Transverse Waves
Waves are disturbances that transfer energy through a medium without a net transport of the medium itself. They occur in many contexts: ripples on water, seismic waves travelling through the Earth, sound waves in air and waves on a stretched string. A pulse is a single disturbance; a wave is a periodic, continuous succession of such pulses.
What is a transverse wave?
A transverse wave is a wave in which the particles of the medium move perpendicular to the direction of propagation of the wave. In other words, the displacement of each particle of the medium is at right angles to the direction in which the wave travels.
Activity (demonstration): Take a rope or a slinky and stretch it horizontally between two people. One person flicks the rope up and down continuously to create a train of pulses. Observe the following:
- Describe the motion you see in the rope.
- Draw the shape of the rope as pulses travel along it.
- Which direction do the pulses travel?
- Tie a small ribbon to a point on the rope to represent a particle. When the pulses pass, what happens to the ribbon?
- Draw the path traced by the ribbon (show the ribbon as a dot and use arrows to indicate its motion).
From the activity you will notice that the wave moves along the rope but each particle (the ribbon) moves only up and down and returns to its equilibrium position. Thus there are two distinct motions: the motion of the particles of the medium and the motion of the wave (the disturbance) itself.
Crests and troughs
On a transverse wave the highest points are called crests and the lowest points are called troughs.
Definition - crest: a point on the wave where the displacement of the medium is a maximum.
Definition - trough: a point on the wave where the displacement of the medium is a minimum.
Amplitude
The amplitude of a wave is the maximum displacement of a particle from its equilibrium position. It is the distance from the equilibrium line to a crest or from the equilibrium line to a trough. The usual symbol for amplitude is A. The SI unit of amplitude is the metre (m).
Activity (measurement): On a drawn transverse wave mark the equilibrium line and measure the distance from the equilibrium to several crests and troughs. You will find that these distances are equal in magnitude; that is, the distance from equilibrium to a crest equals the distance from equilibrium to a trough. That distance is the amplitude.
QUESTION
If the crest of a wave measures 2 m above the still-water mark in a harbour, what is the amplitude of the wave?
SOLUTION
The still-water mark represents the equilibrium position of the water. The crest is 2 m above equilibrium, so the amplitude A = 2 m.
Fact: A tsunami is a series of sea waves caused by an underwater earthquake, landslide or volcanic eruption. In deep ocean water tsunamis may be less than 0.3 m high and travel at speeds up to about 700 km·h-1. In shallow water near the coast they slow down and the wave height can increase dramatically, sometimes reaching tens of metres. A tsunami wave can have a wavelength as long as 100 km and a period of up to an hour.
Example (historical): The 2004 Indian Ocean tsunami was triggered by a very large undersea earthquake. Killer waves radiated from the earthquake and struck coastlines of several countries within hours, causing very large loss of life and extensive destruction.
Wavelength
The wavelength is the distance between any two adjacent points on a wave that are in phase. Commonly wavelength is measured as the distance between two successive crests or two successive troughs. The usual symbol for wavelength is the Greek letter λ (lambda). The SI unit of wavelength is the metre (m).
QUESTION
The total distance between four consecutive crests of a transverse wave is 6 m. What is the wavelength of the wave?
SOLUTION
Four consecutive crests span three wavelengths, so 3λ = 6 m and λ = 6 m / 3 = 2 m.
Points in phase
Two points on a wave are said to be in phase if they have identical displacement and motion at the same instant. Points in phase are separated by an integral number of wavelengths (1λ, 2λ, 3λ, ...). Points that are not separated by a whole number of wavelengths are out of phase.
Alternate definition: The wavelength λ is the distance between any two adjacent points that are in phase.
Period and frequency
The period T of a wave is the time taken for one complete cycle (for example, the time between two successive crests passing a fixed point). The SI unit of period is the second (s).
The frequency f of a wave is the number of complete cycles (crests or troughs) that pass a fixed point per unit time - usually measured per second. The unit of frequency is the hertz (Hz), and 1 Hz = 1 s-1.
The period and frequency are related by
f = 1 / T and T = 1 / f.
QUESTION
What is the period of a wave of frequency 10 Hz?
SOLUTION
T = 1 / f = 1 / 10 Hz = 0.1 s.
Wave speed and the wave equation
The speed v of a wave is the distance travelled by a wave disturbance per unit time. If a wave of wavelength λ takes time T to move one wavelength, then
v = distance / time = λ / T.
