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## Waves

### N-Mass Systems vs. Distributed Systems:

• 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.

### The Wave Equation (for a stretched string):

• 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.

### Traveling Waves:

• 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.

### Reflections at Boundaries:

• 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, .

### Sound Waves

• 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.

### The Doppler Effect

• 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.