WAVE EQUATIONS

 

WAVE EQUATIONS

MATHEMATICAL REPRESENTATION OF WAVE MOTION  -  PROGRESSIVE WAVE

We can represent a wave mathematically by the equation Y = A sin2Pif/landa ; where Y= vertical displacement of vibrating particles;  = wavelength of wave; A = Amplitude of wave; x = horizontal coordinate of the vibrating particle from the origin.

y = ASin(wt -0  )

Y = A sin2pi/landa  (x – vt) where v is the velocity and t is the time.

Y = A sin (2pix/landa - 2pi vt/la

Example

1.  The displacement of a plane progressive wave is given by the equation Y = 12.5 sin (420  - 0.50 x). where the symbols have their usual meanings, Y and x are in metres and t in seconds. For the wave determine the : i. frequency ii. Wavelength  iii. Speed. (G.CE 2009)

Solution

Y = 12.5 sin (420  - 0.50 x)  Comparing the given equation with the standard equation   Y = A sin ( )

i.  = 420  ; 2f =420; f = 420/2 = 210Hz   ii.  = 0.50 x;  = 2/0.50 = 4m  iii.  V =f  = 210Hz x 4m = 840ms-1.

2. The equation of transverse progressive wave is y = 3 sin 2 (  - ), where the length is expressed in cm and the time in seconds. Calculate the wavelength , frequency, amplitude and the speed of the wave.

Solution

Equation of a given wave is y = 3 sin 2 (  - ) Comparing the given equation with the standard equation                      

 y = A sin 2 (  - )

 Wavelength, , = 40cm ; f = 1/T =1/0.04 =25Hz ; Amplitude =3cm ; speed,v, = f  =25Hz x 40cm= 1000cm/s =10m/s

PROPERTIES OF WAVES

Waves has the following properties:  reflection, refraction, diffraction, interference and polarization of  light. Application of plane for polarized light

1. Diffraction of waves is a phenomenon in which waves spread out as they pass through an aperture or round a small obstacle.  Diffraction can be defined as the ability of waves to bend around obstacles in their path.



The effect of diffraction is obvious only if

a) the size of the aperture or obstacle is small enough.  b) the wavelength is large enough
 Characteristics of diffracted waves:
a) Frequency, wavelength and speed of waves do not change. b) Changes in the direction of propagation and the pattern of the wave.
Diffraction of Light
1. Light is diffracted if it passes through a narrow slit comparable in size to its wavelength.
However, the effect is not obvious as the size of the slit increases. This because the wave-lengths of light are very short.

2. Diffraction of light is hardly noticeable compared with diffraction of sound waves and water waves because the wavelength of light is very short or small (approx: 10-7 m)
3. Light waves will be diffracted if
a) Light is propagated through a pin hole or a tiny slit where its size is similar to that of the light wavelength (around one hundredth of a millimetre or less)
b) the light source is monochromatic, i.e. light of one colour, and therefore one wavelength only.

2. interference of Waves

Principle of Superposition
 The principle of superposition states that at any instant or moment, the wave displacement of the combined motion of any number of interacting waves at a point is the sum of the displacements off all the component waves at that point.
Interference is the effect produced when two waves of the same frequency, amplitude and wavelength travelling in the same direction in a medium are superposed – as they simultaneously pass through a given point.

Interference of Waves
i. Interference is the superposition of two waves originating from two coherent sources. Sources which are coherent produce waves of the same frequency,f, amplitude,a, and are in phase.
ii. The superposition of two waves emitted from coherent sources gives either constructive or destructive interference.
iii. Constructive interference occurs when the crests or throughs of both waves coincide to produce a wave with crests and troughs of maximum amplitude.



iv. Destructive interference occurs when the crest of one wave coincides with the trough of the other wave, thus cancelling each other with the result that the resultant amplitude is zero.
v. An antinode is a point where a constructive interference occurs, whereas a node is a point where destructive interference occurs. The antinodal lines join all antinodes and the nodal line joins all nodes.

3. polarization

A wave is said to be plane-polarized if its vibration occur only in one plane.

Polarization is the phenomena of producing transverse vibrations which are only in one plane. Only transverse waves can be plane –polarized. Polarization is therefore occurs with light waves and other electromagnetic waves e.g radio waves, x-rays, infrared radiation, but not with sound waves because sound waves is longitudinal waves.

Polarized light can be produced using tourmaline crystals, quartz or Polaroid.

Application of polaroids

The Polaroid is used in sunglasses to reduce the intensity of incident light and eliminate reflected light or glare.

Assignment

1. The equation of a wave can be represented by y= 0.02 sin  (320t –x) where x and y are in metres and t in seconds. Find the amplitude, frequency, wavelength and velocity of propagation of wave.

2.  The equation of a wave can be represented by y= 0.002 sin 2 (5t - ) where all the quantities are in S.I units. Calculate the displacement of the particle at a distance of 5m from the origin after 0.2s.

 

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