Two speakers that are 15.0 m apart produce in-phase sound waves of frequency 250.0 Hz in a room where the speed of sound is 340.0 m/s. A woman starts out at the midpoint between the two speakers. The room’s walls and ceiling are covered with absorbers to eliminate reflections, and she listens with only one ear for best precision. (a) What does she hear: constructive or

destructive interference? Why? (b) She now walks slowly toward one of the speakers. How far from the centre must she walk before she first hears the sound reach a minimum intensity? (c) How far from the centre must she walk before she first hears the sound maximally enhanced?

Distance between speakers is-$15.0\mathrm{m}$

Frequency produced-$250.0\mathrm{Hz}$

Speed of sound-$340\mathrm{m}/\mathrm{s}$

The sound is reinforced to its maximum when the path difference $\left({\mathrm{r}}_2-{\mathrm{r}}_1\right)$ between the sound waves is an integral multiple of the wavelength$\left(\lambda \right)$. This is also known as Constructive interference. The condition is,

${\mathrm{r}}_2-{\mathrm{r}}_1=\mathrm{m\lambda },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{m}=0,\pm 1,\pm 2,...$ ...(i)

The sound is canceled when the path difference between the sound waves is a multiple of the odd number of half wavelengths. This is also known as destructive interference. The condition is,

${\mathrm{r}}_2-{\mathrm{r}}_1=(\mathrm{m}+1/2)\mathrm{\lambda },\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{m}=0,\hspace{0.33em}\pm 1,\hspace{0.33em}\pm 2,...$ ...(ii)

When the listener is standing in the middle, there is zero path difference which makes the interference constructive.

The wavelength is calculated as follows,

$$\begin{gathered} \mathrm{\lambda }=\frac{\mathrm{V}}{\mathrm{f}} \\ =\frac{340.0\mathrm{m}/\mathrm{s}}{250.0\mathrm{Hz}} \\ =1.36\mathrm{m} \end{gathered}$$

Minimum intensity refers to destructive interference. Thus, according to equation (ii) the path difference is,

$$\begin{gathered} 2\mathrm{d}=\frac{\mathrm{\lambda }}{2} \\ \mathrm{d}=\frac{\mathrm{\lambda }}{4} \\ =\frac{1.36\mathrm{m}}{4} \\ =34.0\mathrm{cm} \end{gathered}$$

Therefore, the distance is $34.0\mathrm{cm}$.

Maximum intensity refers to constructive interference. Thus, according to equation (i), the path difference is,

$$\begin{gathered} 2\mathrm{d}=\lambda \\ \mathrm{d}=\frac{\mathrm{\lambda }}{4} \\ =\frac{1.36\mathrm{m}}{4} \\ =68.0\mathrm{cm} \end{gathered}$$

Therefore, the distance traveled to hear maximum intensity is$68.0\mathrm{cm}$.