Wave Motion

Chapter 16 deals with the general properties of waves. We will be considering several different types of waves; however, here we’ll concentrate on those characteristics that are common to all waves.

First: What is a Wave?

Consider waves on a pond

General Definition of a Wave:

“A wave is an organized disturbance that travels at a well defined speed”

Note: the disturbance propagates, the medium (if there is one) has no bulk motion. (The medium can oscillate, but it doesn’t move with the wave.)

Also, waves are very different than particles which are localized (i.e. they exist at a point). Waves are spread out in space, but they do share a feature with particles: Waves carry energy and momentum.

Sometimes things move by travelling oscillations – Waves.

Types of Waves We will consider three general types of waves:

1.) Mechanical Waves

• Require a medium to propagate • The wave speed is determined by the elastic properties and inertia

of the medium • Oscillations can be transverse or longitudinal to the wave direction • Video of mechanical waves (be honest, is this video too boring?) • Examples: wave on a string (PhET), sound waves (PhET), water waves

(me playing on a wave last summer), stadium wave(?).

2.) Electromagnetic Waves

• Requires no medium to propagate (. . . interesting) • Disturbances in the electromagnetic fields that travel in vacuum at the

speed of light, c = 3 X 108 m/s. • e.g. visible light, radio waves, x-rays, etc. (show on PhET)

3.) Matter Waves

• At the level of fundamental particles, like electrons, particles have wave properties; This is Quantum Mechanics; What’s doing the waving?

Wave Speed for a String Consider a transverse wave pulse on a string:

Equilibrium position of the string

PheT

As derived in your text, the speed of the wave is:

Where:

PheT

Whiteboard Problem 16-1

The wave speed on a string is 150 m/s when the tension is 75N. What tension will give a speed of 180 m/s? (LC)

Travelling Sinusoidal Wave

A sinusoidal (or harmonic) disturbance creates a sinusoidal travelling wave.

At a given time, this wave is a sine wave in space, and at a given point in space, a point has harmonic motion in time.

Where D(x,t) is the general disturbance from the equilibrium state. Note: it is a function of two variables.

PheT

The Equation of Travelling Sinusoidal Wave

“Fundamental Relation for Waves”

General Equation of a sinusoidal travelling wave:

Travelling in +x direction

Travelling in -x direction

(watch the k’s !)

One Other Thing About the Travelling Sinusoidal Wave

Remember what we had for the simple harmonic oscillator:

The differential equation: The solution:

There’s something similar for waves:

The solution: The differential equation:

Computer Activity: Expression for a Travelling Wave

When you’re done with the exercise, make sure your Group’s name and all of your names are on it, and turn it in.

Using the group’s computer, one of you log onto Masteringphysics and begin the exercise: Expression for a Travelling Wave

Use the computer to complete the exercise, but record your answers on the sheet to hand in – that’s what will be graded. As you answer the questions, start up the PhET simulation Wave on a String, and try to create and examine waves of different amplitude, frequency, etc. Make sure that you have “No End” selected and turn the damping to “None”.

http://phet.colorado.edu/en/simulation/wave-on-a-string

Whiteboard Problem 16-2

The displacement of a wave traveling in the positive x direction is

D(x,t) = (3.5 cm) sin (2.7x – 124 t) where x is in meters and t is in seconds.

What are the a) frequency? b) wavelength? c) speed of the wave? d) displacement D at x = 5.2m and t = 3.6s? (LC)

Whiteboard Problem 16-3

Write the displacement equation for a sinusoidal wave that is traveling in the negative y-direction with a wavelength of 50 cm, speed of 4.0 m/s, and an amplitude 5.0 cm. (assume the phase constant is zero.)

Waves in 2 and 3 Dimensions

Wave fronts

In 2D or 3D, the amplitude of the wave will decrease since the energy is spread out over a larger circle (in 2D) or a sphere (in 3D). So a sinusoidal wave looks like:

Source

Note: this is for an outgoing wave.

waves on a pond again.

Wave Phase

For any sinusoidal wave (e.g. 1D):

The wave phase determines where you are on the wave, i.e. a peak, zero, trough, or somewhere in between.

In chapter 17, we’ll look at combining waves, and the difference in phase will be very important.

Phase Difference Between Two Points at the Same Time.

Consider a 1D single wave travelling to the right at some time t:

Phase difference between points 1 and 2:

(1 full cycle)

(1/2 cycle)

Whiteboard Problem 16-4

A spherical wave with a wavelength of 2.0 m is emitted from the origin. At one instant of time, the phase at r = 4.0 m is radians. At this same time, what is the phase at

a) r = 3.5 m? (LC) b) r = 4.5 m?