Two loudspeakers emit sound waves along the x -axis. The...

Two loudspeakers emit sound waves along the $x$ -axis. The sound has maximum intensity when the speakers are $20 \mathrm{cm}$ apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of $60 \mathrm{cm}$ a. What is the wavelength of the sound? b. If the distance between the speakers continues to increase, at what separation will the sound intensity again be a maximum?

Two loudspeakers emit sound waves along the $x$ -axis. The sound has maximum intensity when the speakers are $20 \mathrm{cm}$ apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of $60 \mathrm{cm}$
a. What is the wavelength of the sound?
b. If the distance between the speakers continues to increase, at what separation will the sound intensity again be a maximum?

Key Concepts

Interference

Interference is the phenomenon that occurs when two or more waves overlap and combine to form a resultant wave. The combined effect can be constructive, where the waves reinforce each other to produce regions of higher amplitude, or destructive, where they cancel each other to produce regions of lower or even zero amplitude. This concept is fundamental in understanding how waves from coherent sources interact.

Path Difference and Phase Difference

When waves emitted by different sources reach a given point, any difference in the distance traveled (the path difference) leads to a phase difference. Constructive interference occurs when the phase difference is an integer multiple of 2?, corresponding to a path difference of n? (where n is an integer), while destructive interference happens when the phase difference is an odd multiple of ?, corresponding to a path difference of (n + 1/2)?. This relationship is key to predicting the locations of intensity maxima and minima.

Determining Wavelength from Interference Conditions

By comparing conditions that yield constructive and destructive interference, one can deduce the wavelength of a wave. In scenarios where changing the separation between coherent sources switches the interference from maximum to zero intensity, the difference in separation is directly related to a half-wavelength shift. This method of using interference conditions to calculate the wavelength is a standard technique in wave physics.

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