Using f = 1 / T we can write the fundamental wave equation
v = f · λ.
Here,
- v is the wave speed in m·s-1,
- λ is the wavelength in m,
- f is the frequency in Hz (s-1).
QUESTION
When a string is vibrated at a frequency of 10 Hz, a transverse wave of wavelength 0.25 m is produced. Determine the speed of the wave as it travels along the string.
SOLUTION
Given: f = 10 Hz, λ = 0.25 m.
Use v = f · λ.
v = (10 s-1)(0.25 m) = 2.5 m·s-1.
The wave travels at 2.5 m·s-1 along the string.
QUESTION
A cork on the surface of a swimming pool bobs up and down once every second on some ripples. The ripples have a wavelength of 20 cm. If the cork is 2 m from the edge of the pool, how long does it take a ripple passing the cork to reach the edge?
SOLUTION
Given: f = 1 Hz, λ = 20 cm = 0.20 m, distance D = 2 m.
Wave speed v = f · λ = (1 s-1)(0.20 m) = 0.20 m·s-1.
Time to travel distance D is t = D / v = 2 m / 0.20 m·s-1 = 10 s.
It takes 10 s for the ripple to reach the edge.
Exercises
- When the particles of a medium move perpendicular to the direction of the wave motion, the wave is called a .......... wave.
- A transverse wave is moving downwards. In what direction do the particles in the medium move?
- Consider the diagram below and answer the questions that follow:
- the wavelength of the wave is shown by letter ______.
- the amplitude of the wave is shown by letter ______.
- Draw 2 wavelengths of the following transverse waves on the same graph paper. Label all important values.
- Wave 1: Amplitude = 1 cm, wavelength = 3 cm.
- Wave 2: Peak to trough distance (vertical) = 3 cm, crest to crest distance (horizontal) = 5 cm.
- You are given the transverse wave below. Draw the following:
- A wave with twice the amplitude of the given wave.
- A wave with half the amplitude of the given wave.
- A wave travelling at the same speed with twice the frequency of the given wave.
- A wave travelling at the same speed with half the frequency of the given wave.
- A wave with twice the wavelength of the given wave.
- A wave with half the wavelength of the given wave.
- A wave travelling at the same speed with twice the period of the given wave.
- A wave travelling at the same speed with half the period of the given wave.
- A transverse wave travelling at the same speed with an amplitude of 5 cm has a frequency of 15 Hz. The horizontal distance from a crest to the nearest trough is measured to be 2.5 cm. Find:
- the period of the wave.
- the speed of the wave.
- A fly flaps its wings back and forth 200 times each second. Calculate the period of a wing flap.
- As the period of a wave increases, the frequency increases / decreases / does not change.
- Calculate the frequency of rotation of the second hand on a clock.
- Microwave ovens produce radiation with a frequency of 2 450 MHz (1 MHz = 106 Hz) and a wavelength of 0.122 m. What is the wave speed of the radiation?
- Study the following diagram and answer the questions:
- Identify two sets of points that are in phase.
- Identify two sets of points that are out of phase.
- Identify any two points that would indicate a wavelength.
- Tom is fishing from a pier and notices that four wave crests pass by in 8 s and estimates the distance between two successive crests is 4 m. The timing starts with the first crest and ends with the fourth. Calculate the speed of the wave.
Summary
- A wave is formed when a succession of pulses is transmitted through a medium.
- A crest is the highest point a particle of the medium reaches; a trough is the lowest point.
- In a transverse wave, particles move perpendicular to the direction of the wave.
- Amplitude A is the maximum displacement from equilibrium (unit: metre, m).
- Wavelength λ is the distance between adjacent in-phase points (unit: metre, m).
- Period T is the time for one cycle (unit: second, s). Frequency f is the number of cycles per second (unit: hertz, Hz = s-1).
- f = 1 / T and T = 1 / f.
- The wave equation is v = f · λ (equivalently v = λ / T), where v is wave speed (m·s-1).
Physical quantities and units
| Quantity | Unit name | Unit symbol |
|---|---|---|
| Amplitude (A) | metre | m |
| Wavelength (λ) | metre | m |
| Period (T) | second | s |
| Frequency (f) | hertz | Hz (s-1) |
| Wave speed (v) | metre per second | m·s-1 |
