Question Detail
Question
For a sound wave described by particle displacement y = y_m sin(kx - ωt), the pressure variation p can be expressed as p = P cos(kx - ωt), where P is the pressure amplitude. According to the document, which of the following is the correct expression for the pressure amplitude P?
Correct Answer
Option A is the correct answer.
Detailed Explanation
The document derives the pressure variation p = -B(∂y/∂x). Given y = y_m sin(kx - ωt), then ∂y/∂x = k y_m cos(kx - ωt).So, p = -B k y_m cos(kx - ωt).From a previous derivation, the speed of sound v = √(B/ρ₀), so B = ρ₀v².Substituting B into the pressure equation: p = - (ρ₀v²) k y_m cos(kx - ωt).The pressure amplitude P is the maximum value of this variation, so P = ρ₀v²k y_m (ignoring the negative sign which indicates phase). The document explicitly states: 'P = ρ₀v²kym'.
Hint
Review the section 'Pressure variation in sound waves' and the derived formulas for pressure (p) and pressure amplitude (P) in relation to bulk modulus (B) and wave parameters.