Let $Y$ be a smooth manifold, let $I \subset \mathbb{R}$ be an open interval with $0 \in I$, and let $V \subset T^*Y \setminus 0$ be an open conic set. Let $U(t)$ be a properly supported scalar Fourier integral parametrix, modulo a smooth Schwartz kernel on $I \times Y \times Y$, for the full wave propagator microlocally over $V$. Let $K_U \in \mathcal{D}'(I \times Y \times Y)$ denote the Schwartz kernel of $U(t)$, and let $C \subset T^*(I \times Y \times Y) \setminus 0$ be the associated bicharacteristic canonical relation, with the standard Schwartz-kernel convention that the input covector appears with the opposite sign. If

\begin{align*} \operatorname{WF}(K_U) \subset C, \end{align*}

then

\begin{align*} \operatorname{sing\,supp}(K_U) \subset \pi_{I \times Y \times Y}(C), \end{align*}

where $\pi_{I \times Y \times Y}: T^*(I \times Y \times Y) \to I \times Y \times Y$ is the cotangent bundle base projection.

In the Euclidean case $Y = \mathbb{R}^n$, for the full wave propagator $U(t) = \cos(t\sqrt{-\Delta})$, this gives

\begin{align*} \operatorname{sing\,supp}(K_U) \subset \{(0,x,x) : x \in \mathbb{R}^n\} \cup \{(t,x,y) : t \in \mathbb{R} \setminus \{0\}, |x-y| = |t|\}. \end{align*}

In the Riemannian case, let $(Y,g)$ be a Riemannian manifold, let $\Omega \subset Y$ be a geodesically convex [open set](/page/Open%20Set), and let $T>0$ be smaller than the relevant injectivity radius on $\Omega$. For $I = (-T,T)$ and the localized kernel on $I \times \Omega \times \Omega$ of the full wave propagator $U(t)=\cos(t\sqrt{\Delta_g})$, where $\Delta_g$ is taken with the nonnegative spectral convention, one has

\begin{align*} \operatorname{sing\,supp}(K_U) \subset \{(0,y,y): y \in \Omega\} \cup \{(t,y,z): 0<|t|<T,\ y,z \in \Omega,\ d_g(y,z)=|t|\}. \end{align*}