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Normal Mode Vibration

Thus, vibrations of a continuous system in general obeys, if the system vibrates in a normal mode with frequency ω, then we have

y(x, t) = A ( x ) cos ω t

A normal mode of vibration is also called a standing wave.

The above solution is the equivalent for a continuous system. Substituting this solution, we find

where k = ω/v , the equation represents simple harmonic motion in space. Its solution, in general, is

A ( x ) = A sin kx + B cos kx

The constant l represents wave number; we write k = 2π/λ where λ is called wavelength of standing wave. The value of λ gives the distance after which the changing value of A ( x ) repeats itself in space. (The amplitude variation depicts a harmonic curve in space; hence the name standing wave.)

We have still not determined the value of frequency ω of oscillations. This is done by imposing the boundary conditions on the vibrating system.