Two speakers are arranged as shown in figure 8.24. The distance between them is 3 m and they emit a constant tome of 344 Hz. A microphone P is moved along a line parallel to and 4.00 m from the line connecting the two speakers. It is found that the tone of maximum loudness is heard and displayed on the CRO when the microphone is in the center of the line and directly opposite each speaker. Calculate the speed of sound. (Ans: 344 m s⁻¹)




Construction:


ΔS₁S₂P


Data Given:


Distance between speakers S₁ and S₂ = `\bar {S₁S₂}` = 3 m

Distance between speaker S₂ and point P  = `\bar {S₂}` P= 4 m

Frequency = f = 344 Hz


To Find:


Speed of Sound  = v = ?


Solution:

The general formula to find the speed of a sound is 

v = f λ  -----------(1)

But here we do not know the wavelength Î». So first we have to find the wavelength Î».

For this, we need constructive interference ie. producing a large sound at point P, and for constructive interference, the path difference is integral multiples of wavelength Î»  so

`\bar {S₁P}` - `\bar {S₂P}` =  Î»  ----------(2)

Using Pythagoras' Theorem to find `\bar {S₁P}` 

from ΔS₁S₂P

(`\bar {S₁P}`)² = (`\bar {S₁S₂}`)² + (`\bar {S₂P}`)²

(`\bar {S₁P}`)² = (3m)² + (4 m)²

(`\bar {S₁P}`)² = 9m² + 16 m²

(`\bar {S₁P}`)² = 25 m² 

By taking square root on both sides

`\bar {S₁P}` = 25 m² 

Now equation (2) 

5 m - 4m =  Î»

or

λ = 1 m -----------(3)

Now using equation (1)

v = f λ

putting the corresponding values

v = 344 Hz ï½˜ 1 m 

or

v = 344 m s⁻¹ ---------------Ans.







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