MECHANICAL WAVES: SOUND AND WATER WAVES

INTRODUCTION

A wave is basically energy in motion. Waves fall under two broad categories: 

  1. Electromagnetic (EM) waves a
  2. Mechanical (elastic) waves. 

The main difference between the two types of waves is that EM waves do not require a medium for propagation while mechanical waves require a medium for propagation. Examples of EM waves are radio waves, light and x-rays. Examples of mechanical waves are sound waves and water waves. 

NECHANICAL WAVES

There are  two types:

  1. Longitudinal waves 
  2. Transverse waves.

 Longitudinal waves occur when the direction of vibration of particles is parallel to the direction of motion of the waves, example sound wave. 

Transverse waves on the other hand occur when particles vibrate in a direction perpendicular to the direction of wave travel, for example water waves.

Characteristics of mechanical waves

 (i) Wave equation

Waves obey the wave equation given as;

Where v is the velocity of the wave,, λ  the wavelength and f is the frequency. Velocity is the distance travelled by a wave in unit time while wavelength is the distance between two consecutive particles that are in phase. Particles are said to be in phase when they occupy identical positions and are travelling in the same direction. Frequency refers to the number of complete wavelengths passing a particular point in unit time;

Period (T) is the time taken by a complete wavelength to pass a particular point (time for one complete oscillation). In other words, when N = 1, then t = T. Thus;

Amplitude (A) of a wave is defined as the maximum displacement of a particle from its mean position. Mean position refers to the position of a particle when not vibrating. For example, the surface of water before waves are formed.

(ii) Are graphically represented by a sinusoidal waveform

The sinusoidal waveform comprises of crests (maximum displacement in the positive direction) and troughs (maximum displacement in the negative direction).

 

Particles at the apex of crests are in phase with each other. Particles at the base of troughs are also in phase with each other. Thus, wavelength can also be defined as the distance between two adjacent crests, or distance between two adjacent troughs. 

The locus of all points in a wave having the same phase of oscillation, e.g. apexes of crests, is called a wavefront. Wavefronts can be;

  1. curved (spherical wavefronts) – e.g. those from a point source. They originate from a point and spread outwards 
  2. Straight (plane wavefront) e.g. from a far off source or  straight source. 

Wavefronts are represented by equally spaced lines with the distance between two consecutive wavefronts being equal to the wavelength λ   of the wave. 

A ray is a line drawn perpendicular to the wavefront that indicates the direction of travel of the wave. 

(iii) The waves associated with the following properties (among others)

  • Interference
  • Reflection
  • Refraction
  • Diffraction

SOUND WAVES

Sound waves are mechanical longitudinal waves characterised by compressions (vibrating particles are close together) alternating with rarefactions (vibrating particles are further away from each other). A sound wave in air is sort of a ‘pulsating’ air column with the ‘pulsating’ occurring in the direction of wave travel.

Reflection of sound 

Sound undergoes reflection when it hits an obstacle. The reflected sound is called an echo. Reflection of sound is most pronounced where hard smooth surfaces are involved. For example, the walls of a studio are usually padded for instance with woollen materials. If for example the walls of a studio (say music studio) are hard and smooth, the desired sound (say song) would produce an echo which would not only act as background noise, but would also interfere with the original sound leading to further background noise, this time noise of different intensity thereby affecting the quality of the final product. Padding reduces the production of echo leading to clean and clear sound. The padding absorbs most of the sound energy falling on it which reduces sound reflection.

Echo has advantages as well.  Echoes are used by animals such as bats and dolphins for navigation and in search of food. These animals let out ultrasonic sounds which are reflected by obstacles. By analysing the reflected sound, the animals brain forms a picture of the obstacle (shape), determine its speed as well as its distance. In medicine, ultrasonic sound is used to image organs such liver, and also in ultrasound to map the image of a foetus. Echoes are also used to measure distance for example depth (d) ) of the ocean. 

Suppose the speed of sound in ocean water is, say,  vwater  (speed of sound depends on the medium of transmission), and time interval between sending the sound into the water and receiving the echo is t. In this time, the sound will have travelled to the bottom of the ocean and back, covering a distance 2d. Form the definition of speed:

                                        

Interference of sound waves

Sound waves also undergo interference. Interference refers to the superposition of waves from different sources. Interference of sound waves leads to both enhancement (loud sound) as well as sound of low intensity. Enhancement occurs when rarefactions (or compressions) from different waves arrive at the same place at the same time (arrive in phase). This form of interference is referred to as constructive interference. If on the other hand a rarefaction from one wave and a compression from another arrive at the same place at the same time, the intensity of sound reduces. This is referred to as destructive interference.

