notes-week2 - Physics 1B Waves Second Week LecturesTM...

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1 Physics 1B Second Week Lectures Walter Gekelman Waves Chapter 15 University Physics + MORE Let us study the wave equation. Suppose someone yanks upwards on a rope which is fixed at one end. The impulse will propagate along the length of the rope which is nearly in the x direction except for the small impulse that travels along it. What is the equation for this? There is tension T in the rope represented by the black arrows. Suppose this is a piece of rope that a wave is moving along. The angle of the rope is highly exaggerated and is very small. The difference in the tension between the left and right sides, which is the force on the rope is: T sin θ + d ( ) T sin = m 2 y x 2 Using the Trig identity sin a + b ( ) = sin a cos b + cos a sin b = sin cos d + sin d cos Let us next assume that the wave has a very small amplitude so that the angle theta is small and d is smaller still. Since cos(0) = 1 sin + d ( ) sin d + sin we also from the diagram get sin d d tan d = y x
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2 T sin θ + d ( ) T sin (sin d + sin sin ) = Td T x = μ 2 y t 2 = T 2 y x 2 2 y x 2 = T 2 y t 2 T + d ( ) = Td = dx 2 y t 2 T x = 2 y t 2 = T 2 y x 2 2 y x 2 = T 2 y t 2 (1) This is the wave equation
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3 The figure above shows why at small angles the curve of the tangent (x) (red) the sine (x) (blue) and a straight-line y=xs all become the same at small angles as shown in the bottom curve. Note that sin( θ + δθ ) = sin cos + cos sin . If both , are very small angles then cos 1, and cos 1 sin so that sin( + ) = + . T x = μ 2 y t 2 = T 2 y x 2 2 y x 2 = F 2 y t 2 F has dimension M / L 3 ML / T 2 (1) 2 y x 2 = 1 v 2 2 y t 2 wave equation Note the general solution to the wave equation can also be written as: y =
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notes-week2 - Physics 1B Waves Second Week LecturesTM...

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