• Travelling waves?are defined as follows:

Oscillations that transfer energy from one place to another without transferring matter

• Energy is transferred by the waves, but matter is not
• The direction of the motion of the wave is the direction of the energy transfer
• Travelling waves can be of two types:
  • Mechanical Waves, which propagate through a medium and cannot take place in a vacuum
  • Electromagnetic Waves, which can travel through a vacuum

• Waves are generated by?oscillating sources
  • These oscillations travel?away?from the source

• Oscillations can propagate through a?medium?(e.g. air, water) or in?vacuum?(i.e. no particles), depending on the type of wave

• The key properties of?travelling waves?are as follows:

• 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
  • Measured in metres (m)

• Wavelength?(λ) is the length of one complete oscillation measured from same point on two consecutive waves
  • For example, two crests, or two troughs
  • Measured in metres (m)

• Amplitude?(x0) is the maximum displacement of an oscillating wave from its equilibrium position (x?= 0)
  • Amplitude can be positive or negative depending on the direction of the displacement
  • Measured in metres (m)

• Period?(T) is the time taken for a fixed point on the wave to undergo one complete oscillation
  • Measured in seconds (s)

• Frequency?(f) is the number of full oscillations per second
  • Measured in Hertz (Hz)

• Wave speed?(c)?is the distance travelled by the wave per unit time
  • Measured in metres per second (m s-1)

Diagram showing the amplitude and wavelength of a wave

• The frequency,?f, and the period,?T, of a travelling wave are related to each other by the equation:

Period T and frequency f of a travelling wave

(i) Identify the amplitude?A?of the wave on the graph

• • The amplitude is defined as the maximum displacement from the equilibrium position (x?= 0)

• • The amplitude must be converted from centimetres (cm) into metres (m)

A?= 0.1 m

(ii) Calculate the frequency of the wave

Step 1: Identify the period?T?of the wave on the graph

• • The period is defined as the time taken for one complete oscillation to occur

• • The period must be converted from milliseconds (ms) into seconds (s)

T?= 1 × 10^–3 s

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

Step 3: Substitute the value of the period determined in Step 1

f?= 1000 Hz

• The wave equation describes the relationship between the wave speed, the wavelength and the frequency of the wave

c = fλ

• Where
  • c?= wave speed in metres per second (m s^?1)
  • f?= frequency in hertz (Hz)
  • λ?= wavelength in metres (m)

Deriving the Wave Equation

• The wave equation can be derived using the equation for speed

• Where
  • v?= velocity or speed in metres per second (m s^?1)
  • d?= distance travelled in metres (m)
  • t?= time taken in seconds (s)
• When the source of a wave undergoes one complete oscillation, the travelling wave propagates forward by a?distance?equal to one wavelength?λ
• The travelling wave covers this distance in the?time?it takes the source to complete one oscillation, the time period?T
• Therefore, combining these equations gives the wave equation

c = fλ

Step 1: Write down the known quantities

• • Period,?T?= 1.0 μs = 1.0 × 10^–6?s
  • Velocity,?c?= 100 cm s^–1?= 1.0 m s^–1

Note the conversions:

• • The period must be converted from microseconds (μs) into seconds (s)
  • The velocity must be converted from cm s^–1?into?m s^–1

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

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

f?= 1.0 × 10⁶?Hz

Step 4: Write down the wave equation

c = fλ

Step 5: Rearrange the wave equation to calculate the wavelength?λ

Step 6: Substitute the numbers into the above equation

λ?= 1 × 10^–6?m