Properties of Waves

We have already said that a wave is a disturbance that transports energy. We can use this as the basis to discuss some properties of waves. As energy is transported by a wave, the particles in the wave undergo simple harmonic motion. The energy for this is given as E = 1/2 kA2, where A is the amplitude. This should look a lot like elastic potential energy! Intensity (I) is defined as the power (energy per unit time) transported across an area perpendicular to the energy flow. In other words,

or

If the source is a point source, then the energy radiates in a spherical manner equally in all directions. The area of the surface of a sphere is given by the equation A = 4pr2, where r is the radius or distance from the center. Going back to our equation above,

or

This is very important, because it shows us that waves obey an inverse-square relationship. This is the same type of relationship we discuss when talking about gravity or electric forces. Thus, if you double the distance
from an object, the strength of the wave decreases by a factor of 22, or 4. Triple the distance, and the wave intensity decreases by a factor of 9.
It should also be noted, although we won't prove it, that intensity is proportional to the square of the frequency as well as the square of the amplitude, or

Reflection of a wave occurs when a wave is traveling through a medium and either strikes a boundary or reaches the end of the medium. Some of the wave is reflected back along the path it was taking. We've observe this when we hear an echo or see water waves bounce off a rock. When the wave pulse reflects off a fixed surface, or when there is reflection from a wave traveling to a denser medium, the pulse actually comes back inverted. If the wave is reflected off the free end of a rope, or there is reflection from a less dense medium than the wave was originally traveling in, the pulse will not be inverted. This leads to:

When a wave hits a surface and is reflected, the angle in = the angle out, or the angle of inceidence is equal to the angle of reflection. This is the Law of Reflection.

When two waves pass through the same area at the same time, the wave amplitudes will add point for point and a new wave is formed temporarily while the waves are in contact. After the contact, the waves will pass along as if they had never met. This is called the principle of superposition. When waves algebraically add to make a bigger wave, we call this constructive interference. This is the example shown on the left. The two waves on top will pass each other, creating a bigger wave as seen in the middle, and then continue on as the original waves as seen in the bottom.

When they add to make a smaller wave, we call this destructive interference. Again, we see this on right. The two waves at the top will combine to make the line in the middle - it appears that the wave is gone. But then the waves will pass each other and continue on their ways. It should be noted that these are discrete snapshots of an event that is constantly changing. Check out the applets at the end of this page to play with the principle of superposition some more. We use the term phase to describe the relative positions of 2 wave crests. When the interference is constructive, we say that the waves are in phase. If the interference is destructive, we say that the waves are out of phase.

When a wave is transmitted and reflected back, and the incident and the reflected wave are at the right frequency, we get what appears to be a standing wave. It appears to not be moving.

A node occurs where the interference of the incident and reflected wave is totally destructive, and an anti-node (point of maximum amplitude) occurs at the point where the interference is totally constructive. These occur at natural or resonant frequencies. The lowest frequency that a standing wave will occur is called the fundamental frequency. This occurs at half a wavelength, or l/2. As the frequency increases, we can obtain higher order standing waves, called harmonics.

For more on standing waves, try: http://hyperphysics.phy-astr.gsu.edu/hbase/waves/standw.html

Forced Vibrations (resonance) of an object occurs when an object is vibrating at its natural frequency. This generally happens when something transfers energy to another object, such as a guitar string transferring energy to the acoustic body and setting up a resonant frequency (which is how we hear acoustic guitars without an amplifier), or when an earthquake or high winds cause buildings or bridges to vibrate at natural frequencies and start to shake. This latter can eventually lead to the destruction of the structure, as was most vividly seen in the Tacoma Narrows Bridge incident. An object is said to be in resonance when it is vibrating at its natural frequency.

Refraction of a wave is a bending of a wave due to a change of speed of the wave. This happens when a wave passes from one medium to another and the speed of the wave in the two media is different. We will discuss refraction in great detail in optics.

Diffraction is the bending of a wave around an object. Waves will generally bend around an object if the wavelength is big compared to the object. Otherwise, there is a shadow region behind the object.

But, to read what others have to say on the subject, try:

http://www.physicsclassroom.com/Class/waves/

The NTNU Virtual Physics Laboratory provides several excellent applets that demonstrate principles of Physics. Click Here for an applet you can run from the NTNU Virtual Physics Laboratory that will show reflection and refraction in a wave.

Click Here to see an applet that shows the Superposition Principle of Waves.

Click Here to see an applet that shows interference in a wave from two point sources.

For Practice Problems, Try: Giancoli Multiple Choice PracticeQuestions (Questions 15 - 25)