Equation (1) describes the relationship between the displacement and time for simple harmonic motion where when time equals zero, the displacement is a maximum.
The various forms of this equation show that displacement can be calculated using the frequency, period or angular frequency of the wave.
Equation 1 (three forms):
| x = Xcos 2πf t | | x = Xcos | 2π | t | | x = Xcos ω t |
| T |
The amplitude of the wave in equation (1) is depends on X and the phase (the point in the cycle at which the wave starts) of the wave depends on the 2π
f t term (or its equivalent). At time equals 0 for a wave described by equation 1, the displacement is maximum.
A more general form of the relationship between displacement and time for simple harmonic motion is given below.
The Φ in the phase term is the adjustment that must be made to allow for describing waves where the motion starts from a position other than the point at which the displacement a maximum.
| Equation 2: x = Xcos( | 2πt | + Φ) |
| T |
In case 2 the timing of the motion starts from the point where displacement from the rest (equilibrium) position is zero.
Φ for case 2 is ±π/2
In case 3 the timing of the motion starts from the point where the displacement is a minimum.
Φ for this case is ±π.