diff --git a/wave.lua b/wave.lua
index d8383ed..bd97a5d 100644
--- a/wave.lua
+++ b/wave.lua
@@ -59,6 +59,7 @@ end
 --Derivative of the interpolation.
 local Derivative = function( wave, x )
   local re, im = wave.dftre, wave.dftim
+  local y = 0
 
   for k = 1, N / 2 do
     local c, s = k * math.cos( x * k ), k * math.sin( x * k )
@@ -76,25 +77,45 @@ end
 --Second derivative of the interpolation.
 local SecondDerivative = function( wave, x )
   local re, im = wave.dftre, wave.dftim
+  local y = 0
 
   for k = 1, N / 2 do
     local c, s = k * k * math.cos( x * k ), k * k * math.sin( x * k )
     y = y
     - c * re[k + 1]
-    + s * im[k + 1]
+    - s * im[k + 1]
     - c * re[N - k + 1]
-    - s * im[N - k + 1]
+    + s * im[N - k + 1]
 
   end
-
   return y
 end
 
+--Bandlimited impulse located at angle theta.
+local function AliasedSinc( theta, x )
+  local n, d = math.sin( N * 0.5 * ( x - theta ) ), N * math.sin( 0.5 * ( x - theta ) )
+  if d < 0.0001 and d > -0.0001 then return 1.0 end
+  return n / d
+end
+
+--Apply bandlimited impulse to wave.
+local Impulse = function( wave, location, magnitude )
+  local speed = wave.vrad
+  
+  for i = 0, N - 1 do
+    speed[ i + 1 ] = speed[ i + 1 ] + magnitude * AliasedSinc( 2.0 * math.pi * i, location )
+  end
+  
+end
+
 local mt = { __index = {
     DFT = DFT,
     Interpolate = Interpolate,
     Derivative = Derivative,
-    SecondDerivative = SecondDerivative } }
+    SecondDerivative = SecondDerivative,
+    Impulse = Impulse } }
+
+
 
 local function Wave( )
   local t = {
@@ -138,12 +159,26 @@ local function Draw()
     love.graphics.setColor( 0, 0, 0, 0.5 )
     love.graphics.circle( "fill", cx, cy, 0.02 )
 
-    love.graphics.setColor( 1.0, 0, 0, 0.5 )
+    
     for k = 0.1, 1.0, 0.1 do
+      --Interpolant.
+      love.graphics.setColor( 1.0, 0, 0, 0.5 )
       th = ( i - 1 + k ) * 2.0 * math.pi / N
       r = cur:Interpolate( th )
       x, y = r * math.cos( th ), r * math.sin( th )
       love.graphics.circle( "fill", x, y, 0.01 )
+      
+      --First derivative.
+      love.graphics.setColor( 0, 1.0, 0, 0.5 )
+      r = 1.0 + cur:Derivative( th )
+      x, y = r * math.cos( th ), r * math.sin( th )
+      love.graphics.circle( "fill", x, y, 0.01 )
+      
+      --Second derivative.
+      love.graphics.setColor( 0, 0, 1.0, 0.5 )
+      r = 1.0 + cur:SecondDerivative( th )
+      x, y = r * math.cos( th ), r * math.sin( th )
+      love.graphics.circle( "fill", x, y, 0.01 )
     end
   end
 
@@ -154,6 +189,19 @@ local function Draw()
 end
 
 local function Update()
+  local t = love.timer.getTime()
+  for i = 1, N do
+    local rxx = old:SecondDerivative( 2.0 * math.pi * (i - 1) / N )
+    cur.vrad[i] = 0.7 * old.vrad[i] + 0.5 * step * rxx
+    cur.radii[i] = old.radii[i] + step * cur.vrad[i]
+  end
+  
+  cur:DFT( )
+  
+  if love.math.random( 120 ) < 3 then
+    cur:Impulse( 0, 0.2 )
+  end
+  
   --Deep copy of current state to old state.
   for name, t in pairs( cur ) do
     for i = 1, 13 do
@@ -161,14 +209,6 @@ local function Update()
     end
   end
 
-  local t = love.timer.getTime()
-  for i = 1, N do
-    cur.radii[i] = cur.radii[i] + old.vrad[i] * step
-    cur.vrad[i] = cur.vrad[i] - cur:SecondDerivative( ( i - 1 ) / N )
-  end
-
-  cur:DFT( )
-
 end
 
 local function DetectCollision( px, py, vpx, vpy )