SAT Physics Waves - Traveling Waves

SAT Physics Waves - Traveling Waves

TRAVELING WAVES
A wave is an organized disturbance consisting of many individual oscillators. Although each individual oscillator barely moves, the resulting wave can travel great distances at a constant speed in a specific medium. While traveling, this disturbance transmits energy from one place to another without moving any physical objects over the same distance. The energy propagates outwardly from the source of the oscillations.
      There are two types of traveling waves: transverse and longitudinal. Figure 15.1 illustrates the two types.

Figure 15.1. Transverse (a) and longitudinal (b) waves

In transverse waves, the oscillators vibrate perpendicularly to the direction of wave propagation (they transverse/cross equilibrium). In longitudinal waves, the oscillators vibrate parallel (along or longitudinal) to the direction of wave propagation.
          Mathematically, both transverse and longitudinal waves can be graphically represented as sine wave functions. The longitudinal wave has a unique appearance that is difficult to depict in diagrams. As a result, diagrams used to analyze longitudinal waves depict a sinusoidal form that resembles the appearance of transverse waves. Figure 15.2 identifies the principal components of both transverse and longitudinal waves. The figure depicts an instantaneous view of a wave. The amplitude of the oscillations is shown on the y-axis, while the horizontal position of the wave is frozen on the x-axis.

Figure 15.2. principal Wave Computers

The amplitude, A, is the maximum displacement, A = x_max. It can be measured from equilibrium (x= 0) to either +A or -A. The wave illustrated in Figure 15.2 is a sine wave because the wave begins at the origin and then proceeds upward toward the positive maximum displacement, +A However, the wave might start at the positive maximum displacement (making it a cosine wave), at the origin and proceed downward, or at the negative maximum displacement and proceed upward. Regardless, the wave will exhibit a repeated sinusoidal pattern between the two maximum displacements.
         When the wave pattern repeats, it is known as a wavelength, X. Wavelength is the distance measured between two successive identical portions of a wave. It is often easiest to see the wavelength between two successive crests or two successive troughs. However, the wavelength can also be determined as the distance between three crossings of the equilibrium line.

Medium and Wave Speed
The medium is the substance through which a wave propagates. For example, sound waves most often move in the medium air. Motion of a wave through one, and only one medium, is at constant speed. When a wave changes medium, the wave speed most often changes to a new constant speed in the new medium. The speed of the wave, v, in a specific medium is the product of the wave’s frequency, f, and its wavelength, A. In addition, the constant speed formula is also applicable when waves travel in a constant medium.

As long as the wave travels in only one medium, the wave speed will remain constant and the frequency and wavelength will vary inversely.
          An example of this phenomenon is sound waves. Frequency of sound is perceived as pitch. As the frequency increases, the sound wave itself does not travel any faster. If increasing frequency did affect the wave speed, then high-pitched notes would reach the ears of an observer before those of low-pitched notes. This, however, is not the case. An observer detects multiple frequencies produced by a single source simultaneously. This means that high- and low-pitched sounds must have different wavelengths. Therefore, frequency and wavelength must vary inversely. High-pitched (high-frequency) sounds have short wavelengths. Low-pitched (low-frequency) sounds have long wavelengths.
          Another example of this phenomenon is light waves. The frequency of light is perceived as color. Low-frequency light is perceived as red, and high-frequency light is perceived as blue. If increasing frequency did affect the wave speed, then different colors would arrive at the eye of an observer at different times. This is not the case. Red and blue frequencies, for example, produced by a single source are viewed simultaneously by an observer and appear magenta. Red frequencies simply have longer wavelengths, while blue frequencies have shorter wavelengths.
          You should also note that if a wave changes mediums as it travels, its speed is affected. However, when a wave changes medium, the frequency remains the same. As an example, a yellow swimming suit will appear yellow both above and below water. If changing mediums changes wave speed while keeping frequency constant, then wavelength must be changing. The relationship between wave speed and wavelength is directly proportional. If a wave speeds up when entering a new medium, the wavelength will become longer. If a wave slows down when entering a new medium, the wavelength will become shorter.

Wave Equation

A wave travels through a medium with velocity v, frequency f, and wavelength λ.
If the wave then enters another medium that increases the velocity to 2v, what will be the corresponding frequency and wavelength?
 
WHAT'S THE TRICK?

The velocity of a wave is dependent upon the medium in which it travels. When waves change mediums, only the velocity and wavelength are affected. Frequency remains unchanged. Therefore, according to the wave equation, v = f λ, if velocity doubles, wavelength must also double and frequency will remain the same.

Effect of Amplitude on a Wave
The amplitude of a wave is the maximum displacement of the oscillating particles composing the wave as measured from their equilibrium positions. The amplitude of a wave does not affect the wave speed, wavelength, or frequency of the wave. The amplitude affects only the energy of the wave. A good example of this is the effect of amplitude on a sound wave. In terms of sound, amplitude is the volume of a sound. If, for example, a note of a certain frequency is being played through a loudspeaker, the note will not change if the volume is increased. Similarly, if the amplitude (volume) is increased, the speed of the sound coming out of the loudspeaker will not travel any faster to its intended observer.

