Question

Imagine you are in an open field where two loudspeakers are set up and connected to the same amplifier so that they emit sound waves in phase at 688 Hz. Take the speed of sound in air to be 344 m/s. If you are 3.00 m from speaker A directly to your right and 3.50 m from speaker B directly to your left. What is the shortest distance d you need to walk forward to be at a point where you cannot hear the speakers?

Answer 1

Concepts and reason

The concept required to solve the given problem is interference of sound waves.

Initially calculate the wavelength of sound wave by using the relationship between the speed, frequency and wavelength. Finally, calculate the shortest distance where speaker voice cannot be heard by using the equation of path difference.

Fundamentals

Wave interference- It is the phenomenon that occurs when two waves superimposed while traveling along the same medium.

Constructive interference: If the amplitude of sound waves is superimposed in such a manner that their amplitudes is add up, then the interference is known as constructive interference. In constructive interference, the waves are in phase with each other.

Destructive interference: If the amplitude of sound waves is superimposed in such a manner that their amplitudes are out of phase, then the interference is known as destructive interference.

The wavelength of sound wave is,

$\lambda = \frac{{{\rm{Speed of sound wave}}}}{{{\rm{frequency}}}}$ …… (1)

Substitute 344 m/s for speed of sound wave and 688 Hz for frequency of sound wave in equation (1).

$\begin{array}{c}\\\lambda = \frac{{{\rm{Speed of sound wave}}}}{{{\rm{frequency}}}}\\\\ = \frac{{344{\rm{ m/s}}}}{{688{\rm{ Hz}}}}\\\\ = \left( {\frac{{344{\rm{ m/s}}}}{{688{\rm{ Hz}}}}} \right)\left( {\frac{{1\;{\rm{Hz}}}}{{1/{\rm{s}}}}} \right)\\\\ = 0.5{\rm{ m}}\\\end{array}$

The figure shows the relationship between source and observer.

By using Pythagoras, from the figure ABC,

$\begin{array}{l}\\{r_R} = {\left( {x_R^2 + {d^2}} \right)^{1/2}}\\\\{r_L} = {\left( {x_L^2 + {d^2}} \right)^{1/2}}\\\end{array}$

Here, ${x_L}$ and ${x_R}$ are the position of observer from speaker A and B respectively.

If the path difference is an odd multiple of wavelength then the destructive interference occurs, and if the path difference is an even multiple of wavelength then constructive interference occurs.

The path difference is,

${r_R} - {r_L} = \Delta r$

For destructive interference the path difference is,

$\begin{array}{c}\\{r_R} - {r_L} = \Delta r\\\\{r_R} - {r_L} = \frac{1}{2}\lambda \\\end{array}$

Substitute ${\left( {x_R^2 + {d^2}} \right)^{1/2}}$ for ${r_R}$ and ${\left( {x_L^2 + {d^2}} \right)^{1/2}}$ for ${r_L}$ in ${r_R} - {r_L} = \frac{1}{2}\lambda$ .

${\left( {x_R^2 + {d^2}} \right)^{1/2}} - {\left( {x_L^2 + {d^2}} \right)^{1/2}} = \frac{1}{2}\lambda$

Substitute 3.5 m for ${x_r}$ , 3.0 m for ${x_L}$ in ${\left( {x_R^2 + {d^2}} \right)^{1/2}} - {\left( {x_L^2 + {d^2}} \right)^{1/2}} = \frac{1}{2}\lambda$ .

$\begin{array}{l}\\\left[ {{{\left( {{{\left( {3.5{\rm{ m}}} \right)}^2} + {d^2}} \right)}^{1/2}}} \right] - \left[ {{{\left( {3.0{\rm{ m}}} \right)}^2} + {d^2}} \right] = \frac{1}{2}\left( {0.5{\rm{ m}}} \right)\\\\{\left( {12.25{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)^{1/2}} - {\left( {9{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)^{1/2}} = 0.25{\rm{ m}}\\\end{array}$

Squaring on both sides,

$\begin{array}{c}\\\left( {12.25{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right) + \left( {9{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right) - 2\sqrt {\left( {12.25{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)\left( {9{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)} = 0.0625{\rm{ }}{{\rm{m}}^{\rm{2}}}\\\\2{d^2} + 21.25{\rm{ }}{{\rm{m}}^{\rm{2}}} - 2\sqrt {\left( {12.25{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)\left( {9{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)} = 0.0625{\rm{ }}{{\rm{m}}^{\rm{2}}}\\\\2{d^2} + 21.19{\rm{ }}{{\rm{m}}^{\rm{2}}} = 2\sqrt {\left( {12.25{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)\left( {9{\rm{ }}{{\rm{m}}^{\rm{2}}} + {d^2}} \right)} \\\\d = 5.62{\rm{ m}}\\\end{array}$

Thus, the shortest distance is 5.62 m.

Ans:

The shortest distance where loud speaker voice cannot be heard is 5.62 m.