Calculating waves

Pyth, 46 18 bytes

ucR2sV+hGGt+GeG.*Q

This code expects input in the form iterations, [wave1, wave2, wave3 ...], as seen at the first link below.

Demonstration. Test Harness.

The program works by applying the code within the reduce (u) function to the input list, as many times as the number of iterations.

I will demonstrate the wave propagation function on the list [0, 1, 2, 3, 4, 5], which is in G:

+hGG prepends G's first element to G, forming [0, 0, 1, 2, 3, 4, 5].

t+GeG appends G's last element to G and removes its first element, forming [1, 2, 3, 4, 5, 5].

sV first forms pairs from the lists, giving [[0, 1], [0, 2], [1, 3], [2, 4], [3, 5], [4, 5]] with the final element of the first list truncated away. Then, the pairs are summed via the s function, giving [1, 2, 4, 6, 8, 9].

cR2 uses floating point division to divide all of the numbers by 2, giving the desired result, [0.5, 1.0, 2.0, 3.0, 4.0, 4.5].

Snowman 1.0.0, 219 chars

{vg" "aS:10sB;aM0aa,AAg**-:|al|'NdE'0nRal@(%}{->:1*?{0AaG;:?{;bI:dUNiNwR'NdEwRaC;aM(~:?{(()1wR]0wRaC*))#'wRaC|*#|(()#aLNdEdUNdEwR]wR]aCwR*))#aC;:0wRdUaCwR*?{#aC;#bI:*#0aA'|aa|'!*+'(()#1aA*))#|,aa|'*`nA2nD;aM|*0*;bR|tSsP

With linebreaks for "readability":

{vg" "aS:10sB;aM0aa,AAg**-:|al|'NdE'0nRal@(%}{->:1*?{0AaG;:?{;bI:dUNiNwR'NdEwRaC;
aM(~:?{(()1wR]0wRaC*))#'wRaC|*#|(()#aLNdEdUNdEwR]wR]aCwR*))#aC;:0wRdUaCwR*?{#aC;#
bI:*#0aA'|aa|'!*+'(()#1aA*))#|,aa|'*`nA2nD;aM|*0*;bR|tSsP

Ungolfed / unminified version:

{vg" "aS:10sB;aM  // input space-separated list of numbers
0aa,AAg           // get first element and array of all-but-first elements
**                // discard original input and the 0

// execute the following (input[0]) times
-:
    |al|'NdE'0nR               // get range from (0..input.length-1]
    al@(%}{->:1*?{0AaG;:?{;bI  // chop off first element if any
    :dUNiNwR'NdEwRaC;aM        // map n -> [n-1 n+1]
    // if the input consisted of a single element, append [0 0]
    // otherwise prepend [0 1] and append [len-2 len-1]
    (~:?{(()1wR]0wRaC*))#'wRaC|*#|(()#aLNdEdUNdEwR]wR]aCwR*))#aC;
        :0wRdUaCwR*?{#aC;#bI
    // map indeces to avg(input[i1], input[i2])
    :*#0aA'|aa|'!*+'(()#1aA*))#|,aa|'*`nA2nD;aM
    // replace old input, reset permavar
    |*0*
;bR

|tSsP  // output result

Sample I/O format:

llama@llama:...Code/snowman/ppcg53799waves$ snowman waves.snowman 
4 32 32 32 16 64 16 32 32 32
[33 27 40 22 44 22 40 27 33]

Pyth - 25 24 bytes

Uses enumerate, reduce to do iterations.

u.ecs@L++hGGeG,khhk2GvzQ

Try it online here.