Wave Relation Cross-Scans

Amplitude-Square Proxy Three-Row Scan

square the relative height

Three dimensionless amplitude rows show a square relation without pretending to compute full physical sound intensity.

waves Relative amplitudes 1, 2, and 3 give relative square values 1, 4, and 9.

Example

A dimensionless amplitude proxy makes the square relation visible without pretending to compute full physical sound intensity. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Use a dimensionless amplitude proxy

Real sound intensity needs more physics than this first waves book. Here the scan uses a dimensionless relative amplitude so the square relation is honest: relative intensity proxy equals relative amplitude squared.

Irel=Arel2I_{\text{rel}} = A_{\text{rel}}^{2}

Small relative amplitude

Relative amplitude 1 (dimensionless) gives proxy intensity 1 (dimensionless).

Irel=Arel2=12=1I_{\text{rel}} = A_{\text{rel}}^{2} = 1^{2} = 1
Wave snapshot rowA spatial wave snapshot makes the row relation visible.

Middle relative amplitude

Relative amplitude 2 (dimensionless) gives proxy intensity 4 (dimensionless).

Irel=Arel2=22=4I_{\text{rel}} = A_{\text{rel}}^{2} = 2^{2} = 4
Wave snapshot rowA spatial wave snapshot makes the row relation visible.

Large relative amplitude

Relative amplitude 3 (dimensionless) gives proxy intensity 9 (dimensionless).

Irel=Arel2=32=9I_{\text{rel}} = A_{\text{rel}}^{2} = 3^{2} = 9
Wave snapshot rowA spatial wave snapshot makes the row relation visible.

The square grows faster than the amplitude

The amplitude column grows by equal steps. The proxy-intensity column follows the square, so the last row is not just three times the first.

ArelIrel112439\begin{array}{c|c}A_{\text{rel}} & I_{\text{rel}} \\ \hline 1 & 1 \\ 2 & 4 \\ 3 & 9\end{array}