Waves Study Guide (page 2)

Introduction

This lesson studies the general characteristics of waves such as frequency, speed, and wavelength; sound waves and propagation in a medium; and electromagnetic waves and electromagnetic spectrum.

Characteristics of a Wave

The physical appearance of a water wave or of a string that's fixed at one end but pulsed vertically at the other is very similar to the time dependence of the displacement we have studied in the previous lesson. The quantities we have defined there will be reused in the case of waves.

Waves are disturbances produced in matter by the interaction with a source that supplies an energy. This energy comes from the wind in the case of the ocean waves, from an earthquake in the case of tsunami waves, or from your finger in the case of guitar strings.

Also, a wave has the disturbance propagated through the material with no bulk flow of matter; therefore, a wave is not the motion of masses of water but simply the propagation of the energy from one position to another. The motion of a particle of water in a wave is actually circular, as shown in Figure 17.1.

The water particle is seen to move up and down and not in the direction of propagation of the wave. In conclusion, waves have some characteristics that are worth summarizing: They represent a disturbance in the matter, they carry energy through the matter, and they do not involve bulk flow of matter.

The way the water particles travel while acted on by a wave is the more complex part of the idea of a wave. We can break the behavior down into two more simple types of propagation. According to the relative direction of the disturbance with respect to the propagation of the wave, there are longitudinaland transversal waves.

Longitudinal waves have the disturbance and the wave propagation parallel to each other, such as in the case of a slinky set on a table and pushed back and forth. Regions along the slinky will be compressed, and others will be stretched. See Figure 17.2.

Transversal waves have the disturbance and the wave propagation perpendicular to each other, such as in the case of a slinky set on a table and pulsed up and down.

If you now take a string and fix it at one end and leave the other one free as seen in Figure 17.3, and at the free end, you start applying an up and down pulse, the string will be disturbed in a vertical direction while the wave will propagate to the right as in Figure 17.4.

As one can see, the waves discussed here have the same cyclic behavior as encountered previously in the simple harmonic motion. We will call these waves periodic waves, and we will define their period, frequency, amplitude, and speed, as we did for harmonic motion. See Figures 17.5 and 17.6 for graphic representations of these quantities.

The period will be the time it takes a point acted on by the disturbance to repeat motion, and it is the inverse of the frequency (which represents the number of complete cyclic motions that a point describes in a second). The wavelength is the distance between two identical positions that the wave has reached. And the amplitude is the maximum displacement of the particle relative to the equilibrium position.

The speed of the wave is defined as the ratio of the wavelength to the period:

v =

But because the period and frequency are inversely proportional, we have also:

v = f · λ

Example

The laboratory motion sensors used in mechanics experiments are ultrasonic sensors that send a wave out and measure the amount of time to its return. By knowing the time and the speed of the sound, you can measure the position of different objects relative to the position of the sensor. Considering the speed of sound to be 343 m/s in a room where the temperature is 20° C, find the mass of a molecule of the gas present in the room. (γ is 1.2.)

Solution

First, inventory your known data and then set the equation necessary to find the mass.

v = 343 m/s

T = t + 273.15 K = 20° C + 273.15 K = 293.15 K

γ = 1.2

m = ?

v = =

v =

v² =

m =

m =

m = 1.41 · 10^–23 kg

In the case of a liquid, the speed of sound is:

v =

where K is a coefficient that depends on the pressure change with volume and ρ is the density of the liquid. Both factors are material dependent.

And lastly, the speed of sound is also affected by the nature of the solid in which it travels. The expression is:

v =

where E is Young's modulus that measures the elastic properties of an object subject to a force, and ρ is the density of the solid.

Electromagnetic Waves

Mechanical sources produce mechanical waves that require a medium in which to propagate, but there are other sources that produce waves that can travel through a vacuum. The fluctuation of a magnetic and electric field can produce a propagating wave called an electromagnetic wave.

How is it possible to have propagation without mass? Remember a few lessons ago, we learned that a current can induce a magnetic field and a magnetic field can induce a current. The two factors together can propagate through space with no requirement of matter. The magnetic and electric field fluctuation was studied by James Maxwell (1831-1879), and the diagram of the effect is shown in Figure 17.8.

Although electromagnetic waves propagate through media, the maximum speed is achieved in a vacuum. And for the purpose of this lesson, we will work with a value of:

c = 3.0 · 10⁸ m/s

The electromagnetic spectrum contains the visible spectrum, or light as we call it. But this is only a small part of the range of electromagnetic frequencies, which vary from 4.05 · 10¹⁴ to 7.9 · 10¹⁴ Hz. Other components of the electromagnetic spectrum are listed in Table 17.l.

The wavelength for the ranges of frequency can be easily found by using the definition of the speed of the wave:

v = f · λ

λ =

For a vacuum, the speed of an electromagnetic wave is c:

λ =

Example 1

Find the range of wavelength for visible light if the light spectrum corresponds to the range from 4.05 · 10¹⁴ to 7.9 · 10¹⁴Hz.

Solution 1

Set up the previous equation with the proper data and solve for the frequency.

λ = = = 740.7 · 10^–9m

A unit very often used in optics is the nanometer: 1 nm = 10^–9 m.

λ = = = 379.7 · 10^–9m

Hence, we have a range of 380 nm to 741 nm.

The energy carried by the electromagnetic wave is dependent on the properties of the two propagating fields, and its expression is:

u = · ε₀ · E² + · B²

where E and B are the electric and magnetic fields, and ε₀ and μ₀ are the permittivity and permeability of vacuum.

ε₀ = 8.85 · 10^–12 C²/N · m² and μ = 4 · π · 10 ^–7 T · m/ A

In a vacuum, the electric and magnetic fields are related by:

E = c · B

And the speed in vacuum is given by:

c =

Example 2

Reduce the expression of the energy of the electromagnetic waves to a simpler form using the connection between the electric field and the magnetic field.

Solution 2

Start with the definition of the energy, and, using the previous expression, try to simplify it.

u = · ε₀ · E² + · B²

u = · ε₀ · (c · B)² + · B²

u = · B²

Practice problems of this concept can be found at: Waves Practice Questions