1. LO 6 Standing Waves Adria Lwin 2. What are standing waves? The interference of two waves with equal wavelength, frequency, and amplitude that are moving in OPPOSITE directions. 3. Demonstration of a Standing Wave Notice that the travelling blue and green waves are moving in opposite directions. The red wave is the sum of the blue and green waves (the standing wave). When the two travelling waves are aligned, the resultant wave achieves maximum amplitude (at antinodes). When the two travelling waves are directly opposite each other, the resultant wave is flat. Click on the picture above to play the GIF http://upload.wikimedia.org/wikipedia/commons/6/62/Harmonic_Standing_Wave.gif 4. Equation of a Standing Wave D(x,t) = 2A sin(kx) cos(t) Let us define a position-dependent amplitude: A(x) = 2A sin(kx) This lets us rewrite the previous equation: D(x,t) = A(x) cos(t) 5. Parts of a Standing Wave Node: Points on a standing wave with zero amplitude that remain at rest at all times. Each segment between two nodes oscillates with simple harmonic motion. Its energy continuously transforms between kinetic and potential energy. Antinode: Points on a standing wave which achieve maximum amplitude of 2A. http://www.revisesmart.co.uk/images/resources/physics/nodes-antinodes.png 6. Locations of Nodes and Antinodes At a node, A(x) = 0, therefore, x = 0, /2, 3/2, 2, At an antinode, A(x) = 2A, therefore , x = /4, 3/4, 5/4, (Check page 224 for full derivations.) The distance between two consecutive nodes and antinodes is half a wavelength (/2). The distance between an adjacent node and antinode is a quarter of a wavelength apart. 7. Travelling Waves vs. Standing Waves Carries energy Does not carry energy Has a fixed amplitude, A Amplitude varies as a function of position, between 0 and 2A Not confined to a given space Is confined to a given space (ie. between two walls) Can have any frequency Frequency is quantized Wavelength is the shortest distance over which the waveform repeats itself Wavelength is two times the distance between two nodes or two antinodes D(x,t) = A sin(kx t + ) D(x,t) = 2A sin(kx) cos(t) http://media.tumblr.com/tumblr_m8keuyIx7l1qf3sji.gif http://media.tumblr.com/tumblr_m8kevdFuZn1qf3sji.gif 8. Practice Questions 1. A 20 m long standing wave has three nodes (not including the first and last nodes). What is the wavelength of this standing wave? 2. A standing wave is given by D(x, t) = 10 sin(4x) cos(4t). What is: A. The amplitude, frequency and equations of the constituent waves. B. The distance between two nodes or antinodes. C. The amplitude of a particle whose position along the x-axis is 36 cm. 9. Answer 1 The wavelength is 10 m. We know that: The length is 20 m There are 3 nodes Recall: The distance between two consecutive nodes is /2 10. Solution 1 11. Answer 2 A. Amplitude = 5m, frequency = 2 Hz, wavelength = /2 m B. /4 m C. 9.91 m 12. Solution 2