = 1 4 π c 2 δ (τ) r G 5 = 1 8 π 2 c 2 (δ (τ) r 3 + δ ′ (τ) c r 2) G 7 = 1 16 π 3 c 2 (3 δ (τ) r 4 + 3 δ ′ (τ) c r 3 + δ ′ ′ (τ) c 2 r 2) {\displaystyle {\begin{aligned}&G_{1}={\frac {1}{2c}}\theta (\tau)\\&G_{3}={\frac {1}{4\pi c^{2}}}{\frac {\delta (\tau)}{r}}\\&G_{5}={\frac {1}{8\pi ^{2}c^{2}}}\left({\frac {\delta (\tau)}{r^{3}}}+{\frac {\delta ^{\prime }(\tau)}{cr^{2}}}\right)\\&G_{7}={\frac {1}{16\pi ^{3}c^{2}}}\left(3{\frac {\delta (\tau)}{r^{4}}}+3{\frac {\delta ^{\prime }(\tau)}{cr^{3}}}+{\frac {\delta ^{\prime \prime }(\tau)}{c^{2}r^{2}}}\right)\end{aligned}}} where τ = t − r {\displaystyle \tau =t-r}, and the wave speed c {\displaystyle c} is restored.

In even dimensions, the forward solution is nonzero in r ≤ t {\displaystyle r\leq t}, the entire region behind the wavefront becomes nonzero, called a wake. The wake has equation: G D (t, x) = (− 1) 1 + D / 2 1 (2 π) D / 2 1 c D θ (t − r / c) (t 2 − r 2 / c 2) (D − 1) / 2 {\displaystyle G_{D}(t,x)=(-1)^{1+D/2}{\frac {1}{(2\pi)^{D/2}}}{\frac {1}{c^{D}}}{\frac {\theta (t-r/c)}{\left(t^{2}-r^{2}/c^{2}\right)^{(D-1)/2}}}} The wavefront itself also involves increasingly higher derivatives of the Dirac delta function.

This means that a general Huygens' principle – the wave displacement at a point (t, x) {\displaystyle (t,x)} in spacetime depends only on the state at points on characteristic rays passing (t, x) {\displaystyle (t,x)} – only holds in odd dimensions. A physical interpretation is that signals transmitted by waves remain undistorted in odd dimensions, but distorted in even dimensions.

Hadamard's conjecture states that this generalized Huygens' principle still holds in all odd dimensions even when the coefficients in the wave equation are no longer constant. It is not strictly correct, but it is correct for certain families of coefficients
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