Two loudspeakers, 1.0 m apart, emit sound waves with the...

Two loudspeakers, $1.0 \mathrm{m}$ apart, emit sound waves with the same frequency along the positive $x$ -axis. Victor, standing on the axis to the right of the speakers, hears no sound. As the frequency is slowly tripled, Victor hears the sound go through the sequence loud-soft-loud-soft-loud before becoming quiet again. What was the original sound frequency?

Key Concepts

Constructive and Destructive Interference Conditions

These conditions specify when the interference between waves results in increased intensity (constructive interference) or decreased intensity (destructive interference). Constructive interference occurs when the path difference is an integer multiple of the wavelength, while destructive interference occurs when the path difference is an odd multiple of one-half wavelength. These criteria are key in analyzing how changes in frequency lead to alternating patterns of loud and soft sounds as observed by a listener.

Path Difference

Path difference refers to the difference in distances traveled by waves from two separate sources to a common point. It determines the phase difference between the waves when they meet. When the path difference corresponds to specific fractions of a wavelength (such as half a wavelength for destructive interference), it can result in cancellation of sound, which is critical in predicting regions of silence or reinforcement.

Wave Interference

This concept describes how two or more waves superpose to produce a resultant wave whose amplitude is determined by the phase and amplitude of the individual waves. Depending on their phase difference, the waves can interfere constructively (amplitudes add) or destructively (amplitudes cancel), producing regions of high or low intensity respectively. This is central to understanding how sound from two sources can combine to create audible patterns of loud and soft sounds.

Relationship Between Frequency, Wavelength, and Wave Speed

This fundamental principle, expressed as v = f?, states that the wave speed is the product of frequency and wavelength. In a medium like air where the speed of sound is relatively constant, changes in the frequency of emitted sound directly alter the wavelength. This relationship is vital for understanding how modifications in frequency affect the interference pattern created by the sound waves.

Transcript

Please give Ace some feedback