General
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Unit 4: PROPAGATION OF MECHANICAL WAVES
Unit 4: PROPAGATION OF MECHANICAL WAVES
Topic Area: OSCILLATIONS AND WAVES
Sub-Topic Area: Waves
Key unit competence: By the end of the unit I should be able to evaluate the propagation of mechanical waves.
Unit Objectives:
By the end of this unit learners will be able to;
◊ Explain the terms, concept and characteristics of waves properly.
◊ Explain the properties of waves.
◊ Explain the behavior of waves in vibrating strings and applications of waves properly.
4.0 INTRODUCTION
When we think of the word “wave”, we usually visualize someone moving his hand back and forth to say ‘hello’ or maybe we think of a tall curling wall of water moving in from the ocean to crash on the beach.
In physics, a wave is a disturbance that occurs in a material medium and in such process, energy is transferred from one place to another. When studying waves, it’s important to remember that they transfer energy, not matter.
There are lots of waves all around us in everyday life. Sound is a type of wave that moves through matter and then vibrates our eardrums and we hear. Light is a special kind of wave that is made up of photons that helps us to see. You can drop a rock into a pond and see wave formation in the water. We even use waves (microwaves) to cook our food really fast. Application of this concept is extensively used in telecommunication and music.
Classroom demonstration
(a) Arrange the students in the form of a circle with their right shoulders pointing towards the centre.
(b) Ask one student to raise arms and then lower them. Then the next student raises arms and lowers them, and so on around the circle. It should be like the “wave” in a football stadium.
(c) After the students have the hang of it, ask them what the disturbance in the wave was.
(d) Ask them if the disturbance travels up and down or horizontally around the circle.
(e) Let one student gently push the back of the next student and then the pushed student should gently push the next student and so on, which will make a wave travel around the ring.
(f) Ask students: What is the disturbance? Is the disturbance travelling up and down or around the ring? Which way does the wave travel? Because this disturbance travels in the same direction as the wave, it is a longitudinal wave.
4.1 THE CONCEPT OF WAVES
Waves can be defined as a disturbance in a medium that transfers energy from one place to another, although the medium itself does not travel.
The term wave is often intuitively understood as referring to a transport of spatial disturbances that are generally not accompanied by a motion of the medium occupying this space as a whole. In a wave, the energy of a vibration is moving away from the source in the form of a disturbance within the surrounding medium.
Other properties, however, although usually described in terms of origin, may be generalized to all waves. For such reasons, wave theory represents a particular branch of physics that is concerned with the properties of wave processes independently of their physical origin.
4.2 TERMS USED AND CHARACTERISTICS OF WAVES
All waves are characterized by the following terms;
The Time period (T) of the wave is the time it takes for one wavelength of the wave to pass a point in space or the time for one cycle to occur. It is also defined as the time taken between two successive wave crests or trough. It is measured in seconds (s).
The frequency (f) is the number of wavelengths that pass a point in space per second. In another words, it can be defined as the number of complete oscillations or vibrations per second. Its SI unit is hertz (Hz). Mathematically;
The wavelength
is the horizontal distance in space between two nearest points that are oscillating in phase (in step) or the spatial distance over which the wave makes one complete oscillation. Its SI unit is metre (m). The wave speed
is the speed at which the wave advances. Its SI unit is m/s. 
That is, wave speed = wavelength × frequency.
This is the relationship between wavelength, frequency and velocity.
Amplitude is defined as the maximum distance measured from equilibrium position (mean position). The amplitude is always taken as positive and is measured in metres.
Phase difference (phase angle) is the angular difference between two points on the wave or between two waves. Consider, two points O and P on the wave as shown in Fig. 4-4.
Phase difference is a whole number and is calculated using simple proportions;
The wave number, also called the propagation number k, is the spatial frequency of a wave, either in cycles per unit distance or radians per unit distance. It can be envisaged as the number of waves that exist over a specified distance (analogous to frequency being the number of cycles or radians per unit time). Its unit is per metre (m–1). Mathematically;
The Intensity (I) of a wave or the power radiated by a source are proportional to the square of the amplitude (x).
Wavefront is a line or surface in the path of the wave motion on which the disturbance at every point have the same phase. This can also be defined as the surface which touches all the wavelets from the secondary sources of waves. Consider the Huygens construction principle for the new wavefront.
4.3 TYPES OF WAVES
There are two types of waves called mechanical waves and electromagnetic waves. These waves are classified based on conditions necessary for the wave to propagate.
4.3.1 Mechanical waves
These waves are produced by the disturbance in a material medium and they are transferred by particles of the medium. These waves include waves in strings, water waves and sound waves. Mechanical waves are classified as progressive or standing waves.
4.3.1.1 Progressive waves
A progressive wave is also called a travelling wave which consists of a disturbance moving from one point to another. As a result, energy is transferred between points. Progressive mechanical waves can be categorised according to the direction of the effect of the disturbance relative to the direction of travel. Progressive waves are classified as longitudinal and transverse waves.
