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Section 14.8 Superposition of Waves

When you play two nearby notes on the piano you hear a low frequency beat that was not there before. How could a new note develop from two other notes? Similarly, in a pond when you drop two stones separated by a distance, then at the places the two circular waves meet you find a completely new wave pattern.

Both of these phenomena and many like them result from sum of individual waves. The principle that waves add to make resulting wave is called the principle of superposition of waves. This is similar to the way you get net force by a superposition of individual force vectors.

Suppose there are \(N\) waves, say, \(\psi_1\text{,}\) \(\psi_2\text{,}\)\(\cdots\text{,}\) \(\psi_N\text{,}\) in a region of space at some time \(t\)>. Then, according to the superposition principle, their sum will give us the net wave we will observe in that region.

\begin{equation} \psi = \psi_1+\psi_2+\cdots+\psi_N.\tag{14.8.1} \end{equation}

The superposition of waves leads to some strange and unexpected phenomena. The superposition in time leads to the beating of waves and the beat phenomenon, and the superposition in space leads to interference patterns and diffraction. We will look into them briefly here. The interference and diffraction will be studied in detail in the Optics part of this book.