Two speakers each generate sound of frequency f = 240 Hz. The speakers are in ph

Question

Two speakers each generate sound of frequency f = 240 Hz. The speakers are in phase with each other and separated by a distance of 3.0 m. Find all points along the line perpendicular to the speaker on the left at which constructive interference occurs. Express your answers as values of x, the distance from the speaker on the left. If there are only a few locations where constructive interference occurs, give the numeric values of x for all of them. Otherwise provide an equation that describes the locations.

Explanation / Answer

for constructive interference, path difference should be integral multiple of wavelength.

here frequency=240 Hz

speed of sound=343 m/s

then wavelength=speed/frequency

=1.4292 m

let at distance x from the left speaker , constructive interference occurs.

then path difference=distance from right speaker-distance from left speaker

=sqrt(x^2+3^2)-x

=sqrt(x^2+9)-x

this value should be integral multiple of wavelength

hence sqrt(x^2+9)-x=k*1.4292

==>sqrt(x^2+9)=x+1.4292*k

squaring both sides:

x^2+9=x^2+1.4292^2*k^2+2*1.4292*k*x

==>9-2.0426*k^2=2.8584*k*x

==>x=(9-2.0426*k^2)/(2.8584*k)

value of k        value of x

1           2.434

2           0.14512

3           -1.09425

as negative value of x is not possible,

only two values of x possible.

those two values are:

2.434 m

0.14512 m

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