Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m>s. A woman starts out at the midpoint between the two speakers. The room's walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (a) What does she hear: constructive or destructive interference? Why?

Verified step by step guidance

1. Calculate the wavelength of the sound waves using the formula \(\lambda = \frac{v}{f}\), where \(v\) is the speed of sound and \(f\) is the frequency of the sound.
2. Determine the path difference between the sound waves reaching the woman from each speaker. Since she starts at the midpoint and the speakers are in-phase, the path difference will be zero.
3. Understand that when the path difference is zero or a multiple of the wavelength, constructive interference occurs. This is because the waves add up in phase, amplifying the sound.
4. Check if the path difference (zero in this case) is a multiple of the wavelength calculated in step 1. If it is, then constructive interference will occur.
5. Conclude that the woman hears a louder sound due to constructive interference, as the sound waves from both speakers arrive in-phase and amplify each other.

Key Concepts

Interference of Waves

Interference occurs when two or more waves overlap, resulting in a new wave pattern. This can be constructive interference, where wave amplitudes add together, or destructive interference, where they cancel each other out. The type of interference depends on the phase relationship between the waves, which is influenced by their path lengths and the frequency of the waves.

Path Length Difference

The path length difference is the difference in distance traveled by two waves arriving at a point. For constructive interference to occur, this difference must be an integer multiple of the wavelength. Conversely, for destructive interference, it must be a half-integer multiple of the wavelength. In this scenario, the woman is positioned at the midpoint, which affects the path length difference from each speaker.

Wavelength and Frequency Relationship

The wavelength of a sound wave is the distance between successive crests and is inversely related to its frequency. It can be calculated using the formula: wavelength = speed of sound / frequency. In this case, knowing the frequency of 250 Hz and the speed of sound at 340 m/s allows us to determine the wavelength, which is essential for analyzing the interference pattern experienced by the woman.