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Standing Waves

Thus the sine function with a time dependent phase

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Unformatted text preview: x is ym. Thus, the sine function with a time-dependent phase corresponding to the oscillation of a string element, and Figure 2 Diagram of the wavelength of a wave the amplitude of the function corresponds to the http://cimss.ssec.wisc.edu/satmet/modules/spectrum/wa velength.html extremes of the element’s displacement. 2.4.2 Wavelength and Angular Wave Number The wavelength λ of a wave is the distance (parallel to the direction of the wave’s travel) between repetitions of the shape of the wave (or wave shape). A typical wavelength is shown in Figure 2. The description of the wave shape from equation (4) can be expressed as ,0 sin (5) By definition, the displacement y is the same at both ends of this wavelength. This means that (6) (7) Thus by equation (5), we get sin = sin (8) (9) A sine function begins to repeat itself when its angle (or argument) is increased by 2π rad, so equation (9) must have kλ=2π or 2 (10) We call k the angular wave number of the wave: its SI unit is the radian per meter, or the inverse meter. (note that the symbol k here does NOT represent a spring constant as in the Simple Harmonic Motion Lab.) 3 2.5 The Principle of Superposition for Waves It often happens that two or more waves pass simultaneously through the same region. When we listen to a concert, for example, sound waves from many instruments fall simultaneously on our eardrums. Suppose that two waves travel simultaneously along the same stretched string. Let y1(x,t) and y2(x,t) be the displacements that the strings would experience if each wave traveled alone. The displacement of the string when the waves overlap is the algebraic sum , , , (11) This summation of displacements along the string means that Overlapping waves algebraically add to produce a resultant wave (or net wave) This is another example of the principle of superposition which Figure 1 Superposition of waves says that when several effects occur simultaneously, their net www.wfu.edu/physics/pira/PhysicsDrawings.htm effect is the sum of the individual effects. Figure 1 shows the overlapping of two waves. As we can see, when the pulses overlap, the resultant pulse is their sum. Moreover, each pulse moves through the other wave as if the other wave was not present: Overlapping waves do not in any way alter the travel of each other. 2.6 Standing Waves In the previous section, we discussed two sinusoidal waves of the same wavelength and amplitude traveling in the same direction along a stretched string. What if they travel in opposite directions? We again can find the resultant wave by applying the superposition principle. The outstanding feature of the resultant wave is that there are places along the string, called nodes, where the string never move...
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