Answer to Question #164200 in Optics for wqae

Question

  Explain how audible beats arise when two tuning forks of slightly different frequencies f and fo are sounded together. Derive the expression for the number of beats heard per second.

  Briefly describe an experiment using beats to determine f if fo is known.

Answer 1

Wave interference is the phenomenon that occurs when two when two tuning forks of slightly different frequencies f and$f_o$ are sounded together.

Consider sound from two different sources whose medium is air is meeting at a point P. Let us assume that one source has a shorter time period as T_S and high frequency as $f$ whereas other source time periods and frequency are T_L and $f_o$ . With the help of these frequencies, we represent beat frequency by showing the relationship between these two frequencies.

Now, after an amount of time T_S  passes, the next crest from the shorter-period source arrives, the corresponding crest from the longer-period source won’t arrive for an amount of time 

$\Delta T =T_L − T_S$

​ In fact, with the arrival of each successive short-period crest, the corresponding long-period crest is another $\Delta T$ behind. Eventually, after some number n of short periods, the long-period crest will arrive a full long period TL after the corresponding short period crest arrives.

$n\times ΔT=T_L$ ​ - (1)

This means that as the short-period crest arrives, the long-period crest that precedes the corresponding long-period crest is arriving. This results in constructive interference (loud sound). The time it takes, starting when the interference is maximally constructive, for the interference to become maximally constructive is the beat period.

$T_{beat} ​=n×T_s$ – (2)

From 1 & 2, We get,

$T_{beat} ​= \dfrac{T_L}{\Delta T}T_s$

Here,  $\Delta T = T_L – T_S$  

$ΔT=T_L​ –T_S$

$T_{beat} = \dfrac{T_LT_S}{T_L - T_S}$

$T_{beat} = \dfrac{1}{T_S} - \dfrac{1}{T_L}$

Now, by using the relation between $f = \dfrac{1}{T}$

$f_{beat} = f - f_o ​$

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