Frequency of Beats
Theorem
Let $W_1$ and $W_2$ be harmonic waves whose frequencies are $f_1$ and $f_2$.
Let the superpositon of $W_1$ onto $W_2$ exhibit the phenomenon of beats.
The frequency $f_b$ of those beats is:
- $f_b = \dfrac {\size {f_2 - f_1} } 2$
Proof
Let $\omega_1$ and $\omega_2$ denote the angular frequency of $W_1$ and $W_2$ respectively.
Let us consider the harmonic waves that are $W_1$ and $W_2$ as they disturb the medium at $x = 0$.
Without loss of generality, therefore, let $W_1$ and $W_2$ be be expressed as:
| \(\ds \map {\phi_1} t\) | \(=\) | \(\ds \sin \omega_1 t\) | ||||||||||||
| \(\ds \map {\phi_2} t\) | \(=\) | \(\ds \sin \omega_2 t\) |
where:
| \(\ds \omega_1\) | \(=\) | \(\ds 2 \pi f_1\) | Definition of Angular Frequency | |||||||||||
| \(\ds \omega_2\) | \(=\) | \(\ds 2 \pi f_2\) |
Then:
| \(\ds \map {\phi_1} t + \map {\phi_2} t\) | \(=\) | \(\ds \sin \omega_1 t + \sin \omega_2 t\) | superimposing the two waves | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map \sin {\dfrac {\omega_1 t + \omega_2 t} 2} \map \cos {\dfrac {\omega_1 t - \omega_2 t} 2}\) | Sine plus Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map \sin {\dfrac {\paren {\omega_1 + \omega_2} t} 2} \map \cos {\dfrac {\paren {\omega_1 - \omega_2} t} 2}\) | simplifying | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \map \sin {\dfrac {\paren {\omega_1 + \omega_2} t} 2} \map \cos {\dfrac {\size {\omega_1 - \omega_2} t} 2}\) | Cosine Function is Even |
Hence the superimposition of the two waves is equivalent to forming the product of the waves of frequency $\dfrac {\omega_1 + \omega_2} 2$ and $\dfrac {\size {\omega_1 - \omega_2} } 2$.
In the above diagram, there are $3$ waves superimposed:
- $(1) \quad \pm 2 \map \cos {\dfrac {\size {\omega_1 - \omega_2} t} 2}$ in red
- $(2) \quad 2 \map \sin {\dfrac {\paren {\omega_1 + \omega_2} t} 2}$ in green
- $(3) \quad 2 \map \sin {\dfrac {\paren {\omega_1 + \omega_2} t} 2} \map \cos {\dfrac {\size {\omega_1 - \omega_2} t} 2}$ in blue
Hence we see:
| \(\ds \size {\map {\phi_1} t + \map {\phi_2} t}\) | \(\le\) | \(\ds \size {2 \map \cos {\dfrac {\size {\omega_1 - \omega_2} t} 2} }\) | as $\map \sin {\dfrac {\paren {\omega_1 + \omega_2} t} 2} \le 1$ throughout |
Hence the angular frequency $\omega_b$ of the slower component is seen to be:
- $\omega_b = \dfrac {\size {\omega_1 - \omega_2} } 2$
By definition of angular frequency:
| \(\ds f_b\) | \(=\) | \(\ds \dfrac {\omega_b} {2 \pi}\) | Definition of Angular Frequency | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 {2 \pi} \dfrac {\size {\omega_1 - \omega_2} } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\size {\frac {\omega_1} {2 \pi} - \frac {\omega_2} {2 \pi} } } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\size {f_1 - f_2} } 2\) |
Hence the frequency of the slower component is seen to be:
- $\dfrac {\size {f_1 - f_2} } 2$
The result is apparent.
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): beats
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): beats