◄UNIT 3:FORCED OSCILLATIONS AND RESONANCE OF A SYSTEMUNIT 5: INTERFERENCE OF LIGHT WAVES►

UNIT 4.PROPAGATION OF MECHANICAL WAVES

  • UNIT 4.PROPAGATION OF MECHANICAL 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 I 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.

    Introductory Activity
     a. Arrange yourselves the form of a circle with your right 
    shoulders pointing towards the centre.
     b. Ask your friend to raise arms and then lower them. Then the 
    next friend raises arms and lowers them, and so on around the 
    circle. It should be like the “wave”.
     c. Describe the type of the disturbance formed.
     d. Is the disturbance travelling up and down or horizontally 
    around the circle?
     e. Let one of your friend gently push the back of the next student 
    and then the pushed member should gently push the next 
    member and so on, which will make a wave travel around the ring. 
    f. From what you have done, can you describe what a disturbance 
    is? Is the disturbance travelling up and down or around the ring? 

    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.

     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).

    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. 4-12. 

    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).
                                                I ∝ x2 
    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.

    Crest is the highest point above the equilibrium position while trough is 
    the lowest point below then equilibrium position.
     
    The angular frequency ω represents the frequency in radians per second. It 

    is related to the frequency by

                     

    A node is a point half way between the crest and the trough. The line that 
    connects the nodes is the nodal line. The nodal line shows the original 

    position of the matter carrying the wave.

    Application Activity 4.1

    1. Requirements: a manila paper with the drawing of the wave 

     shown below

    a) How do you call the distance represented by arrow z?
     b) What letter is labelling the wave’s trough?
     c) What letter is labelling a wave’s crest?
     d) The number of waves that pass the poster  per second is called 
        the …………….. of the waves.
     e) If the knot (w) travels 2 meters in 1 second, we say that it has …………
        ….. of 2 m/s.
     f) If the wavelengths were shortened, would the frequency be higher 
       or lower?
     g) The greatest distance the knot (w) travels from its resting position 
        is called…………….. of the wave.

     h) What kind of wave are these in the rope?

    2. Use the following descriptions in waves ad fill in the crossword

      puzzle bellow:


    Across 
    1. How fast something is moving or how much distance is covered 
        in a certain amount of time. 
    3. 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 
    1. Waves that do not require a medium 
    2. The bouncing back of a wave when it meets the surface or boundary 
    3. The matter through which a wave travels 
    4. Distance in a given direction 

    5. The vertical distance between the line of origin and the crest of a wave

    4.3 TYPES OF WAVES 
    Waves are of three main types: Mechanical wave, electromagnetic wave 
    and matter wave. 
    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. 
    The matter through which mechanical waves travel is called the medium. 

    All mechanical waves require (1) some source of disturbance, (2) a medium 
    that can be disturbed, and (3) some physical mechanism through which 
    elements of the medium can influence each other.
     Mechanical Waves are divided into two types according to the direction 
    of the displacements in relation to the direction of the motion of the wave 
    itself (wave form):
     
    a) 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. 

     b) 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.

     4.3.1.4 Examples of mechanical waves
     Mechanical waves, being progressive and stationary, are seen in different 
    forms as described in this section. 

    Sound waves
     Sound waves are longitudinal waves. Sound waves travel fastest in solids, 
    slower in liquids and slowest in gases. This means the air particles (or 
    particles of the medium) move back and forth on paths that are parallel to 
    the direction of wave propagation and thus take the form of compressions 

    and rarefactions of the molecules in the air itself.

    Water waves
     Water waves are a combination of both transverse and longitudinal waves. 
    These waves are periodic disturbances that move away from the source and 

    carry energy as they go.

    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. 

    Surface Waves
     When waves occur at or near the boundary between two media, a transverse 
    wave and a longitudinal wave can combine to form a surface wave. 
    Examples of surface waves are a type of seismic wave formed as a result of 

    an earthquake and water 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.

    4.3.3 Matter Waves
     If we perform the double slit diffraction experiment using a beam of electrons instead of light,
    we still get a diffraction pattern. The interpretation 
    of this is that matter travels as a wave. Thus “matter acts as both a
    par    ticle and as a wave.” If we can sometimes consider an electron to be a wave, 
    what is its wavelength? Louis de Broglie postulated that all particles with 

    momentum have a wavelength 

    The matter waves describe the wavelike characteristics of atomic-level 
    particles.
     For mechanical waves, the speed of the wave is a property of the medium, 
    speed does not depend on the size or shape of the wave.
     
