Phys2212_21.1+to+21.7

# Phys2212_21.1 to 21. - Physics 2212 Waves Lecture 2 Superposition and Standing Waves Waves vs Particles Particles can collide Superposition fails

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Physics 2212 Waves Lecture 2 Superposition and Standing Waves

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10/06/09 Physics 2212 - Lecture 2 Waves vs. Particles Particles can collide. Superposition fails. Waves pass through each other unimpeded. Superposition works!
10/06/09 Physics 2212 - Lecture 2 The Principle of Superposition The principle of superposition: When two or more waves are simultaneously present at a single point in space, the displacement of the medium at that point is the sum of the displacements of each individual wave. In other words, wave displacements add linearly, and essentially ignore each other. net 1 2 i i D D D D = + + = L We note that if the total displacement D net becomes too large, superposition could fail and the wave could go non- linear and “flat-top”.

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10/06/09 Physics 2212 - Lecture 2 Clicker Question 1 Two pulses on a string approach each other at speeds of 1 m/s. Which of these shapes represents the pulses at t=6 s.
10/06/09 Physics 2212 - Lecture 2 Standing Waves Suppose you point two loud speakers at each other or shake both ends of a string. Assume that the two waves produced travel in opposite directions and have the same frequency and amplitude. The superimposed waves form a standing wave that is alternately flat due to destructive interference and double height due to constructive interference.

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10/06/09 Physics 2212 - Lecture 2 Nodes and Antinodes If we plot all the snapshots together, we get the figure shown at the right. There are points called nodes than have zero displacement. These are spaced a distance λ / 2 apart, and they include the point at which the string is anchored to the wall. These are points of maximum destructive interference. Between the nodes are points of maximum displacement. These are called antinodes . These are also spaced λ / 2 apart and are points of maximum constructive
10/06/09 Physics 2212 - Lecture 2 More Nodes and Antinodes We had previously defined the intensity of the oscillation as I = CA 2 . If we plot the intensity of the waves, we see that the maximum intensity occurs at the antinodes. If the superimposed traveling waves have equal intensity, the points of zero intensity occur at the nodes.

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10/06/09 Physics 2212 - Lecture 2 The Mathematics of Standing Waves R sin( ) D a kx t ϖ = - L sin( ) D a kx t = + R L ( , ) sin( ) sin( ) [sin( ) sin( )] [2 sin( )]cos( ) ( )cos( ) D x t D D a kx t a kx t a kx t kx t a kx t A x t = + = - + + = - + + = = sin( ) sin cos cos sin α β ± = ± ( , ) ) D x t A x t = ( ) 2 sin A x a kx =
10/06/09 Physics 2212 - Lecture 2 Example : Node Spacing on a String A very long string has a linear density of μ =5.0 g/m and is stretched with a tension of T s =8.0 N. Waves with frequency 100 Hz and amplitude α =2.0 mm are generated at the ends of the string. (a) What is the node spacing along the resulting standing wave? (b) What is the maximum displacement of the string? (8.0 N) 40 m/s (0.0050 kg/m) s T v μ = = = (40 m/s) 0.40 m 40 cm (100 Hz) v f λ= = = = max 2 4.0 mm A α = =

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10/06/09 Physics 2212 - Lecture 2 Reflection and Transmission When a traveling wave moving on a string reaches a point where the mass density ( and velocity) change, the wave “splits” into a reflected wave and an ongoing wave.
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## This note was uploaded on 06/07/2009 for the course PHYSICS 2212 taught by Professor Geist during the Fall '09 term at Georgia Perimeter.

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Phys2212_21.1 to 21. - Physics 2212 Waves Lecture 2 Superposition and Standing Waves Waves vs Particles Particles can collide Superposition fails

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