Previous: Discrete-Time/Frequency Analysis
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.
- 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.
- 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.