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Lesson_Text_5.3

# Lesson_Text_5.3 - 1 Lesson 5.3 Harmonic Waves Lesson...

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1 Lesson 5.3 Harmonic Waves Lesson Objectives: At the end of this lesson students will be able to (i) describe the characteristics harmonic waves and derive a mathematical expression describing harmonic waves. (ii) describe and interpret interference of harmonic waves and solve problems involving this concept. 1. Harmonic Waves : A wave generated by the simple harmonic vibrations of the particles of a medium is called a harmonic wave. You have seen in lesson 1 that the displacement y (in lesson 1 we used x for displacement) of a particle in simple harmonic motion is given by the equation y = A cos( ϖ t ) …..(i) This equation is obtained by assuming that the particle is at the extreme position when t = 0. This assumption means that y = A when t = 0. On the other hand if the particle is at the equilibrium position when t = 0, then y = 0 at t = 0. This situation is more appropriate to describe a wave. This condition will be satisfied if we replace the cosine function by a sine function in the above equation. Therefore, the equation for a harmonic wave can be represented as y = A sin( ϖ t) …….(ii) A A y = A cos ϖ t y = A sin ϖ t We have seen from our wave model that the state of vibration of a particle is a function of how far the wave has advanced. If the particle vibration along the y-axis causes the propagation of the wave along the x-axis, then the displacement y of the particle is a function of x, the distance the wave has reached In equation (ii) above,

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2 ϖ π π π λ = = = 2 2 2 T f v where v is the wave velocity and λ the wavelength of the wave. Therefore, 2 t vt π ϖ λ = But vt = x. Therefore, the equation for a harmonic wave can be written as: 2 sin sin y A x A kx π λ = = …….(iii) Here 2 k π λ = and is called the wave number of the wave. In order to describe a wave that is travelling with a velocity v in the positive x-direction, we replace x by (x – vt) in the above equation. Similarly in order to describe a wave that is travelling to the left (in the negative x-direction) with a velocity v, we replace x by (x + vt). We will continue describing a wave that is advancing to the right. Thus the equation of a harmonic wave advancing to the right with a velocity v is given by y = A sin{k(x – vt)} = A sin(kx –kvt) ………(iv) kv f f = = = 2 2 π λ λ π ϖ . Therefore, the equation of a harmonic wave advancing to the right can be written as a function of x and t as: y(x,t) = A sin(kx - ϖ t) ……(v) similarly the equation of a harmonic wave advancing to the left can be written as: y(x,t) = A sin(kx + ϖ t) …..(vi) We can also write other forms to this equation as k = 2 π λ and ϖ π = 2 T 2 2 sin sin 2 x t y A x t A T T π π π λ λ = - = - ……(vi)
3 Example 1: The wave y(x,t) = A cos k(x + 34 t) represent a traveling wave where x is in meters and t in second. What are the direction and the speed of the wave?

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Lesson_Text_5.3 - 1 Lesson 5.3 Harmonic Waves Lesson...

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