How wavy is an array?
JavaScript (ES6), 172 171 169 bytes
The output is 0-indexed.
a=>a.map((_,i)=>(D=m=>+m||m.reduce((s,[v],i)=>s+(i&1?-v:v)*D(m.map(([,...r])=>r).filter(_=>i--)),0))(a.map((v,y)=>a.map((_,x)=>x-i?x-~-Math.abs(y%(2*x)-x):v)))&&(j=i))|j
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How?
This is essentially an implementation of Cramer's rule, except that the denominator is not computed.
Given the input vector of length \$n\$, we define \$W_n\$ as the matrix holding the waves, stored column-wise.
Example:
$$W_5=\begin{pmatrix} 1&1&1&1&1\\ 1&2&2&2&2\\ 1&1&3&3&3\\ 1&2&2&4&4\\ 1&1&1&3&5\end{pmatrix}$$
We compute all values \$v_i=\det(M_i)\$ with \$0\le i<n\$, where \$M_i\$ is the matrix formed by replacing the \$i\$-th column of \$W_n\$ with the input vector.
Example:
For the 5th test case and \$i=3\$, this gives:
$$v_3=\det(M_3)=\begin{vmatrix} 1&1&1&\color{red}1&1\\ 1&2&2&\color{red}3&2\\ 1&1&3&\color{red}3&3\\ 1&2&2&\color{red}3&4\\ 1&1&1&\color{red}1&5\end{vmatrix}=0$$
We return the highest index \$j\$ such that \$v_j\neq0\$.
Commented
The helper function \$D\$ uses Laplace expansion to compute the determinant:
D = m => // m[] = matrix
+m || // if the matrix is a singleton, return it
m.reduce((s, [v], i) => // otherwise, for each value v at (0, i):
s // take the previous sum s
+ (i & 1 ? -v : v) // based on the parity of i, add either v or -v
* D( // multiplied by the result of a recursive call:
m.map(([, ...r]) => r) // pass m[] with the 1st column removed
.filter(_ => i--) // and the i-th row removed
), // end of recursive call
0 // start with sum = 0
) // end of reduce()
The matrix \$M_i\$ is computed with:
a.map((v, y) => // for each value v at position y in the input vector a[]:
a.map((_, x) => // for each value at position x in the input vector a[]:
x - i ? // if this is not the i-th column:
x - // compute the wave of power x+1 at position y:
~-Math.abs( // x - (|y mod (2*x) - x| - 1)
y % (2 * x) - x // (where y mod 0 is coerced to 0)
) //
: // else:
v // append the value a[y] of the input vector
) // end of inner map()
) // end of outer map()
APL (Dyalog Unicode), 34 33 27 bytes
⊢{⊢/⍋0≠⍺⌹⍉↑(1+⍵⍴⍳,⊢-⍳)¨⍳⍵}≢
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Lots of golfing thanks to ngn.
How it works
g←⊢{⊢/⍋0≠⍺⌹⍉↑(1+⍵⍴⍳,⊢-⍳)¨⍳⍵}≢ Accept a vector V of length N
⊢{ }≢ Call the inner function with ⍺=V, ⍵=N
⍳⍵ Make 0..N-1
( ⍳,⊢-⍳)¨ For each i of above, make 0..i-1,i..1 and
1+⍵⍴ repeat/truncate to length N and +1
⍺⌹⍉↑ Reform above as matrix and solve equations
0≠ Test nonzero on each item (nonzero→1, zero→0)
⊢/⍋ Find the last index of 1
APL (Dyalog Unicode), 34 33 bytes
⍴-0(⊥⍨=)⊢⌹∘⍉∘↑1+⍴⍴¨(⊢,1+⌽)∘⍳¨∘⍳∘⍴
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Uses ⎕IO←0 and an alternative way to generate the matrix W.
g←⍴-0(⊥⍨=)⊢⌹∘⍉∘↑1+⍴⍴¨(⊢,1+⌽)∘⍳¨∘⍳∘⍴ Accept a vector of length n
⍳∘⍴ Make a vector 0..n-1
(⊢,1+⌽)∘⍳¨ For each k of above, chain 0..k-1 and k..1
⍴⍴¨ and repeat or truncate to length n
The first row has 0 elements, repeat gives n zeros
1+ Increment all elements
⍴-0(⊥⍨=)⊢⌹∘⍉∘↑ The rest is the same as the previous answer
APL (Dyalog Unicode), 34 bytes
⍴-0(⊥⍨=)⊢⌹∘⍉∘↑⍴⍴¨(⊢,1↓¯1↓⌽)∘⍳¨∘⍳∘⍴
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A monadic train that accepts a numeric vector.
How it works
g←⍴-0(⊥⍨=)⊢⌹∘⍉∘↑⍴⍴¨(⊢,1↓¯1↓⌽)∘⍳¨∘⍳∘⍴ Accept a vector of length n
⍳∘⍴ Make a vector 1..n
(⊢,1↓¯1↓⌽)∘⍳¨ For each k of the above, generate one unit of wave of power k (1..k..2)
⍴⍴¨ Repeat or truncate each row to length n (result is nested vector)
⍉∘↑ Convert nested vector to matrix, and transpose it
⊢⌹ Solve the linear equation
0(⊥⍨=) Count trailing zeros
⍴- Subtract from n
Basically, this solution uses the same W matrix as Arnauld's submission, solves the linear equation, and then finds the index of the last nonzero entry using the good old "count trailing ones" ⊥⍨.