Sound waves from the loudspeaker spread fairly equally in all directions, if the wavelength of the sound is substantially larger than the diameter of the loudspeaker,
(a) Calculate the frequency for which the wavelength of sound in air is ten times the diameter of the speaker if the diameter is $20 c m$ .
(b) Sound is essentially transmitted in the forward direction if the wavelength is much shorter than the diameter of the speaker. Calculate the frequency at which the wavelength of the sound is one-tenth of the diameter of the speaker described above. Take the speed of sound to be $340 m s^{- 1}$ .

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Solution

Step 1: Given:
The loudspeaker's diameter is 10 times its wavelength.
Diameter $20 cm$
wavelength $\begin{array}{rcl} (λ) & = & 20 x 10 = 200 c m o r 2 m \end{array}$
Speed of sound $340 m s^{- 1}$
Step 2: Formula Used:
$\begin{array}{rcl} f r e q u e n c y (f) & = & \frac{v e l o c i t y (v)}{w a v e l e n g t h (λ)} \end{array}$
Step 3: Calculating the frequency:
Part (a)
We know that,
$\begin{array}{rcl} f r e q u e n c y (f) & = & \frac{v e l o c i t y (v)}{w a v e l e n g t h (λ)} \\ = & \frac{340}{2} \\ = & 170 H z \end{array}$
Hence, the frequency for which the wavelength of sound in air is ten times the diameter of the speaker if the diameter is $20 c m$ is $170 H z .$
Step 4: Calculating the frequency
Part (b)
It is given that, now the wavelength of the sound is one-tenth of the diameter of the speaker
$\begin{array}{rcl} λ & = & \frac{20}{10} \\ = & 2 c m \\ = & \frac{2}{100} m \\ = & 2 \times 10^{- 2} m \end{array}$
$\begin{array}{rcl} f r e q u e n c y (f) & = & \frac{v e l o c i t y (v)}{w a v e l e n g t h (λ)} \\ = & \frac{340}{2 \times 10^{- 2}} \\ = & 17000 H z \\ o r \\ = & 17 k H z \end{array}$
Hence, the frequency at which the wavelength of the sound is one-tenth of the diameter of the speaker is $17 k H z .$

Thus, the frequency for which the wavelength of sound in air is ten times the diameter of the speaker is $170 H z$ and the frequency at which the wavelength of the sound is one-tenth of the diameter of the speaker is $17 k H z .$

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