Phase Angles in Simple Harmonic Motion (SHM) (HL) (HL IB Physics)

• Two points on a sine wave, or on different waves, are in phase when they are at the same point in their wave cycle
  • The angle between their wave cycles is known as the phase angle

The relationship between a sine wave and phase angle

• If an oscillation does not start from the equilibrium position, then it will be out of phase by an angle of
  • This would be compared to an oscillation which does start from the equilibrium position

• The phase angle of an oscillation (in SHM) is defined as

The difference in angular displacement compared to an oscillator which has a displacement of zero initially (i.e. when )

• The phase angle can vary anywhere from 0 to 2π radians, i.e. one complete cycle
• With the inclusion of the phase angle , the displacement, velocity and acceleration SHM equations become:

• If two bodies in simple harmonic motion oscillate with the same frequency and amplitude, but are out of phase by , then:
  • The displacement of the oscillator starting from the equilibrium position is represented by the equation

• • The displacement of the oscillator which leads by  is represented by the equation

Two oscillators which are out of phase by . The blue-dotted wave represents an oscillator starting from the equilibrium position and the red wave represents an oscillator leading by

• When a sine wave leads by a phase angle of , this is equivalent to the cosine of the wave

• Alternatively, a sine wave can be described as a cosine wave that lags by 

• Notice: 
  • For a wave that lags the phase difference is 
  • For a wave that leads the phase difference is 
• This is the opposite sign to the one you might think.
  • To review this concept, use the notes on the Transformation of Trigonometric Functions

How are sine and cosine functions related?

Sine and cosine functions are simply out of phase by  radians

• The general rules for phase shifts of sine and cosine functions are shown in the table below

Graph	Equation	Phase shift
		Shifts by  to the right (positive direction)
		Shifts by  to the left (negative direction)
		Shifts by  to the right (positive direction)
		Shifts by  to the left (negative direction)