Traveling Waves

To visualize what happens when transverse waves travel in one direction through an elastic medium, try the simulation found at:

http://www.phy.ntnu.edu.tw/java/waveType/waveType.html.

The simulation is a JavaScript "Applet". Concentrate on the top image in the transverse wave simulation. Notice the following points:

Each particle in the medium (such as a stretched string) executes Simple Harmonic Motion, vibrating up and down in place (NOT traveling with the wave). The amplitude A of the wave is the maximum distance each particle is displaced from the equilibrium position (measured vertically from the x axis). The frequency f of the wave is the number of times each particle goes through a full cycle of its motion every second (e.g. from the top to the bottom and back to the top again). Frequency is also the number of wave crests that pass by a given point in the medium per second. Amplitude is in meters, and frequency is in Hz (Hertz, or sec^-1).
The wave moves from left to right with a characteristic speed v which is always slower than the maximum speed of any one particle. In other words, the speed of sound waves is limited by the speed of particles' shaking within the medium. In air, Nitrogen molecules have average speed greater than 500 m/s (due to thermal energy), and the speed of sound is much less-- about 330 m/s.
The wavelength l is defined as the horizontal separation distance between successive wave crests (high points), measured in meters.
If the speed of the wave is constant, f increases if l is decreased (they are inversely proportional). In other words, if you crowd the crests closer together, more of them would pass a given point per second as the wave moves by. Or, if you want to make the crests closer together you should shake the particles up and down at a faster rate (try shaking the end of a rope or garden hose up and down, and watch the spacing of the wave crests as you vary the rate of shaking). The equation which governs this relationship is simply:
v = l f

Practice calculating wavelength or frequency from given information.

If the wave is sound in air, generally v will be given as 330 m/s. Audible sound waves have frequencies in the range from 20 Hz to 20 kHz.
If the wave is AM radio, v automatically is the speed of light, 3.0 x 10⁸ m/s (you may not be told this on a quiz or test-- be prepared to look it up). The frequency will be in the range from 550 kHz to 1600 kHz. AM radio waves have wavelengths in the tens to hundreds of meters.
If the wave is FM radio, v again is the speed of light. The frequency will be in the range 88 MHz to 108 MHz. M stands for mega (10⁶). The wavelength will be a meter or so.
If the wave is visible light, v = 3.0 x 10⁸ m/s. The wavelength will be between 400 nm (violet) and 700 nm (red). n stands for nano (10^-9). Frequencies will be hundreds of THz (T stands for tera, 10¹²).

Factors Affecting Wave Speed

While each particle in the medium can vibrate by itself, the wave is produced by the fact that each particle is connected to its neighbor(s) by elastic forces (bonds). The speed of the wave through the medium is determined by a competition between two factors:

The elasticity factor. If the bonds are strong (like stiff springs), then the motion of one particle will be communicated more rapidly to its neighbors.
The inertia factor. This is a measure of how difficult it is for the particles in the medium to "get out of the way". If there is a high density-- a lot of mass per unit length (mass is the same as inertia), then the medium will be more sluggish in responding to vibration.

In a stretched string, the equation for speed of the wave is

where F_T is tension force in Newtons, and m/L is linear mass density in kg/m.

In a fluid (such as air or water), the equation for speed of the wave has "bulk modulus" in the numerator and "volume density" in the denominator. Water is much more dense than air (obviously, since bubbles float in water), so you might think that sound travels much more slowly in water. In fact, sound travels faster in water-- that's because the elasticity factor is much greater (hydrogen bonds between water molecules in the liquid communicate the wave energy).

While we won't be solving problems with this equation, I might ask you a question such as: "what happens to the speed of waves in a string if we double the mass of the string while keeping its length and tension constant?" Answer: you decrease the speed by a factor of 0.707, or 1/(Ö2).

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