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The convolution is $(f\ast g)(x)={\int }_{-\infty }^{\infty }f(y)g(x-y)dy$. Under the usual integrability hypotheses, Fubini's theorem and $u=x-y$ give

The supplied transform and the Fourier shift theorem giveSincethe shift and differentiation rules yieldThe coincident boundary values make the endpoint convention immaterial.

Zero initial velocity selects the cosine solution of the transformed equation:with the removable value one at $k=0$. Since $\cos (ckt)=(e^{ickt}+e^{-ickt})/2$, the Fourier shift theorem giveswhere $r_{+}=\max (r,0)$. This is triangular pulse splitting under the wave equation, a special case of the D'Alembert formula.

The two half-height triangles move in opposite directions at speed $|c|$. Their sum initially reproduces the full triangle; once $|c|t>1$ their supports are disjoint. The field vanishes outside $|x|\le 1+|c|t$, consistent with finite propagation speed. For $c=0$ it simply remains $f(x)$.

Because the triangular profile has slope discontinuities, this is a weak solution rather than a globally twice differentiable solution. The initial velocity is zero in the distributional or square-integrable sense, and pointwise away from the initial corners. A classical pointwise time derivative need not exist at those corners; for example $u(0,t)=1-|ct|$ near $t=0$. The initial conditions must be interpreted consistently with this nonsmooth displacement.

The characteristic polynomial is $(r+\gamma {)}^2+c^2{k}^2$, with roots $r=-\gamma \pm ick$. For $ck\ne 0$, the transformed general solution isThe repeated-root case $ck=0$ is $e^{-\gamma t}[A_0+B_{0}t]$, rather than two independent copies of the same exponential. An alternative basis regular at $k=0$ is $e^{-\gamma t}[C(k)\cos (ckt)+D(k)\sin (ckt)/(ck)]$, using the limit $t$ for the last factor at zero.

For $c\ne 0$, the Fourier shift theorem turns the two exponential factors into translations. Equivalently, set $v=e^{\gamma t}u$; then $v_{tt}=c^2{v}_{xx}$. The D'Alembert formula gives the general formwhere $F,G$ are arbitrary functions with the required regularity and decay. The arbitrary profiles can accommodate a spatially transformed zero-frequency velocity by their integral components; the isolated repeated root must not be used to discard that degree of freedom. If $c=0$, the equation instead has $u=e^{-\gamma t}[F(x)+tG(x)]$.

This is the exponentially conjugated wave equation. Positive $\gamma$ produces damping, but the algebra also covers negative $\gamma$, which gives exponential amplification. The sign of $c$ is immaterial because the differential equation contains only $c^2$.

In Fresnel diffraction, the propagation wave phase varies quadratically across the aperture and that curvature must be retained. For a transverse scale $w$ at propagation distance $L$, its size is of order $\pi w^2/(\lambda L)$. In Fraunhofer diffraction, that quadratic term is negligible, approximately $w^2/(\lambda L)\ll 1$, while the linear phase encoding the observation angle remains. A lens can also form the far-field angular pattern in its focal plane.

For a normally incident plane wave and aperture transmission $h(x,y)$, the far-field complex amplitude is its Fourier transform:The detected light intensity is proportional to $|\psi {|}^2$. For identical parallel slits, write the transmission as a sum of shifted copies of one slit. The Fourier shift theorem gives one single-slit transform multiplied by the sum of positional phase factors:The single-slit diffraction envelope sets the broad scale, while coherent interference between slit centres produces the finer array pattern. One sums complex amplitudes before squaring, rather than adding independent slit intensities.