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Chapter 16

Chapter 16. Wave Motion (Cont.). Example 16.1. A wave pulse moving to the right along the x axis is represented by the wave function where x and y are measured in cm and t is in seconds. Plot the wave function at t = 0, t = 1.0 s, and t = 2.0 s. Find the speed of the wave pulse.

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Chapter 16

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  1. Chapter 16 Wave Motion (Cont.) PHY 1371

  2. Example 16.1 • A wave pulse moving to the right along the x axis is represented by the wave function where x and y are measured in cm and t is in seconds. • Plot the wave function at t = 0, t = 1.0 s, and t = 2.0 s. • Find the speed of the wave pulse. PHY 1371

  3. Outline • Sinusoidal waves • Basic variables • Wave function y(x, t) • Sinusoidal waves on strings • The motion of any particle on the string • The speed of waves on strings PHY 1371

  4. Basic variables of a sinusoidal wave • Crest: The point at which the displacement of the element from its normal position is highest. • Wavelength : The distance between adjacent crests, adjacent troughs, or any other comparable adjacent identical points. • Period T: The time interval required for two identical points of adjacent waves to pass by a point. Frequency: f = 1/T. • Amplitude A: The maximum displacement from equilibrium of an element of the medium. What is the difference between (a) and (b)? PHY 1371

  5. Wave function y(x, t) of a sinusoidal wave • y(x, 0) = A sin(2x/) • At any later time t: • If the wave is moving to the right, • If the wave is moving to the left, • Relationship between v, , and T (or f): v = /T = f • An alternative form of the wave function: Consider a special case: suppose that at t = 0, the position of a sinusoidal wave is shown in the figure below, where y = 0 at x = 0. (Traveling to the right) PHY 1371

  6. Periodic nature of the wave function • The periodic nature of • At any given time t, y has the same value at the positions x, x+, x +2, and so on. • At any given position x, the value of y is the same at times t, t+T, t+2T, and so on. • Definitions: • Angular wave number k = 2/ • Angular frequency  = 2/T • v = /T = f = /k • An alternative form: y = A sin(kx - t) (assuming that y = 0 at x = 0 and t = 0) • General expression: y = A sin(kx -t+ ) • : the phase constant. • kx -t+ : the phase of the wave at x and at t PHY 1371

  7. Example 16.2 • A sinusoidal wave traveling in the positive x direction has an amplitude of 15.0 cm, a wavelength of 40.0 cm, and a frequency of 8.00 Hz. The vertical displacement of the medium at t= 0 and x = 0 is also 15.0 cm, as shown in the figure. • (a) Find the angular wave number k, period T, angular frequency , and the speed v of the wave. • (b) Determine the phase constant , and write a general expression for the wave function. PHY 1371

  8. Sinusoidal waves on strings • Producing a sinusoidal wave on a string: The end of the blade vibrates in a simple harmonic motion (simple harmonic source). • Each particle on the string, such as that at P, also oscillates with simple harmonic motion. • Each segment oscillates in the y direction, but the wave travels in the x direction with a speed v – a transverse wave. PHY 1371

  9. The motion of any particle on the string • Example (Problem #17): A transverse wave on a string is described by the wave functiony = (0.120 m) sin [(x/8) + 4 t]. • (a) Determine the transverse speed and acceleration at t = 0.200 s for the point on the string located at x = 1.60 m. • (b) What are the wavelength, period, and speed of propagation of this wave? • Assuming wave function is: y = A sin(kx - t) • Transverse speed • vy,max=A • Transverse acceleration • ay,max=2A PHY 1371

  10. The speed of waves on strings • Example 16.4: A uniform cord has a mass of 0.300 kg and a length of 6.00 m. The cord passes over a pulley and supports a 2.00-kg object. Find the speed of a pulse traveling along this cord. • The speed of a transverse pulse traveling on a taut string: • T: Tension in the string. • : Mass per unit length in the string. PHY 1371

  11. Homework • Ch. 16, P. 506, Problems: #10, 12, 13, 16, 21. • Hints: • In #10, find the wave speed v first. • In #12, set the phase difference at point A and B to be: (Phase atB) – (Phase at A) =  /3 rad; then solve for xB. PHY 1371

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