LECTURE 3 TRAVELING WAVES Instructor: Kazumi Tolich
Lecture 3 2 Reading chapter 15-1 to 15-2 Transverse and longitudinal waves Moving displacement of a wave pulse Speed of waves Harmonic waves
3 Quiz: 1
Transverse & longitudinal waves 4 Mechanical wave is the disturbance, that travels through a medium, carrying energy with it. Transverse waves are waves in which the motion of the medium is perpendicular to the direction of propagation of the disturbance. Waves on a string Longitudinal waves are waves in which the motion of the medium is along (parallel to) the direction of propagation of the disturbance. Sound waves
Demo 1 & 2 5 Wave on Rope An example of transverse waves. Longitudinal Waves (Large Spring or Slinky) An example of longitudinal waves.
Speed of waves 6 Speed of waves relative to the medium in which they are traveling depends on the medium. Speed of waves on a string is given by F T is the tension in the string. µ is the mass density (mass per unit length) of the string.
Demo 3 7 Tension Dependence of Wave Speed
8 Quiz: 2 & 3
Harmonic waves 9 If disturbances occur due to periodic motion, they produce a periodic wave. If a harmonic wave is traveling through a medium, each point of the medium oscillates in simple harmonic motion. Wavelength, λ, of a wave is the distance that wave travels in one cycle. Period, T, is the duration of one cycle. The speed of the wave propagation is given by
Moving displacement 10 Suppose we wish to move a parabolic function f x = x & so that it is centered at greater and greater values of the independent variable x. Then we change x to x 2, x 4, etc. Similarly, if we wish to displace some arbitrary function f x by a distance d in the positive xdirection, we replace f x by f x d. If we wish to make the displacement increase with time, i.e., move the curve with speed v, we make d = vt, so that the function becomes f x vt. Similarly, to displace the function in the negative x direction, we use f x + vt.
Harmonic transverse wave function 11 For a transverse harmonic wave traveling in the positive x-direction, the wave function is k is called wave number and is given by ω is the angular frequency. A is the amplitude, the maximum displacement in the y-direction.
Quiz 3-4
Example 1 13 The wave function for a harmonic wave on a string is y x, t = 1.00 mm sin 62.8 m :; x + 314 s :; t. a) In what direction does this wave travel, and what is the wave s speed? b) Find the wavelength, frequency, and period of this wave. c) What is the maximum speed of any point on the string?
Energy transfer via a wave on a string 14 As a wave moves along a string, energy is transferred from one segment to the next. Average power are given by P >? = ; & μvω& A & The average energy flowing at a point P during Δt is This energy is distributed over a length Δx = vδt, so the average energy in length Δx is
Example 2 15 A harmonic wave on a string that has a mass per unit length of µ = 0.050 kg/m and a tension of F T = 80 N has an amplitude of A = 5.0 cm. Each point on the string moves with simple harmonic motion at a frequency of f = 10 Hz. What is the power carried by the wave propagating along the string?
Demo 4 16 Transverse Waves (Bell Labs Wave Machine) Torsional waves sent on a series of rods. The ends of the rods move up and down. The shorter the rods, the faster the wave travels. (Each rod connected to the wire acts like a torsion pendulum. The shorter rod has a smaller I, so the period of oscillation is shorter.)
Harmonic sound waves 17 A harmonic sound wave traveling in the positive x- direction can be described by s 0 is the amplitude, the maximum displacement in the x-direction. The pressure wave is given by p x, t = p D cos kx ωt where p D = ρωvs D. p stands for the change in pressure from the equilibrium pressure p 0 is the pressure amplitude, the maximum value of p
Energy of sound waves 18 The average energy of a harmonic sound wave in a volume element ΔV is given by Energy per unit volume is given by
Demo 5 19 Siren in Vacuum Demonstration of sound wave traveling in the air.