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. What does she hear: constructive or destructive interference? Why?

검증된 단계별 안내

Determine the wavelength of the sound waves using the formula:

λ = \frac{v}{f}

, where

v

is the speed of sound (340.0 m/s) and

f

is the frequency (250.0 Hz).

Calculate the path difference between the sound waves reaching the woman from the two speakers. Since she starts at the midpoint, the distance to each speaker is equal, so the initial path difference is zero.

Recall the condition for constructive interference: the path difference must be an integer multiple of the wavelength (

n λ

, where

n

is an integer). For destructive interference, the path difference must be an odd multiple of half the wavelength (

\frac{1}{2} λ, \frac{3}{2} λ, etc.).

Since the path difference is zero at the midpoint, it satisfies the condition for constructive interference (

0 = 0 λ

). This means the sound waves reinforce each other, leading to constructive interference.

Conclude that the woman hears constructive interference at the midpoint because the sound waves from both speakers arrive in phase, with no path difference.

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주요 개념

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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 Difference

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

Wavelength and Frequency Relationship

The wavelength of a sound wave is inversely related to its frequency, as described by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. For the given frequency of 250 Hz and speed of sound at 340 m/s, the wavelength can be calculated. Understanding this relationship is crucial for determining the conditions for constructive or destructive interference in the scenario presented.