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## Wave Phenomena

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.

### General Wave Properties

• 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.

### Superposition

• 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).

### 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.

### Wave Reflection

• 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.

### Refraction

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

### Diffraction

• 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.

### 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.

### 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.