Dr. Jie ZouPHY 13711 Chapter 18 Superposition and Standing Waves.

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Dr. Jie ZouPHY Chapter 18 Superposition and Standing Waves

Dr. Jie ZouPHY Outline Superposition and interference Superposition principle Constructive and destructive interference Superposition of sinusoidal waves Interference of sound waves

Dr. Jie ZouPHY Superposition principle Superposition principle: If two or more traveling waves are moving through a medium, the resultant value of the wave function at any point is the algebraic sum of the values of the wave functions of the individual waves. Linear waves and nonlinear waves

Dr. Jie ZouPHY Superposition of two pulses Two traveling waves can pass through each other without being destroyed or even altered. Interference: The combination of separate waves in the same region of space to produce a resultant wave is called interference. Constructive interference Destructive interference

Dr. Jie ZouPHY Superposition of sinusoidal waves Two sinusoidal waves traveling in the same linear medium: y 1 = A sin(kx-  t) y 2 = A sin(kx-  t+  ) Resultant wave function: y = y 1 + y 2 = 2A cos(  /2)sin(kx-  t+  /2). If  = 0, 2 , 4  …, y 1 and y 2 are in phase and interfere constructively; y has an amplitude = 2A. If  = , 3  …, y 1 and y 2 are  rads out of phase and interfere destructively; y has an amplitude = 0. If 0 <  < , y has an amplitude between 0 and 2A.

Dr. Jie ZouPHY Interference of sound waves A phase difference may arise between two waves generated by the same source when they travel along paths of unequal lengths. Because a path difference of one wavelength corresponds to a phase angle of 2  rad,  /2  =  r/, or  r= (  /2  ). Conditions for constructive and destructive interference:  r = (2n) ( /2) for constructive interference  r = (2n+1) ( /2) for destructive interference Some definitions: Path length r: the distance along any path from speaker to receiver. Path difference  r = |r 2 – r 1 |

Dr. Jie ZouPHY Graphical example of interference of sound waves

Dr. Jie ZouPHY Example 18.1 A pair of speakers placed 3.00 m apart are driven by the same oscillator. A listener is originally at point O, which is located 8.00 m from the center of the line connecting the two speakers. The listener then walks to point P, which is a perpendicular distance m from O, before reaching the first minimum in sound intensity. What is the frequency of the oscillator?

Dr. Jie ZouPHY Homework Ch. 18, Problems: #1, 5, 8, 11.