Sound Waves and Interference Problem - Check My Work?

[Malavin]

Okay, I have this problem worked out, but I have no clue if my answers are right. Could someone please check my work and give me some confidence or show me where I may have messed up.

Homework Statement

Two loudspeakers, one of them generates sound with 400 W power the other with 100 W. The sound is generated isotropically (the same in all directions) from each loudspeaker. A listener is seated 20m from the 100 W loudspeaker and 40 m from the other loudspeaker. A signal generator drives the two speakers in phase with the same frequency. The frequency is swept through the audible range from 20-20,000 Hz.

a) What is the ratio of the intensities of the two sounds (I(for 400W)/I(for 100W)) at the listener's position?

b) What are the two highest frequencies at which the listener will hear a maximum signal because of constructive interference?

c) What are the two highest frequencies at which the listener will hear a minimum signal because of destructive interference?

d) Assuming a single frequency of sound (f= 5000Hz) from the 400 W loudspeaker. What is the displacement amplitude of the sound waves that arrives at the listener's position from the 400 W loudspeaker only?

Homework Equations

I = P_s/(4πr²)
I = 0.5ρvω²s_m²
ΔL/λ = 0, 1, 2... (fully constructive interference)
ΔL/λ = 0.5, 1.5, 2.5... (fully destructive interference)
v = λf
v_air = 343 m/s
ρ_air = 1.21 kg/m³

The Attempt at a Solution

a) I₁ = P_s/(4πr²) = 400/(4π40²) = 1.99*10^-2 W/m²
I₂ = 100/(4π20²) = 1.99*10^-3 W/m²
I₁/I₂ = 10

b) ΔL/λ = ΔL/(v/f) = fΔL/v = f(40-20)/343 = 0, 1, 2...
f = (0, 1, 2...)*17.15 Hz
f_maxf = 20,000 Hz
Looking for two highest frequencies: (n)17.15 Hz ≤ 20,000 Hz
n ≤ 1166.18
n_max = 1165, 1166
f_max = 19,979.75 Hz, 19,996.9 Hz

c)f = (0.5, 1.5, 2.5...)*17.15 Hz
(n)17.15 Hz ≤ 20,000 Hz
n ≤ 1166.18
n_max = 1164.5, 1165.5
f_max = 19,971.18 Hz, 19,988.33 Hz

d)ω = 2πf = 2π(5000) = 10000π
I = 0.5ρvω²s_m²
s_m = √(2I/(ρvω²))
s_m = 3.18*10^-7 m

 

[Redbelly98]

For (a), try recalculating I₂. I get a different number.

(b), (c) and (d) look good.