- General Wave Properties
- Superposition
*N*-Mass Systems vs. Distributed Systems- The Wave Equation (for a stretched string)
- Traveling Waves
- Wave Reflection
- Refraction
- Diffraction
- The Doppler Effect
- Sound Waves

Wave motion involves the transfer of energy. The behavior of this energy transfer varies with the particular medium of transport and energy form. Mechanical waves travel in a material medium, such as water or a string.

- Wave motion is initiated by a disturbance which subsequently travels through a medium with a fixed velocity (for homogeneous media).
- A disturbance is transported through a medium via internal cohesive forces, though the medium itself is not transported.
- A simple sinusoidal disturbance of frequency will produce periodic motion with a
*wavelength*given by , where is the wave speed of propagation. - The wave speed is determined by the mass (or mass density) and elastic modulus (or tension) of the medium in which it travels.
*Longitudinal*Wave Motion: vibration of particles in the medium is along the same direction as the wave motion.*Transverse*Wave Motion: vibration of particles in the medium is perpendicular to the direction of wave motion.

- When two or more waves pass through the same region of space at the same time, the actual displacement is the vector (or algebraic) sum of the individual displacements.
- By
*Fourier's theorem*, any complex wave can be considered as composed of many simple sinusoidal waves of different amplitudes, wavelengths, and frequencies (Matlab example).

- 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,
.
- In two or three dimensions, the angle an incident wavefront makes with a reflecting surface is equal to the angle of reflection.

- A sudden or progressive change in wave speed will produce a change in propagation direction or a ``bending'' of the waves.

- Waves tend to bend around an obstacle.
- The amount of diffraction depends on the wavelength of the wave and on the size of the obstacle.
- If the wavelength is much larger than the object, the wave bends around it almost as if it isn't even there. When a wavelength is less than the size of an object, a ``shadow'' region will result.

- 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.

- 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.

©1998-2001 CCRMA, Stanford University. All Rights Reserved. Maintained by Gary P. Scavone, gary@ccrma.stanford.edu. |