Q37E - University Physics with Modern Physics

Question

Two loudspeakers, A and B, are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 12.0 m to the right of speaker A. The frequency of the waves emitted by each speaker is 688 Hz. You are standing between the speakers, along the line connecting them, and are at a point of constructive interference. How far must you walk toward speaker B to move to a point of destructive interference?

Step-by-Step Solution

Answer

The distance, you need to walk, is  $0.125\text{\hspace{0.17em}m}$.

Step-1: Given Data

Distance between speakers:  $12 \mathrm{m}$

Frequency emitted:  $688\text{\hspace{0.17em}Hz}$

Step-2: Concept applied

For constructive interference the path difference is an integeral multiple of wavelengths and for destructive interference the path difference is a half-integeral multiple of wavelengths.

Step-3: Calculation

The wavelength is,

$v=\lambda f$

where v is the wave velocity, f is the frequency and $\lambda$ is the wavelength of the wave.

$\begin{array}{c}\lambda =\frac{v}{f}\\ =\frac{344\text{\hspace{0.17em}m/s}}{688\text{\hspace{0.17em}Hz}}\\ =0.5\text{\hspace{0.17em}m}\end{array}$

To shift from point of constructive interference to the point of destructive interference, the path difference must change by $\lambda /2$. If you move a distance x toward speaker B, the distance to B gets shorter by $x$  and the distance to A gets longer by $x$ so the path difference changes by  $2x$.

$\begin{array}{c}2x=\frac{\lambda }{2}\\ x=\frac{\lambda }{4}\\ =\frac{0.5\text{\hspace{0.17em}m}}{4}\\ =0.125\text{\hspace{0.17em}m}\end{array}$

Hence, you must walk $0.125 \mathrm{m}$ toward speaker B to move to a point of destructive interference.