Properties of Oscillations (DP IB Physics: SL)

• An oscillation is defined as follows:

The repetitive variation with time t of the displacement x of an object about the equilibrium position (x = 0)  

A pendulum oscillates between A and B. On a displacement-time graph, the oscillating motion of the pendulum is represented by a wave, with an amplitude equal to x₀

• Displacement (x) of a wave is the distance of a point on the wave from its equilibrium position
  • It is a vector quantity; it can be positive or negative and it is measured in metres (m)

• Period (T) or time period, is the time interval for one complete repetition and it is measured in seconds (s)
  • If the oscillations have a constant period, they are said to be isochronous

Diagram showing the time period of a wave

• Amplitude (x₀) is the maximum value of the displacement on either side of the equilibrium position is known as the amplitude of the oscillation
  • Amplitude is measured in metres (m)
• Wavelength (λ) is the length of one complete oscillation measured from the same point on two consecutive waves
  • Wavelength is measured in metres (m)

• Frequency (f) is the number of oscillations per second and it is measured in hertz (Hz)
  • Hz have the SI units of per second s⁻¹
• The frequency and the period of the oscillations are related by the following equation:

Phase & Phase Difference

• Phase is a useful way to think about waves
• The phase of a wave can be considered in terms of:
  • Wavelength
  • Degrees
  • Radians
• One complete oscillation is:
  • 1 wavelength
  • 360°
  • 2π radians

Wavelength λ and amplitude A of a travelling wave

• The phase difference between two waves is a measure of how much a point or a wave is in front or behind another
• This can be found from the relative position of the crests or troughs of two different waves of the same frequency
  • When the crests of each wave, or the troughs of each wave are aligned, the waves are in phase
  • When the crest of one wave aligns with the trough of another, they are in antiphase

• The diagram below shows the green wave leads the purple wave by ¼ λ

Two waves ¼ λ out of phase

• In contrast, the purple wave is said to lag behind the green wave by ¼ λ
• Phase difference is measured in fractions of a wavelength, degrees or radians
• The phase difference can be calculated from two different points on the same wave or the same point on two different waves
• The phase difference between two points can be described as:
  • In phase is 360^o or 2π radians
  • In anti-phase is 180^o or π radians

Step 1: Write down the frequency of the child's oscillations 

f = 0.2 Hz

Step 2: Write down the relationship between the period T and the frequency f

Step 3: Substitute the value of the frequency into the above equation and calculate the period 

T = 5 s