4.3.1.1.1 Longitudinal waves
When a wave propagates through some medium and the local displacements of the medium that constitute the disturbance are in the direction of travel of the disturbance, then the wave is longitudinal.
An example of a longitudinal wave is the pulse that can be sent along a stretched slinky by shaking one end of the slinky along its length. The pulse moves along the line of the slinky and ultimately makes the other end move. Notice that in this case, the individual coils of the slinky vibrate back and forth about some equilibrium position, but there is no net movement of the slinky itself.
4.3.1.1.2 Transverse waves
These are waves in which the direction of disturbance is perpendicular to the direction of travel of the wave. The particles do not move along with the wave; they simply oscillate up and down about their individual equilibrium positions as the wave passes by.
4.3.1.1.3 Equation of a progressive wave
An equation can performed to represent displacement yof a vibrating particle in a medium in which a wave passes. Suppose a wave moves from left to right and that a particle at the origin moves with displacement given by equation.
EXAMPLE 1
A travelling wave is described by the equation y(x, t) = 0.003 cos (20x + 200t) where y and x are measured in meters and t in seconds. What is the direction in which the wave is travelling? Calculate the following physical quantities:
(a) angular wave number
(b) wavelength
(c) angular frequency
(d) frequency
(e) time period
(f) wave speed
(g) amplitude
particle velocity when x = 0.3 m and t = 0.02 s
(i) particle acceleration when x = 0.3 m and t = 0.02 s
4.3.1.2 Principle of superposition
The displacement at any time due to any number of waves meeting simultaneously at a point in a medium is the vector sum of the individual displacements of each one of the waves at that point at the same time.
This means that when two waves travel in a medium, their combined effect at any point can be determined using this principle. Consider two waves of displacements y1 and y2 passing through the same medium. The resultant displacement after superposition is:
Here, A denotes the antinodes and N denotes the nodes.
Mathematical treatment of standing waves
Consider the displacement y1 = a sin (wt – F) of a progressive sinusoidal wave at time t and at a distance x from the origin and moving to right.
Consider also the displacement y2 of an identical wave travelling in opposite direction given by;
If these waves are superposed, the resultant displacement y is given by;
The only variable part of equation 4.18 is sin wt. This means that the amplitude of the resultant displacement is given by equation
Position of antinodes
Antinodes are points of maximum displacements. So, antinodes are obtained when the value of Equation 4.19 is maximum. This occurs when;
EXAMPLE 2
4.3.1.4 Examples of mechanical waves
Mechanical waves, being progressive and stationary, are seen in different forms as described in this section.
Sound waves
Water waves
Ocean waves
These waves are longitudinal waves that are observed moving through the bulk of liquids, such as our oceans. Ocean waves are powerful forces that erode and shape of the world’s coastlines. Most of them are created by the wind. Winds that blow over the top of the ocean, create friction between the air and water molecules, resulting in a frictional drag as waves on the surface of the ocean.
Earthquake waves
Earthquakes occur when elastic energy is accumulated slowly within the Earth’s crust (as a result of plate motions) and then released suddenly along fractures in the crust called faults. Earthquake waves are also called seismic waves and actually travel as both transverse and longitudinal waves.
The P waves (Primary waves or compressional waves) in an earthquake are examples of longitudinal waves. The P waves travel with the fastest velocity and are the first to arrive.
The S waves (Secondary waves or shear waves) in an earthquake are examples of transverse waves. S waves propagate with a velocity slower than P waves, arriving several seconds later.
Body Waves
Body waves are of two types: compressional or primary (P) waves which are longitudinal in nature and shear or secondary (S) waves which are transverse in nature. P- and S- waves are called ‘body waves’ because they can travel through the interior of a body, such as the Earth’s inner layers, from the focus of an earthquake to distant points on the surface. The Earth’s molten core are only travelled by compressional waves.
4.3.2 Electromagnetic waves
These waves consist of disturbances in the form of varying electric and magnetic fields. No material medium is necessary for their movement and they travel more easily in vacuum than in matter.
Examples of electromagnetic waves are: Radio waves, Microwaves, Infrared radiation, Visible light, Ultraviolet light, X-rays and Gamma rays. These waves vary according to their wavelengths.
4.4 PROPERTIES OF WAVES
This section introduces the properties of waves and wave motion to describe the behavior of waves in detail.
4.4.1 Reflection
This is the property of waves to bounce back from the surface on which they hit. Huygens principle can also be applied to reflection. Consider a parallel beam of light incident on the reflecting surface such that its direction of travel makes an angle i with the normal to the surface.
Consider that side A of an associated wavefront AB has just reached the surface. In the time that light from side B of the wavefront travels to B′, a secondary wavelet of radius equal to BB′ will be generated by A. Because of the reflecting surface, this wavelet is a semicircle above the surface.
