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. In general, vibrations propagate in the form of waves. Mechanical waves travel in a material medium, such as a string or a membrane. Acoustic waves travel in fluids, such as air or water.

General Wave Properties

• Wave motion is initiated by an energetic disturbance that subsequently travels through a medium with a fixed velocity (for homogeneous media). This moving disturbance is referred to as a traveling wave.

• A wave propagates through a medium via internal cohesive forces, though the medium itself is not transported.

• A simple sinusoidal disturbance of frequency f will produce periodic motion with a wavelength given by , where c is the wave speed of propagation. The wavelength represents the distance between successive, periodic movements of a medium.

• The wave speed is determined by the mass (or mass density) and elastic modulus (or tension) of the medium in which it travels. A more massy'' material will have a lower propagation speed. A stiffer'' material will have a higher speed of propagation.

• 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.

• Sound waves are longitudinal disturbances that travel in a solid, liquid, or gas.

• The speed of sound in air is approximately given by c = 331.3 + 0.6 t (meters / second), where t is the temperature of the air in degrees Celsius. A value of 345 meters / second is a good estimate at room temperature.

Wave Reflection

• When a wave encounters a change in the material in which it propagates, wave scattering will occur at that boundary. The way in which waves scatter at a boundary is determined by boundary conditions.

• In two or three dimensions, the angle an incident wavefront makes with a large reflecting surface (over several wavelengths in all directions) is equal to the angle of reflection (specular reflections).

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

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.

• Defraction can be better understood by considering Huygens' Principle.

The Wave Equation (for a stretched string)

• The wave equation provides an analytic description of wave motion over time and through a spatial medium.

• In analyzing the string section above, we make the following assumptions:

• The mass per unit length of the string is constant and the string is perfectly elastic (there is no resistance to bending).
• The tension caused by stretching the string before fixing it at its endpoints is so large that gravitational forces on the string are negligible.
• The string moves only in the transverse direction and these deflections are small in magnitude.

• The mass of the short string section (length ) is , where is the mass per unit length of the string.

• Since there is no horizontal motion, the two horizontal components of tension must be constant: .

• The net vertical force on the section is .

• By Newton's Second Law: .

• Making use of the horizontal tension components, we obtain:
 (1)

• The expressions and are the slopes of the string at the points x and :

• Making these substitutions in Eq. 1 and dividing by ,

• Letting approach zero, we obtain the linear partial differential equation

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

Wave Equation Solutions

• A common approach to solving the wave equation is to assume sinusoidal solutions in the form of complex exponentials.

• It is easy to verify that is a general solution to the wave equation, where A is the (complex) amplitude, is known as the wave number, and is the wavelength.

• This solution can be represented in a more general form, attributed to d'Alembert in 1747, of y = y+(ct - x) + y-(ct + x).

• y+(ct - x) represents a wave traveling in the positive x direction with a velocity c. Similarly, y-(ct + x) represents a wave traveling in the negative x direction with the same velocity.

• The functions y+ and y- are arbitrary.

• Thus, in a one-dimensional medium, waves can propagate in two opposite directions.

• 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.

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