Describing Oscillations (Cambridge (CIE) A Level Physics)

Describing oscillations

• 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)  

Pendulum oscillation on a displacement-time graph

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

• Equilibrium position (x = 0) is the position when there is no resultant force acting on an object

• 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)

     

• Amplitude (x₀) is the maximum value of the displacement on either side of the equilibrium position and 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)

Wavelength and amplitude on a displacement-time graph

Diagram of wavelength and amplitude of a wave

• Period (T) or time period, is the time interval for one complete repetition and it is measured in seconds (s)

  • Simple harmonic oscillations have a constant period

  • Time period can be calculated in terms of both frequency and angular frequency by the equations:

• Where:

  • T = Time period (s)

  • f = frequency (Hz)

  • ω = angular frequency (rad s⁻¹)

Time period on a displacement-time graph

Diagram showing the time period of a wave

• Frequency (f) is the number of oscillations per second measured in hertz (Hz)

  • Hz have the SI units of per second s⁻¹

• Angular Frequency (ω) is the rate of change of angular displacement with respect to time 

  • Angular frequency is measured in rad s⁻¹

  • It is given by the equations:

• Where:

  • ω = angular frequency (rad s⁻¹)

  • T = time period (s)

  • f = frequency (Hz)

Phase difference

• Phase is a useful way to consider wave behaviour

• The phase of a wave can be measured in terms of:

  • Fractions 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 that

  • the green wave leads the purple wave by ¼ λ

  • the purple wave lags behind the green wave by ¼ λ

A phase difference of 1/4 wavelength

Two waves ¼ λ out of phase

• Phase difference can be described as in phase or in anti-phase:

  • In phase is 360^o or 2π radians

  • In anti-phase is 180^o or π radians