# Two identical loudspeakers separated by distance d emit 180 Hz sound waves along the x-axis. As...

## Question:

Two identical loudspeakers separated by distance {eq}d {/eq} emit {eq}180 \ Hz {/eq} sound waves along the {eq}x {/eq} -axis. As you walk along the axis, away from the speakers, you don't hear anything even though both speakers are on.

(a) What are the three lowest possible values for {eq}d {/eq}? Assume a sound speed of {eq}340 \ m / s. {/eq}

## Sound Cancellation:

Any entity having wave nature will have its speed equivalent to the product of the wavelength of that wave times the frequency of that wave. Generally, the speed of the sound wave is observed to be approximately 340 m/s under normal temperature and pressure conditions.

## Answer and Explanation:

From the expression,

{eq}V_{sound} = \lambda \times \nu\\ \Rightarrow 340 = \lambda \times 180\\ \Rightarrow \lambda = \dfrac{340}{180}\\ \Rightarrow \lambda = 1.88\; \textrm{m} {/eq}

The condition for the destructive interference is,

{eq}d_2-d_1 = \dfrac{n \lambda}{2}\;\;\;\;n = 1,3,5...\\ \textrm{where,}\\ d_2-d_1 = \textrm{difference between sound sources}\\ \textrm{For minimum first three position for 'no sound' observation}, n = 1,3,5\\ Therefore, \\ \textrm{First lowest position,} d_2 - d_1 = \dfrac{(1)1.88}{2} = 0.94\; \textrm{m}\\ \textrm{Second lowest position,} d_2 - d_1 = \dfrac{(3)1.88}{2} = 2.82\; \textrm{m}\\ \textrm{Third lowest position,} d_2 - d_1 = \dfrac{(5)1.88}{2} = 4.7\; \textrm{m}\\ {/eq} 