Waves II

This topic deals with the superposition of moving waves. Consider two propagating moves as shown below:

\[\begin{align*}\psi_1(x,t) &= A_1 \sin(k_1x - \omega_1 t)\\\psi_2(x,t) &= A_2 \sin(k_2x - \omega_2 t)\end{align*}\]

If we consider the summation of these waves, \(\psi_3(x,t)\),

\[ \begin{align*} \psi_3(x,t) &= \psi_1(x,t) + \psi_2(x,t) \\ &= A\sin(k_1x - \omega_1t) + A\sin(k_2x-\omega_2 t) \\ &= A\left(2\sin\left(\frac{k_1x-\omega_1t + k_2x-\omega_2t}{2} \right)\cos\left(\frac{k_1x-\omega_1t - k_2x+\omega_2t}{2} \right) \right) \\ \psi_3(x,t) &= 2A\left(2\sin\left(\frac{k_1+k_2}{2}x - \frac{\omega_1+\omega_2}{2}t\right)\cos\left(\frac{k_1-k_2}{2}x - \frac{\omega_1-\omega_2}{2}t\right) \right) \\ \end{align*} \]

The result?

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