    Example 4.1

     1. Find de Broglie wavelength for

     4.4 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. 

    Equation of a progressive wave 
    An equation can performed to represent displacement 
    of 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.

    A particle at P will be out of phase from the particle at O, so, its displacement is given by;

    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 metres 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
     heart 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


    4.5 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:


    When two pulses of equal or different amplitudes on a string approach each 
    other, then on meeting, they superimpose to produce a resultant pulse of 
    amplitude greater than any of the two. After crossing, the two pulses travel 
    independently.
     
    4.5.1 Stationary waves
     A stationary wave (or a standing wave) is a wave which results when 
    two waves travelling in opposite directions and having the same speed, 
    frequency and approximately equal amplitudes are superposed. A standing 

    wave is shown in Fig. 4.6 below.

     4.5.2 Mathematical treatment of superposition
     Position of nodes

     A node is defined as the point of zero amplitude. This means

    Equation (4.21) means that nodes are obtained when the horizontal 
    displacement of waves are odd quarter values of wavelength.
     
    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;

    4.6 PROPERTIES OF WAVES

    This section introduces the properties of waves and wave motion to describe 

    the behaviour of waves in detail.

    4.6.1 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 wavelet 

    and will also contain point B′. The reflected wavefront will be A′B′.

     We conclude by saying that all laws of reflection are obeyed. So, any 
    wavefront can reflect.
     
    4.6.2 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 

     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.

     4.6.3 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.

     4.6.4 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.
     
    a) 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 1: the source and the screen placed at finite distances.

    Case 2: the source is placed at infinite distance from obstacle and the screen is near.


     Case 3: the screen is placed at infinite distance from obstacle and the source is near.

    b) 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.

     4.7 WAVE ON A VIBRATING STRING
     ACTIVITY 4-1: Propagation of Waves
     Learning Objectives
     • To observe the propagation of vibrations through a solid
     • To understand how sound is transmitted through a medium
     Required Materials
     Spoon, string of length 1 m

         Procedure
     (a)  Tie the spoon into the middle of the length of string so that it will hang 
    freely when you hold the string ends.
     (b)  Hold the string ends to your temples or the bone just under your ears 

    as you strike the spoon with a pen or other object.

    Discussion Questions
     1.  What causes the sound to be loud when the string is held to your head?
     2.  Why does the bone in front of your ear transmit vibrations more easily 
          than other bones?
     3.  What is the purpose of the string in this activity?
     Standing wave also known as a stationary wave, is wave pattern that 
    results when two waves of the same frequency; wavelength and amplitude 
    travel in opposite directions along string and interfere. 

    The point at which the two waves cancel are called     node. There no motion in 
    the string at the nodes, but midway between two adjacent nodes, the string 
    vibrates with the largest amplitude. These points are called antinodes. At 

    points between successive nodes the vibrations are in phase.


     A single loop corresponds to either a crest or tough alone, while two loops 
    correspond to a crest and trough together, or one wave length.
     
    Stationary waves are present in the vibrating strings of musical instruments. 
    A violin string, for instance, when bowed or plucked, vibrates as a whole, 
    with nodes at the ends, and also vibrates in halves, with a node at the center, 
    in thirds, with two equally spaced nodes, and in various other fractions, all 
    simultaneously. The vibration as a whole produces the fundamental tone, 
    and the other vibrations produce the various harmonics.

    Standing waves can occur at more than one frequency. The lowest 
    frequency of oscillation that produces a standing wave gives rise to the 
    pattern shown in Fig. 4.24b. The standing waves shown in Figs. 4.24c and 

    4.24d are produced at precisely twice and three times the lowest frequency, 

    respectively, assuming the tension in the cord is the same. The cord can also 
    oscillate with four loops (four antinodes) at four times the lowest frequency, 
    and so on. 

    The frequencies at which standing waves are produced are the natural 
    frequencies or resonant frequencies of the cord, and the different 
    standing wave patterns shown in Fig. 4.24 are different “resonant modes 
    of vibration.” A standing wave on a cord is the result of the interference of 
    two waves traveling in opposite directions. A standing wave can also be 
    considered a vibrating object at resonance. Standing waves represent the 
    same phenomenon as the resonance of an oscillating spring or pendulum, 
    However, a spring or pendulum has only one resonant frequency, whereas 
    the cord has an infinite number of resonant frequencies, each of which is a 
    whole-number multiple of the lowest resonant frequency.




    one antinode (or loop). And as can be seen in Fig. 4.24b, the whole length 
    corresponds to one-half wavelength. 

