Two in-phase loudspeakers emit identical 1000 Hz sound waves...

Two in-phase loudspeakers emit identical $1000 \mathrm{Hz}$ sound waves along the $x$ -axis. What distance should one speaker be placed behind the other for the sound to have an amplitude 1.5 times that of each speaker alone?

Key Concepts

Wavelength Calculation

The wavelength of a wave is the distance over which its shape repeats, and it is calculated using the relation ? = v/f, where v is the speed of the wave and f is its frequency. This concept is essential for relating the physical separation between sources to the phase difference of the waves.

Phase Difference and Path Difference

The phase difference between two waves is directly related to the difference in the paths that the waves travel. This relationship is given by the formula ? = (2?/?) × (path difference), where ? is the wavelength. Adjusting the path difference can control whether interference is constructive or destructive.

Wave Interference

Wave interference is the phenomenon that occurs when two or more waves overlap and combine. The resulting wave pattern depends on the phase relationship between the original waves, leading to areas of enhanced amplitude (constructive interference) or reduced amplitude (destructive interference).

Constructive Interference

Constructive interference happens when waves meet in phase, meaning their peaks and troughs align. In such cases, the amplitudes of individual waves add together, resulting in a higher overall amplitude than that of a single wave.

Amplitude Superposition

Amplitude superposition describes the process by which the total amplitude of overlapping waves is determined by vectorially adding their individual amplitudes. For waves of equal amplitude, the resultant amplitude can be expressed in terms of the phase difference between them.

Transcript

Please give Ace some feedback