Interference occurs when two or more waves are in the same space at the same time (superimposed on each other). Superposition may lead to either enhancement or reduction of the sum effect in accordance with the superposition principle. Superposition principle states that the resultant effect (or net displacement of particles) in wave-transmitting medium at a point in space and time is equal to the algebraic sum of displacement of particles in individual waves. If two waves each of amplitude A interfere constructively, the resultant wave amplitude will be A + A = 2A. If destructive, the amplitude will be A +(-A) = 0 Thus, if crests + crest and trough + trough of two different wave superimpose, one bigger wave is formed (total constructive interference) .

If the superimposed waves are 1800 out of phase, crests of one wave coincide with the troughs of the other and vice versa. Total destructive interference occurs and the wave dies out. 

As an example, if two identical sound systems A and B are placed some distance apart, constructive and destructive interference of sound from the two systems occur. A person walking along a path parallel to, and at a distance from, the sound systems will therefore hear loud sound alternating with low sound as he moves along. 

Partial constructive and destructive interference can also occur. This happens when waves of nearly equal frequencies are in the same place at the same timeSuppose two sound waves of slightly different frequencies are travelling in same direction at the same place at the same time. During the travel, the degree of interference will be different ranging from near complete destructive interference to near complete constructive interference. 

This will cause the intensity of resultant sound to rise and fall periodically. 

The periodic fall and rise of the intensity of sound when two waves of slightly different frequencies are in the same place at the same time is called a beat

Factors affecting the velocity of sound

Sound being a mechanical wave requires a medium for propagation. The speed of the sound depends on a number of factors;

  1. The density of the propagating medium
  2. Elasticity of the propagating medium
  3. Temperature of the propagating medium

Sound travels faster in denser media relative to less dense media (opposite to the effect in EM waves e.g. light where the speed is higher in less dense media). The speed of sound is therefore highest in solids, followed by liquids and lastly gases. Materials with higher elasticity (an elastic material regains its shape after distortion if the elastic limit is not exceeded) conduct sound faster than plastic (get permanently distorted after application of a force). Example of an elastic material used to make telephone cables is copper (since it is elastic, ductile and a good conductor). Sound travels faster when the temperature of the propagating medium is higher than when it is low. When molecules gain heat, their kinetic energy increases leading to the increase in their vibratory motion. The increased vibration leads to an increase in the speed of propagation of sound.

WATER WAVES

Water waves are mechanical transverse waves for which the direction of displacement of the particles is perpendicular to the direction of travel of the waves. The wavefronts may be circular if from a point source, for example when a stone is dropped in water, or plane such as those from a far off source.  Like all waves, water waves are reflected when they hit an obstacle, refracted when between waters of different depth, and diffracted (spread out) when they pass through a small opening or over a relatively small obstacle (barrier).

Reflection of water waves

Like any other waves, water waves obey the laws of reflection;

  1. Angle of incidence equals angle of reflection.
  2. At the point of incidence, the incident ray, the reflected ray and the normal all lie on the same plane.

Also, if water waves hit a plane reflector, the reflected waves are laterally inverted (left becomes right).

 

 

Refraction of water waves

When water waves move from a deep region to a shallow region, their velocity reduces (similar to reduction of velocity when EM waves e.g. light move from a less dense (say air) to a denser region (say water). And by the way the fact that colours of light move with different velocities in water is what causes the rainbow).

If we let the velocity of water waves in the deep and shallow regions be vd and vs respectively, and λd and λs the wavelengths in deep and shallow regions respectively;

                                                  (i)

                                                    (ii)

The frequency f of the wave is independent of the medium of propagation (it depends on the source). Comparing equations (i) and (ii), it follows that;

                                                     (iii)

Equation(ii) implies that if the velocity of a wave reduces (frequency constant), then the wavelength has to reduce proportionately.

Thus, when a wave moves from a deep region to a shallow region, both the velocity and wavelength reduce proportionately while the frequency remains constant. When the wave moves from a shallow to a deeper region, both the velocity and wavelength increase proportionately while the frequency remains constant. If the waves enter a shallow region from a deep region obliquely, they change direction with the refracted wave (wave in the shallow region) bending towards the normal.

If waves moves from a shallow to a deeper region, both the velocity and wavelength increase proportionately while the frequency remains constant, and if they enter obliquely, the wave in the deeper region is refracted away from the normal.

Diffraction of water waves

Diffraction refers to the bending or spreading of waves when they pass through an aperture (slit, hole) or when they encounter an obstacle. When plane waves pass through a thin slit relative to the wavelength, the slit will act as a point source leading to circular waves. If the slit is not very small, then the curvature of the waves that pass through the slit will not be as pronounced.

When the small slit is replaced with a small barrier/obstacle, the waves curve when they pass through the obstacle. A small obstacle acts like a point source. If the barrier is large, the bends are pronounced at the ends with the resulting waves changing direction. 


EXAMPLES


 


Dr. Margaret W. Chege

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