Chapter 18

CHAPTER 18

18.1 Superposition dan Interference

To  analyze  such wave  combinations,  one  can make  use  of  the  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.

Waves that obey this principle are called linear waves. Waves  that  violate  the  superposition principle are called nonlinear waves. One consequence of  the superposition principle  is  that two traveling waves canpass

through each other without being destroyed or even  altered.

Figure  18.1  is  a  pictorial  representation  of  the  superposition  of  two  pulses.  The wave  function  for  the pulse moving  to  the  right  is  y1,  and  the wave  function  for  the pulse moving to the left is y2. The pulses have the same speed but different shapes, and the displacement of the elements of the medium is in the positive y direction for both pulses. When the waves begin to overlap (Fig. 18.1b), the wave function for the resulting  complex  wave  is  given  by  y1 ! y2.  When  the  crests  of  the  pulses  coincide (Fig. 18.1c), the resulting wave given by y1 ! y2 has a larger amplitude than that of the individual pulses. The two pulses finally separate and continue moving in their originaldirections (Fig. 18.1d).

            The combination of separate waves in the same region of space to produce a resultant wave is called interference.. Because the displacements caused by the two pulses are in the same direction, we refer to their superposition as constructive interference. the  two pulses pass through each other; however, because the displacements caused by the two pulses are in opposite directions, we refer to their superposition as destructive interference..

Superposition of Sinusoidal Waves

            If the two waves are traveling to the right and have the  same  frequency,  wavelength,  and  amplitude  but  differ  in  phase,  we  can  express their individual wave functions as

hence the resultant wave is

If  we  let 

This result has  several  important  features. The resultant wave  function y also  is  sinusoidal and has the same frequency and wavelength as the individual waves because the sine function incorporates the same values of k and $ that appear in the original wave functions.

18.2 Standing Waves

A standing wave, such as  the one shown  in Figure 18.8,  is an oscillation pattern with a stationary outline that  results  from  the  superposition of  two  identical waves  traveling  in opposite directions.

The maximum amplitude of an element of  the medium has a minimum  value of zero when x satisfies the condition sin kx = 0, that is, when

Because , these values for kx give

These points of zero amplitude are called nodes. The positions in the medium at which  this maximum displacement occurs are  called antinodes. The antinodes  are  located  at  positions  for which  the  coordinate  x satisfies  the  condition sin kx = ±1, that it when

Thus, the positions of the antinodes are given by

In examining Equations 18.4 and 18.5, we note the following important features of

18.3 Standing Waves in a String Fixed at Both Ends           

Consider a  string of  length L fixed at both ends, as  shown  in Figure 18.10. Standing waves are set up  in the string by a continuous superposition of waves  incident on and reflected from the ends. Note that there is a boundary condition for the waves on the string.  The  ends  of  the  string,  because  they  are  fixed,  must  necessarily  have  zero displacement and are, therefore, nodes by definition. The boundary condition results in the string having a number of natural patterns of oscillation, called normal modes, each of which has a characteristic  frequency  that  is easily calculated. This situation  in which only certain frequencies of oscillation are allowed is called quantization. Quantization  is a common occurrence when waves are  subject  to boundary conditions

The  first  normal  mode  that  is  consistent  with  the boundary conditions, shown in Figure 18.10b, has nodes at its ends and one antinode in the middle. This is the longest-wavelength mode that is consistent with our requirements. This first normal mode occurs when  the  length of  the  string  is half  the wave length  The next normal mode (see Fig.18.10c) of wavelength t occurs when  the wavelength equals  the  length of  the string, that is, when The third normal mode (see Fig. 18.10d) corresponds to the case in which In  general,  the wavelengths  of  the  various normal modes  for  astring of length L fixed at both ends are

where the index n refers to the nth normal mode of oscillation. These are the possible modes of oscillation  for  the  string. The actual modes  that are excited on a  string are discussed shortly. The natural  frequencies  associated with  these modes  are obtained  from  the  relationship f=v/a,  where the wave speed v  is the same for all frequencies. Using Equation 18.6, we find that the natural frequencies  of the normal modes are

These natural frequencies are also called the quantized frequencies associated with the vi-brating string fixed at both ends

The lowest frequency f1, which corresponds to n =1, is called either the fundamental or the fundamental frequency and is given by

The frequencies of the remaining normal modes are  integer multiples of the fundamental frequency. Frequencies of normal modes that exhibit an integer-multiple relationship  such  as  this  form  a  harmonic  series

18.4. Resonance

            We  have  seen  that  a  system  such  as  a  taut  string  is  capable  of  oscillating  in  one  or more normal modes of oscillation. If a periodic force is applied to such a system, the amplitude of the resulting motion is greatest when the frequency of the applied force is equal to one of the natural frequencies of the system. We discussed this  phenomenon,  known  as  resonance, briefly  in  Section  15.7.  Although  a block–spring system or a simple pendulum has  only one natural frequency, standing-wave systems have a whole  set of natural  frequencies,  such as  that given by Equation 18.7 for a string. Because an oscillating system exhibits a large amplitude when driven at any of its natural frequencies, these frequencies are often referred to as resonance frequencies.

            Figure  18.14  shows  the  response  of  an  oscillating  system  to  various  driving  frequencies, where one of the resonance frequencies of the system is denoted by f0 . Note that  the amplitude of oscillation of  the  system  is greatest when  the  frequency of  the driving force equals the resonance frequency. The maximum amplitude  is  limited by friction in the system. If a driving force does work on an oscillating system that is initially at rest, the input energy is used both to increase the amplitude of the oscillation and  to overcome  the  friction  force. Once maximum amplitude  is  reached,  the work done by the driving force is used only to compensate for mechanical energy loss due to friction.

Interference occurs if the following two conditions are met:  
1. These two light waves must be coherent, in the sense that the two waves of light must have a phase difference which is always fixed, and therefore both must have the same frequency 
2. These two light waves must have nearly the same amplitude