# Two speakers are driven by the same amplifier and emit sinusoidal waves in phase. The speaker s...

## Question:

Two speakers are driven by the same amplifier and emit sinusoidal waves in phase. The speaker s are on the x axis a distance {eq}d = 2.5 0\ m {/eq} apart. The frequency of the sound waves produced by the loudspeakers is {eq}200\ Hz {/eq}. For a point on the x axis between the two speakers, at what values of x will

(a) destructive interference occur, and

b) constructive interference occur ?

(c) Can destructive or constructive interference occur at a point on the x axis to the right of both speakers or to the left of both speakers? Explain.

( Assume the temperature is {eq}86 ^\circ F (30 ^\circ C) {/eq} . )

## Patterns of field interference along with a straight line passing through two synchronized sources of harmonic radiation.

On the line segment between the radiation sources, a change in the amplitude of the sum of the waves will be observed, since the signals will arrive at different points of the segment in a different phase.

On a line outside this segment, the phase difference from the two sources will remain constant and there will be no change in the amplitude of the sum of the oscillations.

The amplitude of the sum of two coherent waves has the following form:

{eq}A(x)=2A_0|cos(\pi/\lambda*(|x-x_1|-|x-x_2|))| {/eq} (1)

{eq}A_0 -\text{source oscillation amplitude}, \lambda=v/f=(348.9/200)m=1.74m -\text{wavelength} {/eq}

v=348.9m/s -sound's speed at a given temperature.

For simplicity, let the signal sources be located symmetrically with respect to the origin:

{eq}x_1=-d/2, x_2=d/2 {/eq}

For -d/2<x<d/2 expression (1) will take the following form

{eq}A(x)=2*A_0*|cos(2\pi*x/\lambda)| {/eq} (2)

(a)In accordance with formula (2)

Destructive interference in this segment (-d/2,d/2) will be only in the two points

{eq}x=\lambda/4=0.435m, x=-0.435m {/eq}

b) Contractive interference in this segment occurs only in the three points:

{eq}x=0, x=\lambda/2=0.87m, x=-0.87m {/eq}

c)

Expression (1) outside the specified segment (-d/2,d/2) will take the form

{eq}A(x)=2A_0*|cos(\pi*d/\lambda)| {/eq} (3)

Amplitude in the expression (3) doesn't depend on x and equal to a constant.

It means there is no destructive or constructive interference to the right and to the left sides of the interval limited by speakers. 