Sound waves from a loudspeaker spread nearly uniformly in all directions if the wavelength of the sound is much 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 cm. (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.

#### Solution

Given:

The diameter of the loudspeaker is 20 cm.

Velocity of sound in air *v* = 340 m/s

As per the question,

wavelength (*λ*) of the sound is 10 times the diameter of the loudspeaker.

∴ (*λ*) = 20 cm \[\times 10\]= 200 cm = 2 m

(a) Frequency *f* = ?

As we know, \[v = f\lambda\]

\[\therefore f = \frac{v}{\lambda} = \frac{340}{2} = 170 Hz\]

(b) Here, wavelength is one tenth of the diameter of the loudspeaker.

⇒ *λ* = 2 cm = 2 × 10^{−2} m

\[\therefore f = \frac{v}{\lambda} = \frac{340}{2 \times {10}^{- 2}} = 17, 000 \text{ Hz } = 17 \text{ kHz }\]