Small speakers A and B are driven in phase at 725 Hz by the same audio oscillator. Both speakers start out 4.50 m from the listener, but speaker A is slowly moved away (Fig. E16.34). At what distance d will the sound from the speakers first produce destructive interference at the listener's location?
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Verified step by step guidance

1. Understand that destructive interference occurs when the path difference between two waves is an odd multiple of half wavelengths, i.e., (2n+1)λ/2, where n is an integer.
2. Calculate the wavelength (λ) of the sound using the formula λ = v/f, where v is the speed of sound in air (approximately 343 m/s) and f is the frequency of the sound (725 Hz).
3. Determine the initial path difference between the two speakers and the listener. Initially, both speakers are 4.50 m from the listener, so the path difference is zero.
4. As speaker A is moved away, calculate the new path difference, which is the difference in distance from speaker A to the listener and speaker B to the listener. This is given by (x + 3.20 m) - 3.20 m = x.
5. Set the path difference x equal to the first instance of destructive interference, which is λ/2, and solve for x to find the distance d that speaker A must be moved to achieve this condition.

Key Concepts

Wave Interference

Wave interference occurs when two or more waves overlap and combine to form a new wave pattern. In this scenario, the sound waves from speakers A and B interfere with each other. Destructive interference happens when the waves are out of phase, leading to a reduction in amplitude, which can result in silence or a decrease in sound intensity at the listener's location.

Path Difference

Path difference refers to the difference in the distance traveled by two waves from their sources to a common point. For destructive interference to occur, the path difference must be an odd multiple of half the wavelength (λ/2, 3λ/2, etc.). This condition ensures that the waves arrive out of phase, canceling each other out at the listener's position.

Frequency and Wavelength Relationship

The frequency of a wave is related to its wavelength and the speed of sound in the medium by the equation v = fλ, where v is the speed of sound, f is the frequency, and λ is the wavelength. Given the frequency of 725 Hz and assuming the speed of sound in air is approximately 343 m/s, the wavelength can be calculated, which is crucial for determining the path difference needed for destructive interference.