1.2 Adding harmonic motions

Same frequency, different phase

One of the most important concepts we encounter in vibrations and waves is the principle of superposition. Lets look at a couple of cases starting withadding two motions with the same frequency but different phases. It is easiest to calculate this if you use complex notation $${\displaystyle \begin{array}{c}{x}_1=A_1{e}^{i(\omega t+{\alpha }_1)}\\ x_2=A_2{e}^{i(\omega t+{\alpha }_2)}\\ \\ x=x_1+x_2=A_1{e}^{i(\omega t+{\alpha }_1)}+A_2{e}^{i(\omega t+{\alpha }_2)}\\ x=e^{i(\omega t+{\alpha }_1)}\left[A_1+A_2{e}^{i({\alpha }_2-{\alpha }_1)}\right]\end{array}}$$ This comes up all the time in real life: For example noise canceling headphones use this technique. In headphones there is a membrane vibratingwith the frequency of the sound you are listening two. In a noise canceling headphone there is also a microphone "listening" to the noice coming fromoutside the headphone. This oscillation is inverted and then added to membrane producing the sound you listen to. The net result is a signal that containsthe desired sound and subtracts the noise resulting in quieter operation.

Different frequency

One can also consider the case of two oscillations with the same phase but differentfrequencies: $${\displaystyle \begin{array}{c}{x}_1=A_1{e}^{i({\omega }_{1}t)}\\ x_2=A_2{e}^{i({\omega }_{2}t)}\\ \\ x=x_1+x_2=A_1{e}^{i({\omega }_{1}t)}+A_2{e}^{i({\omega }_{2}t)}\\ x=e^{i({\omega }_{1}t)}\left[A_1+A_2{e}^{i({\omega }_2-{\omega }_1)t}\right]\end{array}}$$ In an acoustical system, this gives beats, which is more easily seen if we take the case where $A_1=A_2\equiv A$ , then: $${\displaystyle \begin{array}{c}x=x_1+x_2=A{e}^{i({\omega }_{1}t)}+A{e}^{i({\omega }_{2}t)}\\ =A{e}^{i\left(\frac{{\omega }_1+{\omega }_2}{2}+\frac{{\omega }_1-{\omega }_2}{2}\right)t}+A{e}^{i\left(\frac{{\omega }_1+{\omega }_2}{2}-\frac{{\omega }_1-{\omega }_2}{2}\right)t}\\ =A{e}^{i\left(\frac{{\omega }_1+{\omega }_2}{2}\right)t}\left[e^{i\left(\frac{{\omega }_1-{\omega }_2}{2}\right)t}+e^{-i\left(\frac{{\omega }_1-{\omega }_2}{2}\right)t}\right]\\ =2A{e}^{i\left(\frac{{\omega }_1+{\omega }_2}{2}\right)t}\cos \left[\left(\frac{{\omega }_1-{\omega }_2}{2}\right)t\right]\end{array}}$$ Where the last step used $$\cos \theta =\frac{e^{i\theta }+e^{-i\theta }}{2}$$ So in an acoustical system we will get a dominant sound that has the average ofthe two frequencies and and envelope of amplitude that slowly oscillates. This will be looked at more closes in the context of mechanical waves.