WAVES AND WAVE PHENOMENA

Physics 12

Source: Giancoli Chapter 11

Objectives

Waves

What are some examples of waves in the “real” world?

Do these waves transmit energy or matter? How do you know?

How does the velocity of the particles that transmit the wave relate to the velocity of the wave itself?

Waves

The source of any wave is a pulse of energy. If the pulse is consistent (as in a vibration) the

wave becomes continuous.

What are some kinds of waves that you can think of?

Waves

Two common types of waves are mechanical waves and electromagnetic waves.

We will focus on mechanical waves.

Waves

Important quantities that describe wavesTerm Definition

displacement, x the movement of particles above and below the mean position

amplitude, A the difference between the maximum displacement and the mean position

period, T the amount of time required to complete one cycle (between two identical positions)

frequency, f the number of complete cycles passing a given point in one second (Hz)

wavelength, λ the distance covered in a complete wave cycle.

wave velocity, v the velocity at which wave crests move

Waves

Wave motion

Suppose the x-axis is time, t. How do we find the velocity, v, at which the wave moves?

Waves

A wave crest travels the distance of one wavelength, λ, in a time equal to one period, T.

Therefore, the wave velocity is v = λ/T

orv = λf

What factors might affect the velocity of a wave?

Waves

The medium in which the wave travels is very important: think of sound through air and sound through water.

For a cord, the velocity varies accordingly:

v =

where FT is the tension in the cord, m/L is mass per unit length

Waves

v =

What happens to v if the cord is tighter? What happens if the cord is heavier? Why is this

so?

Waves

A wave whose wavelength is 0.30 m is traveling down a 300-m long wire whose total mass is 15 kg. If the wire is under a tension of 1000 N, what are the speed and frequency of the wave?

Waves

v = 140 m/sf = 470 Hz

Types of waves

There are two main classifications of waves:1. transverse2. longitudinal

They differ in the relationship of the direction of particle motion and the direction of wave motion.

Types of waves

Transverse wave

What are some examples of a transverse wave?

Types of waves

Longitudinal wave

What is a common example of a longitudinal wave?

Mathematical description

The location of a point on a wave, y, depends on two variables—position and time (x and t), such that:

y = A sin (ωt ± κx)where A is the amplitude, ω is the angular frequency (2πf) and κ is the angular wave

number (2πk)

k is the propagation constant

Superposition of waves

What happens when two waves meet?

Superposition of waves

When two waves meet, they are said to interfere with each other.

There are two types of interference: constructive and destructive.

When do you think each type of interference occurs?

Superposition of waves

When two waves have the same displacement (either both positive or both negative) when they meet, the resultant wave will be greater than each individual wave.

If the two waves are exactly in phase, maximum constructive interference will occur.

The amplitude of the resultant wave will be the sum of the two individual waves’ amplitudes.

Superposition of waves

Constructive interference

Superposition of waves

When two waves have opposite displacements when they meet, destructive interference will occur.

If the waves are out of phase, then maximum destructive interference will occur.

The amplitude of the resultant wave will be the difference of the amplitudes of the individual waves.

Superposition of waves

Destructive interference

Superposition of waves

Two waves, one with an amplitude of 8 cm and the other with an amplitude of 3 cm, travel on a single string and overlap. What are the maximum and minimum amplitudes of the string while these waves overlap?

Superposition of waves

11 cm and 5 cm

Resonance

When you vibrate a cord at just the right frequency, you produce a standing wave.

It is called “standing” because the waves do not appear to be moving.

The points that are still are called nodes and the points with maximum amplitude are called antinodes.

Resonance

Standing waves

Resonance

Standing waves can occur at numerous frequencies.

The frequencies at which standing waves occur are called natural frequencies or resonant frequencies.

Resonance

Resonant standing waves (Harmonics)

What is the relationship between L and λ in each standing wave?

Resonance

L =

where L is the length of the string and n is the number of the harmonic

How do we determine the frequency f that these harmonics occur at?

Resonance

If L = , then λn =

Furthermore, if fn = vn / λn

then fn = = = nf1

Resonance

A piano string is 1.10 m long and has a mass of 9.00 g.

(a) How much tension must the string be under tot vibrate at a fundamental frequency of 131 Hz?

(b) What are the frequencies of the first four harmonics?

Resonance

(a) 679 N(b) 262 Hz, 393 Hz, 524 Hz