MECHANICAL WAVES
Mechanical waves involve the displacement of molecules in a medium from a position of equilibrium. Examples of mechanical waves and their mediums include vibrations on a string, ripples on a pond, and sound moving through air. A medium can therefore be a solid, liquid, or gas. These waves are disturbances created by a source that travels outwardly from the source at a constant velocity for that medium.

Sound Waves
Sound is a form of mechanical waves that travels as longitudinal (compression) waves. Longitudinal sound waves are difficult to illustrate. However, they mathematically graph as sinusoidal functions. As a result, graphs of sound waves often make them appear similar to transverse waves for illustrative purposes. Mechanical waves, such as sound, require a medium in order to propagate and travel. Sound can travel through solids, liquids, and gases. Each medium affects the speed and wavelength of the sound but not the frequency. The speed of a wave in a medium depends on many factors (density, elasticity, temperature, etc.). However, as a general rule, sound travels faster in denser mediums (see Table 15.2). This means sound moves fastest in solids and slowest in gases.

Table 15.2. Speed of Sound in Different Mediums

 

ELECTROMAGNETIC WAVES

Electromagnetic waves are light waves. These include radio waves, microwaves, infrared light, visible light, ultraviolet light, X-rays, and gamma rays. Researchers determined in the early twentieth century that these waves do not require a medium in which to travel and can therefore move through the vacuum of space. However, if they do travel in a medium, such as air, water, or glass, their wave behavior will be affected by that medium.
        All forms of electromagnetic waves travel at the same constant speed in a vacuum. This value is known as the speed of light, c. Its value is 3 × 10⁸ meters per second. For light moving in a vacuum, the speed formula can be modified.

c = f λ

 The speed of electromagnetic waves is the complete opposite of mechanical waves. Electromagnetic waves have their highest speed in a vacuum, where mechanical waves cannot even exist. While mechanical waves typically speed up in denser mediums, electromagnetic waves are slowed as mediums become denser. Table 15.3 shows commonly encountered speeds for light waves.

Table 15.3 Speed of Light in Different Mediums

A practical example of this phenomenon can be seen when white light enters a glass prism. White light is comprised of multiple wavelengths of light that are each slowed down as they enter the prism. When they emerge on the other side of the prism, the colors become separated into the familiar spectrum of the rainbow. Chapter 16, “Geometric Optics,” will cover this concept in more detail.

Speed of Sound vs. Speed of Light

During a thunderstorm, a flash of lightning is seen. Then 5.0 seconds later, a thunderous crack is heard. How far away was the lightning?
 
WHAT'S THE TRICK?

This problem illustrates the disparity between the speed of light and the speed of sound. They are both traveling simultaneously through the medium of air, but each has vastly different speeds. The speed of light in air is nearly the same as in a vacuum. This speed is so fast that the light from the lightning arrives nearly instantaneously. The travel time is so short that it is negligible (near zero) and not worth including in calculations.
               The speed of sound in air is about 340 meters per second. The distance to the lightning strike can be determined using the constant speed of sound in air.

Electromagnetic Spectrum
The oscillations of an electromagnetic field create a spectrum of electromagnetic waves. These waves transmit energy in direct proportion to their frequency. At the low end of the spectrum are radio waves. At the high end are X-rays and gamma rays. In between these ends of the spectrum is visible light. The human eye is sensitive to the frequencies in this range, which are perceived as colors. The electromagnetic spectrum is shown in Figure 15.3. It starts at the left with long-wavelength, low-frequency, and low-energy radio waves. The spectrum progresses in order to the short-wavelength, high-frequency, high-energy gamma rays. Visible light is broken into the colors of the rainbow in their relative order as well.

Figure 15.3. The electromagnetic spectrum

Knowing the exact frequencies and wavelengths of each portion of the electromagnetic spectrum will not be important. However, knowing the relative wavelength, frequency, and energy relationships is helpful. Remembering the order of the visible light spectrum, from low frequency to high frequency, with the acronym ROYGBIV will also help. Note that the energy of the blue end of the spectrum is higher than the red end. This is often mistaken by students who wrongly think that red is the high-energy color, when in reality blue and violet light possess greater energy.

Electromagnetic Spectrum 

Sort the following electromagnetic waves in order from lowest energy to highest energy:
                X-rays, radio waves, ultraviolet, visible light, and microwaves
 

WHAT'S THE TRICK?
 

This trick requires you simply to know the order of the spectrum and to know which end has the highest energy. Remember that visible light is right in the middle of the spectrum, with infrared next to red and ultraviolet next to violet. Ultraviolet causes skin cancer, so it possesses higher energy and will be on the right. X-rays and gamma rays sound dangerous and possess even higher energy, so they are to the right of ultraviolet. This leaves radio waves and microwaves on the low-energy end, to the left of infrared. The correct order for the electromagnetic waves given is:
                    Radio waves, microwaves, visible light, ultraviolet, and X-rays