PX154 - F3b - beats

TLDR:
- two waves of the same amplitude with frequencies, $f_{1}$ and $f_{2}$ , such that $Δ f = | f_{1} - f_{2} |$ is small
- resultant amplitude varies with time, oscillating between constructive and destructive interference $⟹$ beat
- frequency of beat $= Δ f$
- frequency heard $= \frac{f_{1} + f_{2}}{2}$
two waves of the same speed, $c$ , with frequencies, $(ω + Δ ω)$ and $(ω - Δ ω)$ , and wave numbers, $(k + Δ k)$ and $(k - Δ k)$ , where $Δ ω$ and $Δ k$ are small compared to $ω$ and $k$
adding them together: $u_{t o t a l} = u_{1} + u_{2}$
using trig identity in equation $[3]$ :

u_{t o t a l} = 2 A \cos (k x - ω t) \cos (Δ k x - Δ ω t)

here, the two $\cos$ components are two travelling waves with frequencies $ω$ and $Δ ω$ , multiplied together

$ω = 2 π f$ , with $f = 25 H z$ , $Δ f = 1 H z$
$f_{u_{1}} = 26 H z$ ; $f_{u_{2}} = 24 H z$

a beat pattern is formed
furtherreading [onlinetonegenerator.com], videos of beats
in a complete cycle of $Δ f$ , beats are heard with a frequency of $2 Δ f$
volume of beat pattern modulated at $2 H z$
eg: two pianos at $258 H z$ , and $254 H z$
$f = 256 H z; Δ f = 2 H z$
- $f$ is the frequency heard
- $Δ f$ is the amplitude modulated at $2 Δ f = 4 H z ⟹$ beat frequency