Julia performs an experiment to measure the wavelength of four different waves and records her data in the table below.

Julia's Waves
\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{ Wave } & \multicolumn{1}{c|}{ Information } \\
\hline 1 & This wave has a [tex]$3 cm$[/tex] amplitude. \\
\hline 2 & The distance from the midpoint to the crest is [tex]$6 cm$[/tex]. \\
\hline 3 & The distance from the midpoint to the trough is [tex]$12 cm$[/tex]. \\
\hline 4 & This wave has a [tex]$4 cm$[/tex] amplitude. \\
\hline
\end{tabular}

Which accurately ranks the waves from the lowest energy wave to the highest energy wave?

A. [tex]$1 \rightarrow 4 \rightarrow 3 \rightarrow 2$[/tex]

B. [tex]$2 \rightarrow 3 \rightarrow 4 \rightarrow 1$[/tex]

C. [tex]$3 \rightarrow 2 \rightarrow 4 \rightarrow 1$[/tex]

D. [tex]$1 \rightarrow 4 \rightarrow 2 \rightarrow 3$[/tex]

To determine the accurate ranking of the waves from the lowest energy wave to the highest energy wave, we need to understand how wave energy is calculated. The energy $ E $ of a wave is proportional to the square of its amplitude $ A $. In simpler terms, if we know a wave's amplitude, we can compute its energy using the formula:

[tex]\[ E = A^2 \][/tex]

Let's analyze the information provided for each wave in the table and calculate their energies step-by-step:

1. Wave 1:
- Given amplitude: $3 \, cm$
- Energy: $ E_1 = 3^2 = 9 \, (cm^2) $

2. Wave 2:
- Given distance from the midpoint to the crest: $6 \, cm$
- This distance is the amplitude of the wave.
- Energy: $ E_2 = 6^2 = 36 \, (cm^2) $

3. Wave 3:
- Given distance from the midpoint to the trough: $12 \, cm$
- The distance from the midpoint to the trough is the amplitude of the wave.
- Energy: $ E_3 = 12^2 = 144 \, (cm^2) $

4. Wave 4:
- Given amplitude: $4 \, cm$
- Energy: $ E_4 = 4^2 = 16 \, (cm^2) $

Now, let's list the calculated energies for each wave:

- Wave 1: $9 \, (cm^2)$
- Wave 2: $36 \, (cm^2)$
- Wave 3: $144 \, (cm^2)$
- Wave 4: $16 \, (cm^2)$

To rank the waves from the lowest energy to the highest energy, we sort the energies in ascending order:

- $9 \, (cm^2)$ (Wave 1)
- $16 \, (cm^2)$ (Wave 4)
- $36 \, (cm^2)$ (Wave 2)
- $144 \, (cm^2)$ (Wave 3)

Thus, the correct ranking of the waves from the lowest energy to the highest energy is:
[tex]\[ 1 \rightarrow 4 \rightarrow 2 \rightarrow 3 \][/tex]

Therefore, the last option,
[tex]\[ 1 \rightarrow 4 \rightarrow 2 \rightarrow 3 \][/tex]
is the accurate ranking of the waves based on their energies.