LECTURE 4TRANSVERSE AND LONGITUDINAL WAVES

Instructor: Kazumi Tolich

Lecture 4

¨ Reading chapter 15-1 to 15-2¤ Transverse and longitudinal waves¤ Moving displacement of a wave pulse¤ Speed of waves¤ Harmonic waves

2

Quiz: 13

Transverse & longitudinal waves

¨ 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

4

Demo 1 & 2

¨ Wave on Rope¤ An example of transverse waves.

¨ Longitudinal Waves (Large Spring or Slinky)¤ An example of longitudinal waves.

5

Speed of waves

¨ 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

¤ FT is the tension in the string.¤ µ is the mass density (mass per unit length) of the string.

6

Demo 3

¨ Tension Dependence of Wave Speed

7

Quiz: 2 & 38

Harmonic waves

¨ 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

9

Moving displacement

¨ 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 𝑥-direction, 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 .

10

Harmonic transverse wave function

¨ 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.

11

Demo 4

¨ 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.)

12

Quiz: 4

Example 1

¨ The wave function for a harmonic wave on a string is𝑦 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?

14

Energy transfer via a wave on a string

¨ 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

15

Example 2

¨ A harmonic wave on a string that has a mass per unit length of µ = 0.050 kg/m and a tension of FT = 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?

16