# You are standing 5 feet from one speaker and 14 feet from another. Both speakers are producing...

## Question:

You are standing 5 feet from one speaker and 14 feet from another. Both speakers are producing the same sound.

If the combined sound from both speakers is barely audible at your location, which of the following could be the wavelength of the sound?

## Wave Interference

A wave is a disturbance that propagates through medium or even through a vacuum in case of light. Wavelength of a wave ({eq}\lambda {/eq}) is the minimum amount of spatial separation between two points such that the pattern repeats itself. Frequency of a wave ({eq}\nu {/eq}) is defined as the number of wavelengths that pass through a given point in one second.

When two waves generated from two different sources, but of the same frequency and wavelength reaching any particular location interfere with each other producing an interference pattern, a repeated pattern where the waves add up (constructive interference) or cancel each other (destructive interference).

Given:

• Distance to the first and second speakers from your location: {eq}d_1 \ = \ 5 \ ft {/eq} and {eq}d_2 \ = \ 14 \ ft {/eq}

As the sound produced by the speakers are of the same frequency, the resulting sound due to both speakers will be an interference pattern. As the sound is barely audible, they must interfere destructively at your location i.e. the additional distance {eq}d_2 \ - \ d_1 \ = \ 9 \ ft {/eq}, must equal an odd integral multiple of {eq}\dfrac{\lambda}{2} {/eq}, where {eq}\lambda {/eq} is the wavelength of the wave generated.

Therefore,

{eq}(2n \ + \ 1) \ \dfrac{\lambda}{2} \ = \ (d_2 \ - \ d_1) \ = \ 9 \ ft \\ \implies \ \lambda \ = \ \dfrac{18}{(2n \ + \ 1)} \ ft {/eq}

Where:

{eq}n \ = \ 0 \ , \ 1 \ , \ 2 \ , \ 3 \ ...... {/eq}

For {eq}n \ = \ 0 {/eq}, {eq}\lambda \ = \ \dfrac{18}{1} \ = \ 18 \ ft {/eq} 