Longitudinal & Transverse Waves | Physics for Grade 12 PDF Download

In mechanical waves, particles oscillate about fixed points

• A progressive wave is an oscillation that transfers energy and information
  • The substance in which the waves move through are disturbed (eg. water, air)
  • The particles of the substance oscillate about a fixed position
  • This is sometimes called a travelling wave
• There are two types of waves
  • Transverse
  • Longitudinal
• The type of wave can be determined by the direction of the oscillations in relation to the direction the wave is travelling

Transverse Waves

• A transverse wave is defined as:
  A wave in which the particles oscillate perpendicular to the direction of the wave travel (and energy transfer)
• Examples of transverse waves are:
  Electromagnetic waves e.g. radio, visible light, UV
  Vibrations on a guitar string
• Transverse waves can be shown on a rope
• Transverse waves can be polarised

Longitudinal Waves

• A longitudinal wave is defined as:
  A wave in which the particles oscillate parallel to the direction of the wave travel (and energy transfer)
• Examples of longitudinal waves are:
  • Sound waves
  • Ultrasound waves
• Longitudinal waves can be shown on a slinky spring
• Longitudinal waves cannot be polarised

Waves can be shown through vibrations in ropes or springs

General Wave Properties

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

Amplitude (A) is the maximum displacement of a particle in the wave from its equilibrium position
Diagram showing the amplitude and wavelength of a wave

• Wavelength (λ) is the distance between points on successive oscillations of the wave that are in phase
• Displacement, amplitude and wavelength are all measured in metres (m)

A wavelength on a longitudinal wave is the distance between two compressions or two rarefactions

• Period (T) or time period, is the time taken for one complete oscillation or cycle of the wave Measured in seconds (s)

Diagram showing the time period of a wave

• Frequency (f) is the number of complete oscillations or wavelengths passing a point per unit time
  Measured in Hertz (Hz) or s^-1
• Speed (v) is the distance travelled by the wave per unit time, and defined by the wave equation
  Measured in metres per second (m s^-1)

Phase

• The phase difference tells us how much a point or a wave is in front or behind another
• It is defined as:
  How far the cycle of one point is compared to another point on the same wave
• This can be found from the relative positive of the crests or troughs of two different waves of the same frequency
  • When the crests or troughs 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:
  • In phase is 360^o or 2π radians
  • In anti-phase is 180^o or π radians

Example: Plane waves on the surface of water at a particular instant are represented by the diagram below.

The waves have a frequency of 2.5 Hz. Determine: 
(a) The amplitude 
(b) The wavelength 
(c) The phase difference between points A and B

A. The Amplitude
Maximum Displacement from The Equilibrium Position
7.50 mm ÷ 2 = 3.75 mm
B. The Wavelength
Distance Between Points on Successive Oscillations of The Wave That Are in Phase
From Diagram: 25cm = 3³/<sub>4 Wavelengths

C. The Phase Difference Between Points A and B</sub>

Points A and B Have 1/2 λ Difference = 1/2 × 360^o = 180^o

Calculating Frequency

Frequency (f) is defined by the equation:

Where T is the time period, the time taken for one complete oscillation or cycle of the wave

Example: Calculate the frequency of the following wave:

Step 1: List the known quantities
Period, T = 0.28 ms = 0.28 × 10^-3 s

Step 2: Write down the frequency equation

Step 3: Substitute in the values
f = 1 ÷ (0.28 × 10^-3) = 3571.4 = 3.57 kHz

Determining Frequency from an Oscilloscope

• A Cathode-Ray Oscilloscope (CRO) is a laboratory instrument used to display, measure and analyse waveforms of electrical circuits
• The x-axis is the time-base and the y-axis is the voltage (or y-gain)
  • The time-base is important for calculating the frequency of the signal

Diagram of Cathode-Ray Oscilloscope display showing wavelength and time-base setting

• The frequency of a wave is determined from the time period of the wave
• The period can be determined from the time-base
  • This is how many seconds each division represents measured commonly in s div^-1 or s cm^-1
• Dividing the total time by the number of wavelengths will give a value for T
  • Use as many wavelengths shown on the screen as possible to reduce uncertainties
• The frequency is then determined using the equation:

Example: A cathode-ray oscilloscope (c.r.o.) is used to display the trace from a sound wave. The time-base is set at 7 µs mm^-1.

What is the frequency of the sound wave?
(a) 2.4 Hz
(b) 24 Hz
(c) 2.4 kHz
(d) 24 kHz

Ans. c

The Wave Equation

• The wave equation links the speed, frequency and wavelength of a wave
• This is relevant for both transverse and longitudinal waves

The wave equation shows that for a wave of constant speed:

• As the wavelength increases, the frequency decreases
• As the wavelength decreases, the frequency increases

The relationship between frequency and wavelength of a wave

Example: The wave in the diagram below has a speed of 340 m s^–1.

What is the wavelength of the wave?

Graphical Representations of Transverse & Longitudinal Waves

Transverse and longitudinal waves can be represented graphically

Transverse Waves

• Transverse waves show areas of crests (peaks) and troughs

Diagram of a transverse wave

• The peaks are the maximum positive displacements
• The troughs are the maximum negative displacements
• The direction of the energy transfer is perpendicular to the direction of vibration of the particles in the wave

Longitudinal Waves

• Longitudinal waves show areas of compressions and rarefactions

Diagram of a longitudinal wave

• The compressions are areas of high pressure due to particles being close together.
• The rarefactions are areas of low pressure due to the particles spread further apart
• The direction of energy transfer is parallel to the direction of vibration of the particles in the wave

Example: The graph shows how the displacement of a particle in a wave varies with time.
Which statement is correct?
(a) The wave has an amplitude of 2 cm and could be either transverse or longitudinal.
(b) The wave has an amplitude of 2 cm and has a time period of 6 s.
(c) The wave has an amplitude of 4 cm and has a time period of 4 s.
(d) The wave has an amplitude of 4 cm and must be transverse.

Ans. a
The Waves Amplitude is the Displacement from The Equilibrium Position
From The Graph, this is 2 Cm
The Graph is Displacement Against Time, Not Displacement Against Direction of Wave Travel
Therefore, The Wave Could Be Either Transverse or Longitudinal

Demonstrating Waves Using a Ripple Tank

Intensity is proportional to the amplitude squared

• Assuming there’s no absorption of the wave energy, the intensity I decreases with increasing distance from the source
• Note the intensity is proportional to 1 / r²
  • This means when the source is twice as far away, the intensity is 4 times less
• The 1 / r² relationship is known in physics as the inverse square law

Example: The intensity of a progressive wave is proportional to the square of the amplitude of the wave. It is also proportional to the square of the frequency. The variation with time t of displacement x of particles when two progressive waves Q and P pass separately through a medium are shown on the graphs.

The intensity of wave Q is I₀.What is the intensity of wave P?