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.

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

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