Problem — Ranking the powers of four waves

Problem

Ranking. Four transverse sinusoidal waves on the same string with linear density $\mu$ have amplitude $A$ and frequency $f$: (A) $A = 2\;\text{cm}$, $f = 100\;\text{Hz}$. (B) $A = 1\;\text{cm}$, $f = 100\;\text{Hz}$. (C) $A = 3\;\text{cm}$, $f = 200\;\text{Hz}$. (D) $A = 2\;\text{cm}$, $f = 200\;\text{Hz}$. Rank the average powers carried by the waves (which go as $A^2 f^2$) from smallest to largest.

Solution

Setup. The average power goes as $P \propto A^2 f^2$. We compute the indicator $A^2 f^2$ (with $A$ in cm, $f$ in Hz — only the relative values matter):

$P_A \propto 2^2\cdot 100^2 = 4\cdot 10^{4}$

$P_B \propto 1^2\cdot 100^2 = 1\cdot 10^{4}$

$P_C \propto 3^2\cdot 200^2 = 9\cdot 4\cdot 10^{4} = 3{,}6\cdot 10^{5}$

$P_D \propto 2^2\cdot 200^2 = 4\cdot 4\cdot 10^{4} = 1{,}6\cdot 10^{5}$

Ranking (from smallest to largest): $1\cdot10^4 < 4\cdot10^4 < 1{,}6\cdot10^5 < 3{,}6\cdot10^5$, i.e.:

$\ev{\text{B} < \text{A} < \text{D} < \text{C}}$

Collegamenti

Argomenti: Waves Concetti: Wave Intensity Competenze: Ranking Reasoning