Double-slit interference: Two small forward-facing speakers are 2.50 m apart. They are both emitting, in phase with each other, a sound of frequency 1100 Hz in a room where the speed of sound is 344 m/s. A woman is standing opposite the midpoint between the speakers and is initially 35.0 m from the midpoint. As she slowly walks parallel to the line connecting the speakers, at what angle θ (relative to the centerline coming outward from the midpoint between the speakers) will she first hear no sound?
A) 0.063°
B) 3.6°
C) 7.2°
D) 1.8°
E) 11°

Question

Quizplus · Accepted Answer

The first position of destructive interference (no sound) occurs when the path difference between the sound waves from the two speakers is equal to half a wavelength ($\lambda/2$). The wavelength ($\lambda$) can be found using the formula $\lambda = v/f$, where $v$ is the speed of sound (344 m/s) and $f$ is the frequency (1100 Hz). This gives $\lambda = 344 / 1100 = 0.3127$ m. For destructive interference, the path difference should be $\lambda/2 = 0.1564$ m. Using the geometry of the situation, the path difference can also be expressed as $d \sin(\theta)$, where $d$ is the distance between the speakers (2.50 m) and $\theta$ is the angle relative to the centerline. Setting $d \sin(\theta) = \lambda/2$ and solving for $\theta$ gives $\theta = \arcsin(\lambda/2d) = \arcsin(0.1564/2.5) \approx 3.6^\circ$.

Double-Slit Interference: Two Small Forward-Facing Speakers Are 2