- General Wave Properties
- Wave Reflection
- Diffraction
- The Wave Equation (for a stretched string)
- Wave Equation Solutions

- 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:

- 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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