A source of sound operates at 2.0 kHz and 20 W , radiating sound uniformly in all directions. Given that the speed of sound in air is 340 m s ⁻¹ and the density of air is 1.2 kg m ⁻³ , what is the displacement amplitude at a distance of 6.0 m from the source?

Step-by-step solution

The intensity I of a sound wave radiating uniformly in all directions from a point source is given by: I = P 4 r^2 The intensity is also related to the displacement amplitude s₀ of the sound wave by the formula: I = 1 2 v ^2 s₀^2 Since = 2 f , we can write: I = 1 2 v (2 f)^2 s₀^2 = 2 ^2 v f^2 s₀^2 Equating the two expressions for intensity: 2 ^2 v f^2 s₀^2 = P 4 r^2 s₀^2 = P 8 ^3 r^2 v f^2 s₀ = 1 2 r f P 2 v Substituting the given values ( P = 20 W , r = 6.0 m , = 1.2 kg m ⁻³ , v = 340 m s ⁻¹ , f = 2000 Hz ): s₀ =