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Stationary Waves

Here, we describe the free, natural vibrations of a continuous and uniform system of limited length in one dimension. The familiar examples are the transverse vibrations of a fixed length of a string, or the longitudinal vibrations of air (or any other gas) in an organ pipe of fixed length.

Such stationary vibrations are called the standing waves. We know that when stationary vibrations are set up in a string or air column, each particle of the medium oscillates with same frequency and in phase. This also happens to be the characteristic of a normal mode of vibrations of a coupled system. We will now extend that concept and show that stationary vibrations are nothing but normal modes of vibrations of a continuous system.

Further, we observed that a system of two coupled oscillators has two normal modes of vibrations. We will now show that a system of N coupled oscillators exhibit N normal modes. A continuous system is, in fact, a system of N particles coupled together where N is very large – practically N ∞. Hence, in principle, a continuous system can vibrate in a large (infinite) number of normal modes, each mode characterized by a particular frequency. We are familiar with these results for a string or air column; they can oscillate in fundamental note or higher harmonics. Hence, we shall determine the normal modes of a system of N coupled oscillators and see how these results lead to results familiar to us for continuous systems (i.e. in the limit N ∞).