The new wavefront generated by reflection will be the tangent to this
We conclude by saying that all laws of reflection are obeyed. So, any wavefront can reflect.
4.4.2 Refraction
Consider a parallel beam of waves (for example light waves) incident on a refracting surface between two media such that its direction of travel makes angle q1 with the normal to the refracting surface.
Consider side A of the wavefront AB has reached the surface before B. If the ray from the other side B of the beam consequently travels to C at time t,
At the same time, wavelets from A travel distance AD in medium 2. Here, a refracted wavefront CD is formed by many wavelets in the beam. Fig.4-16 above illustrates this description.
Equation 4-32 confirms Snell’s law meaning that waves behave like normal light during reflection.
4.4.3 Interference
In the region where wave trains from coherent sources (sources of the same frequency) cross, superposition occurs giving reinforcements of waves at some points which is called constructive interference and cancellation at others which is called destructive interference. The resulting effect is called interference pattern or the system of fringes.
4.4.4 Diffraction
This is a phenomenon in which waves from one source meet an obstacle and spread around it. Diffraction is normally observed when these waves pass through narrow slits. There are two types of diffraction and these are; Fresnel’s diffraction and Fraunhofer diffraction.
4.4.4.1 Fresnel’s diffraction
This is a type of diffraction in which either the source of waves or screen on which diffraction is observed or both are at finite distances from the obstacle that cause diffraction. Below are different cases to explain this diffraction.
Case 3: the screen is placed at infinite distance from obstacle and the source is near.
4.4.4.2 Fraunhofer Diffraction
This is a type of diffraction in which the source of waves and the screen on which diffraction is observed are effectively at infinite distances from the obstacle. This phenomenon is practically complicated but theoretically understood. To obtain waves to or from infinite source in laboratory, biconvex lenses are used.
4.5 YOUNG’S DOUBLE SLIT EXPERIMENT
In this experiment, a source S of monochromatic waves is used to illuminate two narrow and parallel slits S1 and S2 that are apparent sources of light. The arrangement is shown in Fig.4.22.
From the above experiment, a narrow slit S diffracts light falling on it and so illuminates S1 and S2. Diffraction also takes place at S1 and S2 and interference takes place in the region where light from S1 overlaps the light from S2. Light from S1 and S2 is from a monochromatic source, hence S1 and S2 act as coherent sources (they are also monochromatics).
4.6 WAVE ON A VIBRATING STRING
When a string is fixed at one end and the other end is moved up and down, a transverse wave is formed. The simplest mode of vibration is the one in which both ends are nodes. Let us use l as the wavelength, l as the length of the string, c as the speed of the wave and f as the frequency of the wave.
The figure above shows the simplest mode of vibration which is the first harmonic (fundamental mode) and its frequency f1 is called the fundamental frequency. Higher order frequencies are called overtones. The second harmonic is the first overtone and has the mode shown by Fig.4-24.
EXAMPLE 4
A wire of length 400 mm and mass 1.2 × 10–3 kg is under a tension of 120 N. What is
(a) the fundamental frequency of vibration?
(b) the frequency of the third harmonic?
4.7 APPLICATIONS OF WAVES
1. They are used in radar, broadcasting and radio communication.
2. They are used in MRI in hospitals.
3. They are also used in radio communication which forms an integral part of wireless communication.
Across
1. How fast something is moving or how much distance is covered in a certain amount of time.
2. The time it takes for a wave to repeat itself
4. The lowest point of a wave beneath the line of origin
9. Waves that require a medium
10. The highest point of a wave above the line of origin
11. Particles of light
12. A push or a pull
13. The tendency of an object at rest to remain at rest or in motion until acted upon
Down
2. Waves that do not require a medium
5. The bouncing back of a wave when it meets the surface or boundary
6. The matter through which a wave travels
7. Distance in a given direction
8. The vertical distance between the line of origin and the crest of a wave
Materials to choose from:
3 white screens, 3 biconvex lenses
, 3 biconcave lenses
The question:
Explain how you can perform Fresnel’s diffraction and Fraunhofer diffraction in the laboratory.
Hypothesis:
Write a hypothesis about how diffraction is obtained in the lab.
Procedure
1. Decide which materials you will need (from the list) to test the hypothesis.
2. Plan your investigation.
a. Which arrangements best gives the idea of diffraction?
b. Which adjustments do you care to take care of ?
3. Write a procedure and show it to your teacher. Do not proceed any further until it is approved.
4. Carry out your investigation.
Collecting Data
Make sure you have recorded at least the following information:
◊ the hypothesis
◊ your procedure
Analyzing and Interpreting
Share and compare your results with your classmates. Which idea is important to be used and achieve the proper arrangement of apparatus to achieve your objective?
Forming Conclusions
Make a brief report of your project with neat diagrams. In this project what is needed is the concept not the analysis of the fringes formed.
