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

Dr. Jie ZouPHY 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.

Dr. Jie ZouPHY 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

Dr. Jie ZouPHY 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)?

Dr. Jie ZouPHY 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)

Dr. Jie ZouPHY 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

Dr. Jie ZouPHY 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.

Dr. Jie ZouPHY 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.

Dr. Jie ZouPHY 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 = 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?

Dr. Jie ZouPHY 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 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.

Dr. Jie ZouPHY 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.