    The other natural frequencies are called overtones; for a vibrating string 
    they are whole-number (integral) multiples of the fundamental, and then 
    are also called harmonics, with the fundamental being referred to as the 
    first harmonic. The next mode of vibration after the fundamental has two 
    loops and is called the second harmonic (or first overtone), Fig. 4.24c. The 
    length of the string at the second harmonic corresponds to one complete 


    A normal mode of an oscillating system is a motion in which all particles 

    of the system move sinusoidally with the same frequency

    EXAMPLE 4


    EXAMPLE 4


    Application Activity 4.2

    Use the following descriptions in waves and fill the puzzle
     Down:
     1)  The part of a longitudinal wave where the particles of the medium 
         are close together.
     2)  A wave which needs to travel through a medium.
     3)  A repeated back-and-forth or up-and-down motion.
     6)  A wave which moves the medium in a direction across the direction.
         the energy is traveling.
     8)  The ability to do work.
     
    Across:
     4)  A disturbance that transfers energy from place to place.
     5)  The highest point of a wave.
     7)  The part of a longitudinal wave where the particles of the mediu
         are far apart.
     9)  A wave which moves the medium in the same direction as the energy 
         is traveling.
     10)  The lowest part of a transverse wave.

     11)  The material through which a wave travels.


    END OF UNIT PROJECT

    Materials to choose from: 
    3 white screens, 3 biconvex lenses egg, 3 biconcave lensesegg, 3 biconvex 
    mirrorsegg, 3 biconcave mirrorsegg, 3 boards with a hole, 3 laser pens, 3 big 
    torches, 3 very bright open lamps, 1 plane mirror.

     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.

    END OF UNIT ASSESSMENT
     1. The string shown in Figure below is driven at a frequency of 5.00 Hz. 
    The amplitude of the motion is 12.0 cm, and the wave speed is 20.0 m/s. 
    Determine the angular frequency and wave number k for this wave, and 

    write an expression for the wave function.

    2. The wave shown in Fig. below is being sent out by a 60 Hz vibrator.

    3. A string of length 3 m and mass density 0.0025 kg/m is fixed at both 
    ends. One of its resonance frequencies is 252 Hz. The next higher res
    onance frequency is 336 Hz. Find the fundamental frequency and the 
    tension in the string.
     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?
     5. A sinusoidal wave traveling in the positive x direction has an amplitude 
    of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The 
    vertical position of an element of the medium at t = 0 and x = 0 is also 

    15.0 cm, as shown in Figure below.

     (A) Find the wave number k, period T, angular frequency  and speed v of the wave.
    (B) Determine the phase constant and write a general expression for the wave function.

    UNIT SUMMARY
     Waves can be defined as a disturbance in a material medium that transfers 
    energy from one place to another.
     The time period (T) of the wave is the time it takes for one complete 
    vibration of the wave.
     The frequency f is the number of wavelengths that pass a point in space 
    in one second. 
    The wavelength is the horizontal distance in space between two nearest 
    points that are oscillating in phase. 
    The wave speed v is the speed at which the wave advances. 
    Phase difference (phase angle) is the angular difference between two 
    points on the wave or between two waves.
     The wave number  also called the propagation number k is the spatial 
    frequency of a wave.
     The Intensity of a wave or the power radiated by a source are proportional 
    to the square of the amplitude.
     Wavefront is a line or surface in the path of the wave motion on which the 
    disturbance at every point have the same phase. 
    Mechanical waves are waves produced by the disturbance in a material 
    medium.
     A progressive wave consists of a disturbance moving from one point to 
    another.
     Longitudinal wave propagates through some medium with vibrations in 
    the direction of propagation of the disturbance.
     In Transverse waves, the direction of vibrations is perpendicular to the 
    direction of propagation of the wave. 

    Equation of a progressive wave is given by:

    Principle of superposition states that the resultant displacement at any 
    time is the vector sum of the individual displacements.

     Stationary waves are waves which seem to be at rest.

    Electromagnetic waves are disturbances in form of varying electric and 
    magnetic fields.
     All kinds of waves reflect, refract, interfere and also spread around the obstacle.
     Other than the superposition of waves meeting at a point, other conditions 
    for interference     are:
     • The sources of the waves must be coherent, which means they emit 
    identical waves with a constant phase difference.
     • The waves should be monochromatic - they should be of a single 

    wavelength.

    ◄UNIT 3:FORCED OSCILLATIONS AND RESONANCE OF A SYSTEMUNIT 5: INTERFERENCE OF LIGHT WAVES►