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Dr. Jie ZouPHY 13711 Chapter 16 Wave Motion (Cont.)

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Dr. Jie ZouPHY 13712 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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Dr. Jie ZouPHY 13713 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

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Dr. Jie ZouPHY 13714 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)?

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Dr. Jie ZouPHY 13715 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)

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Dr. Jie ZouPHY 13716 Periodic nature of the wave function The periodic nature of (1) At any given time t, y has the same value at the positions x, x+, x +2, and so on. (2) 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

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Dr. Jie ZouPHY 13717 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.

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Dr. Jie ZouPHY 13718 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.

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Dr. Jie ZouPHY 13719 The motion of any particle on the string Assuming wave function is: y = A sin(kx - t) Transverse speed v y,max = A Transverse acceleration a y,max = 2 A Example (Problem #17): A transverse wave on a string is described by the wave function y = (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?

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Dr. Jie ZouPHY 137110 The speed of waves on strings The speed of a transverse pulse traveling on a taut string: T: Tension in the string. : Mass per unit length in the string. 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.

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Dr. Jie ZouPHY 137111 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 at B) – (Phase at A) = /3 rad; then solve for x B.

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