diff --git a/slides/analytical_solutions.md b/slides/analytical_solutions.md
index bb52484..e2897d7 100644
--- a/slides/analytical_solutions.md
+++ b/slides/analytical_solutions.md
@@ -51,6 +51,8 @@ $$
 ## Short wave wavenumber (cont.)
 
 <img class="r-stretch" src="assets/fig_wavenumber_modulation_by_ak.png">
+
+* But, phase-averaged $\widetilde k/k$ *should* be 1 if the crests are conserved.
 </section>
 
 
diff --git a/slides/governing_equations.md b/slides/governing_equations.md
index 831e4ad..c501bba 100644
--- a/slides/governing_equations.md
+++ b/slides/governing_equations.md
@@ -36,17 +36,15 @@ $$
 $$
 \eta = a_L \cos\psi
 $$
-
 $$
 U = a_L \omega_L e^{\varepsilon_L \cos\psi} \cos\psi
 $$
-
 $$
 W = a_L \omega_L e^{\varepsilon_L \cos\psi} \sin\psi
 $$
 
 * Assume no vertical shear; Stewart and Joy (1974) would be straightforward here
-  but not sure whether it's applicable.
+  but perhaps not applicable.
 </section>
 
 
diff --git a/slides/numerical_solutions.md b/slides/numerical_solutions.md
index 6b7c420..1219172 100644
--- a/slides/numerical_solutions.md
+++ b/slides/numerical_solutions.md
@@ -20,6 +20,7 @@ $$
   - Space differencing: 2$^{nd}$ order centered
   - Time integration: 4$^{th}$ order Runge-Kutta
   - 100 grid points in phase with periodic boundary conditions
+  - $k_L = 1$, $k_s = 10$, $\varepsilon_L = 0.1$
 </section>
 
 
@@ -29,7 +30,7 @@ $$
 
 * Initial conditions seem to be very important
 * Peureux et al. (2021) found:
-  - Stable if initialized with the L-H&S solution for $\widetilde k$
+  - Stable solution if initialized with the L-H&S solution for $\widetilde k$
   - Unstable growth if initialized uniformly
   - _"...sudden appearance of a long wave perturbation in the middle of a homogeneous
 short wave field."_
@@ -43,7 +44,6 @@ short wave field."_
 <video
   width=600
   controls
-  data-autoplay
   loop
   src="assets/modulation_movie_infinite_train.mp4"
   type="video/mp4">
diff --git a/slides/title.md b/slides/title.md
index e85694b..cc0aefc 100644
--- a/slides/title.md
+++ b/slides/title.md
@@ -18,7 +18,7 @@ _WISE 2024, Cargèse, France_
 ## Problem statement
 
 * Short waves are modulated by ambient longer waves.
-* Well studied but not yet fully understood; many processes going on at once.
+* Well studied but not fully understood; many processes going on.
 * Modulation is important for at least:
   - Remote sensing (e.g. *EM return from Bragg waves carry information about long waves*)
   - Numerical wave prediction (e.g. *can different wave bands be treated independently in wind input functions?*)
@@ -26,6 +26,14 @@ _WISE 2024, Cargèse, France_
 </section>
 
 
+<section>
+
+## Dynamics of a linear wave
+
+<img class="stretch" src="assets/wave_diagram.jpeg">
+</section>
+
+
 <section>
 
 ## One possible look at hydrodynamic modulation