Beats of two harmonic waves
When two harmonic motions, with frequencies close to one another, are added, the resulting motion exhibits a phenomenon known as beats.
Sound of the sine wave can be heard by clicking the buttons Play A (first wave) and Play B (second wave). The parameters of the sine waves can be varied using the sliders (amplitude, frequency and phase). The phase can be set to zero by clicking the button 'Set phase zero'.
The sound of the wave obtained by adding the first and second waves can be heard by clicking the button Play C.
The sound of beat phenomenon can be heard when the difference between the frequencies of the two waves are very small. By clicking the button Play beat, the beat sound can be heard with default beat frequency of 12 Hz (difference between the frequencies of A and B). Nevertheless, the beat frequency can be varied with frequency sliders and the resulting beat sound can be heard by clicking the button Play C.
The scale factor for time (horizontal axis) can be varied from 1 to 5 using the slider 'Scaling factor for time'. It must be noted that 'Scaling factor' does not change the equation of the resulting wave, rather it can be used only for changing the time scale.
Sound of the sine wave can be heard by clicking the buttons Play A (first wave) and Play B (second wave). The parameters of the sine waves can be varied using the sliders (amplitude, frequency and phase). The phase can be set to zero by clicking the button 'Set phase zero'.
The sound of the wave obtained by adding the first and second waves can be heard by clicking the button Play C.
The sound of beat phenomenon can be heard when the difference between the frequencies of the two waves are very small. By clicking the button Play beat, the beat sound can be heard with default beat frequency of 12 Hz (difference between the frequencies of A and B). Nevertheless, the beat frequency can be varied with frequency sliders and the resulting beat sound can be heard by clicking the button Play C.
The scale factor for time (horizontal axis) can be varied from 1 to 5 using the slider 'Scaling factor for time'. It must be noted that 'Scaling factor' does not change the equation of the resulting wave, rather it can be used only for changing the time scale.
Link to Geogebra file (Use this link if play buttons do not work to make sounds)
Equations:
\(A=a\sin(\omega_at+\phi_a)\)
\(B=b\sin(\omega_bt+\phi_b)\)
\(C=A+B=a\sin(\omega_at+\phi_a)+b\sin(\omega_bt+\phi_b)\)
If amplitude and phase of the harmonic waves \((A\,\&\,B)\) assumed to be \(X\) and \(0\) respectively, the resulting wave \((C)\) becomes
\(C=2X\cos\bigg(\dfrac{\omega_a-\omega_b}{2}t\bigg)\sin\bigg(\dfrac{\omega_a+\omega_b}{2}t\bigg)\)
with beating frequency \(\omega_{\text{beat}}=\omega_a-\omega_b\)
Ref: S.S. Rao, "Mechanical vibrations", 6th Edition, Pearson (2017)
Equations:
\(A=a\sin(\omega_at+\phi_a)\)
\(B=b\sin(\omega_bt+\phi_b)\)
\(C=A+B=a\sin(\omega_at+\phi_a)+b\sin(\omega_bt+\phi_b)\)
If amplitude and phase of the harmonic waves \((A\,\&\,B)\) assumed to be \(X\) and \(0\) respectively, the resulting wave \((C)\) becomes
\(C=2X\cos\bigg(\dfrac{\omega_a-\omega_b}{2}t\bigg)\sin\bigg(\dfrac{\omega_a+\omega_b}{2}t\bigg)\)
with beating frequency \(\omega_{\text{beat}}=\omega_a-\omega_b\)
Ref: S.S. Rao, "Mechanical vibrations", 6th Edition, Pearson (2017)