*N*-Mass Systems vs. Distributed Systems:- The Wave Equation (for a stretched string):
- Traveling Waves:
- Reflections at Boundaries:
- Sound Waves
- The Doppler Effect

- An
*N*-mass system has*N*modes per degree of freedom. - As
*N*gets very large, it becomes convenient to view the system as a continuous string with a uniform mass density and tension.

- ``Equation of motion for a wave''.
- The mass of the short string section (length ) is , where is the mass per unit length of the string.
- The net vertical force on the section is .
- For small angles, and , the slope of the string.
- The net vertical force can thus be rewritten: .
- By Newton's Second Law: .
- As
,

- The expression based on Newton's Second Law then becomes

where is the speed of wave motion on the string. This is the one-dimensional wave equation which describes small amplitude transverse waves on a stretched string.

- Solution to the wave equation of the form .
- represents a wave traveling to the right with a velocity . represents a wave traveling to the left with the same velocity.
- The functions and are arbitrary.

- At a fixed end, . If the string is fixed at , then and .
- At a free end, because no transverse force is possible. In this case, .

- Longitudinal waves that travel in a solid, liquid, or gas.
- The speed of sound in air is approximately given by where is the temperature of the air in degrees Celsius.

- Observer moving toward source:

where is the frequency of the source, is the speed of the observer, and is the speed of sound. - Source moving toward observer:

where is the speed of the source.

©1999 CCRMA, Stanford University. All Rights Reserved. Maintained by Gary P. Scavone, gary@ccrma.stanford.edu. |