Sound Waves and Interference Problem - Check My Work?

In summary, the conversation revolved around a problem involving two loudspeakers generating sound with 400 W and 100 W power, respectively. The sound is isotropically generated and a listener is seated at different distances from the two speakers. The questions asked for the ratio of intensities between the two sounds, the highest and lowest frequencies at which the listener will hear maximum and minimum signals due to interference, and the displacement amplitude of the sound waves at a specific frequency from the 400 W loudspeaker. The conversation also included relevant equations and attempts at solutions for each question.
  • #1
Malavin
4
0
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 = Ps/(4πr2)
I = 0.5ρvω2sm2
ΔL/λ = 0, 1, 2... (fully constructive interference)
ΔL/λ = 0.5, 1.5, 2.5... (fully destructive interference)
v = λf
vair = 343 m/s
ρair = 1.21 kg/m3

The Attempt at a Solution


a) I1 = Ps/(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
fmaxf = 20,000 Hz
Looking for two highest frequencies: (n)17.15 Hz ≤ 20,000 Hz
n ≤ 1166.18
nmax = 1165, 1166
fmax = 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
nmax = 1164.5, 1165.5
fmax = 19,971.18 Hz, 19,988.33 Hz

d)ω = 2πf = 2π(5000) = 10000π
I = 0.5ρvω2sm2
sm = √(2I/(ρvω2))
sm = 3.18*10-7 m
 
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  • #2
For (a), try recalculating I2. I get a different number.

(b), (c) and (d) look good.
 

1. What are sound waves and how do they travel?

Sound waves are longitudinal mechanical waves that travel through a medium, such as air or water. They are created when an object vibrates, causing the particles in the medium to vibrate and transmit the energy from the source to the receiver.

2. How does sound interference occur?

Sound interference occurs when two or more sound waves meet and interact with each other. This can result in either constructive interference, where the amplitude of the waves is increased, or destructive interference, where the waves cancel each other out.

3. How can we calculate the wavelength and frequency of a sound wave?

The wavelength of a sound wave can be calculated by dividing the speed of sound in the medium by the frequency of the wave. The frequency of a sound wave can be determined by counting the number of complete cycles of the wave in one second, measured in hertz (Hz).

4. How do different materials affect the speed of sound?

The speed of sound varies depending on the medium it travels through. It is generally faster in solids, due to the particles being closer together and able to transmit energy more efficiently, and slower in gases, due to the particles being more spread out. The density and temperature of the medium also affect the speed of sound.

5. Can sound waves interfere with each other in an open space?

Yes, sound waves can interfere with each other in an open space. This is because even in a seemingly empty space, there are still particles present that can transmit sound waves. However, the interference is typically less noticeable in open spaces compared to closed spaces, where sound waves can reflect and amplify more easily.

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