Question

Derive the value of BEAT FREQUENCY due to superimposition of waves.

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$\star$ Revision_Q18​

Answer 1

$\sf{\green{\underline{\underline{\orange{Answer :-}}}}}$

Basically we have the function

$\tt f(t)=sin(2πν1t)+sin(2πν2t)+sin(2πν3t)$

Let T be the fundamental period of f(t).

Beat frequency is defined as the number of peaks in intensity per unit time for the resultant wave ....

Which means

$\tt f′(t)=0=2πν1cos(2πν1t)+2πν2cos(2πν2t)+2πν3cos(2πν3t)$

This equation is also periodic with period T.

So beat frequency can be defined as the number of roots of f′(t)=0 where

$t∈[0,T)$ divided by T.

Is there any simple way of calculating this? (I don't need a formula ... A simple algorithm will also do)

Does it help if all three frequencies are natural numbers (because that's usually the case when dealing with tuning fork related problems)?

I tried considering three vectors of lengths ν1,ν2,ν3. The vectors are rotating with angular speeds

$\star\tt2πν1,2πν2,2πν3.$

If the frequencies are natural numbers I think

$\tt\star T=1/gcd(ν1,ν2,ν3)$

$\tt HOPE\; IT \;HELPED$

Answer 2

Answer:

• The value of beat frequency (ν) is f₁ - f₂

Explanation:

Numbers of beat per second is called Beat frequency.

Now, let's consider two waves of nearly equal angular frequencies ω₁ and ω₂ respectively. And let's consider that the phase difference is zero, and both have same amplitude.

Then at x = 0 the two waves can be given by, S₁ = A cosω₁ t and S₂ = A cosω₂t .

(Here S₁ & S₂ are longitudinal displacement)

Now, according to the principle of super position.

⇒ S = S₁ + S₂

⇒ S = A cosω₁ t + A cosω₂t

Now from the identity.

• cos A + cos B = 2 cos (A + B / 2) cos (A - B / 2)

⇒ S  = 2A cos (ω₁ + ω₂ / 2)t cos (ω₁ - ω₂ / 2)t

let's assume

• ωₓ = ω₁ + ω₂ / 2
• ωₐ = ω₁ - ω₂ / 2

⇒ S = 2A cosωₓ t cos ωₐ t

So, the resultant has a cosine function whose amplitude is 2A cosωₓ t . The amplitude is maximum whenever cosωₓ t has the value ± 1 (which happens twice).

Therefore, Angular frequency of beat

⇒ ω_{beat} = 2 ωₐ

⇒ ω_{beat} = 2 × (ω₁ - ω₂ / 2)

⇒ ω_{beat} = ω₁ - ω₂

As we know ω = 2πf, therefore,

⇒ 2πν = 2πf₁ - 2πf₂

⇒ ν = f₁ - f₂

∴ The value of beat frequency (ν) is f₁ - f₂.