Sinusoidal Waves A wave source at x = 0 that oscillates with simple harmonic motion (SHM) generates a sinusoidal wave. 2017 Pearson Education, Inc. Slide 16-1 Sinusoidal Waves Above is a history graph for a sinusoidal wave, showing the displacement of the medium at one point in space. Each particle in the medium undergoes simple harmonic motion with frequency f, where f = 1/T. The amplitude A of the wave is the maximum value of the displacement. 2017 Pearson Education, Inc. Slide 16-2 Sinusoidal Waves Above is a snapshot graph for a sinusoidal wave, showing the wave stretched out in space, moving to the right with speed v. The distance spanned by one cycle of the motion is called the wavelength λ of the wave. 2017 Pearson Education, Inc. Slide 16-3 1
QuickCheck A wave on a string is traveling to the right. At this instant, the motion of the piece of string marked with a dot is A. Up. B. Down. C. Right. D. Left. 2017 Pearson Education, Inc. Slide 16-4 Sinusoidal Waves The distance spanned by one cycle of the motion is called the wavelength λ of the wave. Wavelength is measured in units of meters. During a time interval of exactly one period T, each crest of a sinusoidal wave travels forward a distance of exactly one wavelength λ. Because speed is distance divided by time, the wave speed must be or, in terms of frequency: 2017 Pearson Education, Inc. Slide 16-5 The Mathematics of Sinusoidal Waves Define the wave number and angular frequency: ω = 2πv λ = 2π T Wave function: y(x, t) = Asin(kx ωt + ϕ o ) This wave travels at a speed v = ω/k. 2017 Pearson Education, Inc. Slide 16-6 2
Particle velocity and acceleration in a sinusoidal wave Wave equation: 2 y(x, t) x 2 = 1 2 y(x, t) v 2 t 2 All wave behavior obeys the wave equation. Likewise, any physical behavior that satisfies the wave equation can be modeled as a wave. Acos(kx+ωt);. Acos(kx+ωt);. QuickCheck Which of the following equations satisfy the wave equation? 2 y(x, t) x 2 = 1 2 y(x, t) v 2 t 2 A. y(x, t) = Asin(kx ωt) B. y(x, t) = Acos(kx ωt) C. y x, t = Acos kx + Acos(ωt) D. Both A and B. 2017 Pearson Education, Inc. Slide 16-8 Example 1 - A water wave traveling in a straight line on a lake is described by the equation y(x,t) = 2.75cos(0.410x+6.20t) cm (a)how much time does it take for one complete wave pattern to go past a fisherman in a boat at anchor? (b) What horizontal distance does the wave crest travel in that time? (c) What is the wave number? (d) What is the number of waves per second that pass the fisherman? (e) How fast does a wave crest travel past the fisherman? (f) What is the maximum speed of his cork floater as the wave causes it to bob up and down? 3
In-class Activity 1 - Transverse waves on a string have wave speed 8.00 m/s, amplitude 0.0700 m, and wavelength 0.320 m. The waves travel in the -x direction, and at t = 0 the x = 0 end of the string has its maximum upward displacement. (a) Find the frequency, period, and wave number of these waves. (b) Write the wave function describing this wave. The Principle of Superposition If wave 1 displaces a particle in the medium by y 1 and wave 2 simultaneously displaces it by y 2, the net displacement of the particle is y 1 + y 2. It can be easily shown that the superposition of waves still satisfies the wave equation. 2017 Pearson Education, Inc. Slide 17-11 The Principle of Superposition The figure shows the superposition of two waves on a string as they pass through each other. The principle of superposition comes into play wherever the waves overlap. The solid line is the sum at each point of the two displacements at that point. 2017 Pearson Education, Inc. Slide 17-12 4
QuickCheck 17.1 Two wave pulses on a string approach each other at speeds of 1 m/s. How does the string look at t = 3 s? 2017 Pearson Education, Inc. Slide 17-13 QuickCheck 17.2 Two wave pulses on a string approach each other at speeds of 1 m/s. How does the string look at t = 3 s? 2017 Pearson Education, Inc. Slide 17-14 FIGURE 16.20 Constructive interference of two identical waves produces a wave with twice the amplitude, but the same wavelength. 5
FIGURE 16.21 Destructive interference of two identical waves, one with a phase shift of 180 (π rad), produces zero amplitude, or complete cancellation. Superposition of two waves with identical amplitudes, wavelengths, and frequency, but that differ in a phase shift. The resultant wave has a modified amplitude and phase shift but in other ways it is similar to the original waves. y net (x, t) = 2Acos( ϕ o 2 )sin(kx ωt + ϕ o 2 ) Standing waves on a string Waves traveling in opposite directions on a taut string interfere with each other. The result is a standing wave pattern that does not move on the string. Destructive interference occurs where the wave displacements cancel, and constructive interference occurs where the displacements add. At the nodes no motion occurs, and at the antinodes the amplitude of the motion is greatest. 6
Standing Waves Time snapshots of two sine waves. The red wave is moving in the xdirection and the blue wave is moving in the +x-direction. The resulting wave is shown in black. Consider the resultant wave at the points x = 0 m, 3 m, 6 m, 9 m, 12 m, 15 m and notice that the resultant wave always equals zero at these points, no matter what the time is. These points are known as fixed points (nodes). In between each two nodes is an antinode, a place where the medium oscillates with an amplitude equal to the sum of the amplitudes of the individual waves. FIGURE 16.27 y x, t = 2Asin kx cos(ωt) The sine function dictates the position of the standing waves while the cosine function expresses how the shape oscillates with time. Example 2 A guitar string is plucked and creates a standing sinusoidal wave with amplitude 0.750 mm and frequency 440 Hz. The wave velocity is 143 m/s. (a) Find the equation of the standing wave. (b) Locate the nodes (c) Find the maximum speed and acceleration of the string. 7
Standing waves on a string This is a time exposure of a standing wave on a string. This pattern is called the second harmonic. Standing waves on a string As the frequency of the oscillation of the right-hand end increases, the pattern of the standing wave changes. More nodes and antinodes are present in a higher frequency standing wave. Normal modes For a taut string fixed at both ends, the possible wavelengths are and the possible frequencies are f n = n v/2l = nf 1, where n = 1, 2, 3, f 1 is the fundamental frequency, f 2 is the second harmonic (first overtone), f 3 is the third harmonic (second overtone), etc. The figure illustrates the first four harmonics. 8
Example 3 - Adjacent antinodes of a standing wave on a string are 15.0 cm apart. A particle at an antinode oscillates in simple harmonic motion with amplitude 0.850 cm and period 0.0750 s. The string lies along the +x-axis and is fixed at x = 0. (a) How far apart are the adjacent nodes? How far from antinodes? (b) Find the wavelength, amplitude, and speed of the standing wave. (c) Find the wavelength, amplitude, and speed of the traveling waves. (c) Find the max speed of the string. In-class Activity #2 A standing wave on a wire has an amplitude of 2.40 mm, an angular frequency of 934 rad/s, and wave number 0.750π rad/m. The left end of the wire is at x = 0. At what distances from the left end are (a) the nodes of the standing wave? (b) the antinodes of the standing wave? (c) What is the node to antinode distance? 9