Question

Two loudspeakers emit sound waves along the x-axis. The sound has maximum intensity when the speakers are 19 cm apart. The sound intensity decreases as the distance between the speakers is increased, reaching zero at a separation of 29 cm.

What is the wavelength of the sound?
Express your answer using two significant figures.

If the distance between the speakers continues to increase, at what separation will the sound intensity again be a maximum?
Express your answer using two significant figures.

Answer 1

Concepts and reason

The concepts used to solve this problem are weave length, the intensity of sound.

First calculate the wavelength of the sound after that calculate the separation where the intensity will maximum again.

Fundamentals

The wavelength is defined as the distance between two successive crests of wave.

The intensity of sound wave is defined as the power per square meter of the wave.

(A)

Calculate the wavelength of the sound wave.

The separation between the speakers is $19{\rm{ cm}}$leads maximum intensity, but separation $29{\rm{ cm}}$leads to zero intensity.

The waves are in phase $\Delta {x_1}{\rm{ = }}19{\rm{ cm}}$.

And out of the phase when $\Delta {x_2} = 29{\rm{ cm}}$.

Thus, the wavelength is,

$\Delta {x_2} - \Delta {x_1} = \frac{\lambda }{2}$

Rearrange for $\lambda$.

$\lambda = 2\left( {\Delta {x_2} - \Delta {x_1}} \right)$

Substitute $29{\rm{ cm}}$ for $\Delta {x_2}$ and $19{\rm{ cm}}$ for $\Delta {x_1}$.

$\begin{array}{c}\\\lambda = 2\left( {29{\rm{ cm}} - 19{\rm{ cm}}} \right)\\\\ = 2\left( {{\rm{10 cm}}} \right)\\\\ = 20{\rm{ cm}}\\\end{array}$

(B)

Calculate the separation where the sound intensity again be maximum.

As the distance between the speaker continue to increase the intensity will again be maximum when the separation between the speakers that produce a maximum has increased by one wavelength.

So,

$x = \Delta {x_1} + \lambda$

Substitute $19{\rm{ cm}}$for $\Delta {x_1}$, and $20{\rm{ cm}}$for $\lambda$.

$\begin{array}{c}\\x = \left( {19{\rm{ cm}}} \right) + \left( {20{\rm{ cm}}} \right)\\\\ = 39{\rm{ cm}}\\\end{array}$

Ans: Part A

The wavelength of the sound is$20{\rm{ cm}}$.

Part B

The separation where the sound intensity again be maximum is $39{\rm{ cm}}$.