Q36E - University Physics with Modern Physics

Question

Two loudspeakers, A and B (see Fig. E16.35), are driven by the same amplifier and emit sinusoidal waves in phase. Speaker B is 2.00 m to the right of speaker A. The frequency of the sound waves produced by the loudspeakers is 206 Hz. Consider a point P between the speakers and along the line connecting them, a distance x to the right of A. Both speakers emit sound waves that travel directly from the speaker to point P. For what values of x will (a) destructive interference occur at P; (b) constructive interference occur at P? (c) Interference effects like those in parts (a) and (b) are almost never a factor in listening to home stereo equipment. Why not?

Step-by-Step Solution

Answer

(a)  $\mathrm{x}=0.58\text{\hspace{0.17em}m and 1.42\hspace{0.17em}m}$ are the points of destructive interference.

(b) $x=1\text{\hspace{0.17em}m},0.17\text{\hspace{0.17em}m and}1.83\text{\hspace{0.17em}m}$  are the points of constructive interference .

Step1: Formula of speed of wave

Constructive interference occurs when the difference of the distances of each source from point P is an integer number of wavelengths. The interference is destructive when this difference of path lengths is a half integer number of wavelengths.

Step2: Calculations

The wavelength is,

 $\mathrm{v}=\mathrm{\lambda f}$

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

  $\begin{array}{c}\lambda =\frac{v}{f}\\ =\frac{344\text{\hspace{0.17em}m/s}}{206\text{\hspace{0.17em}Hz}}\\ =1.67\text{\hspace{0.17em}m}\end{array}$

The value of  $x$  ranges from 0 to L, where $L=2.00m$ is the distance between the speakers.

The difference in path length is

$\begin{array}{c}\Delta l=(L-x)-x\\ =L-2x\\ x=(L-\Delta l)/2\end{array}$

 

destructive interference occurs for path difference  $\mathrm{\Delta l}=(\mathrm{n}+(1/2\left)\right)\mathrm{\lambda }$ and constructive interference for  $\mathrm{\Delta l}=\mathrm{n\lambda }$.

(a) Destructive interference:  $\mathrm{n}=0\mathrm{gives}\mathrm{\Delta l}=0.835\text{\hspace{0.17em}m}\mathrm{andx}=0.58\text{\hspace{0.17em}m}.$

And at  $\mathrm{n}=-1\text{\hspace{0.17em}gives}\mathrm{\Delta l}=-0.835\text{\hspace{0.17em}m}\mathrm{andx}=1.42\text{\hspace{0.17em}m.}$

For any other value of  $n$, P will not lie between the speakers.

(b)Constructive interference: $\mathrm{n}=0\text{\hspace{0.17em}gives}\mathrm{\Delta l}=0\text{\hspace{0.17em}m and}\mathrm{x}=1.00\text{\hspace{0.17em}m}.$ 

And at $\mathrm{n}=1\text{gives }\mathrm{\Delta l}=1.67\text{\hspace{0.17em}m and }\mathrm{x}=0.17\text{\hspace{0.17em}m}.$  

At  $\mathrm{n}=-1\text{\hspace{0.17em}gives }\mathrm{\Delta l}=-1.67\text{\hspace{0.17em}m and }\mathrm{x}=1.83\text{\hspace{0.17em}m.}$

For any other value of  $n$ , P will not lie between the speakers

(c)Treating the speakers as point sources is a poor approximation for these dimensions, and sound reaches the given points after multiple reflections from the walls, ceiling or floor.