Chapter 1 الباب الأول Wave Motion الحركة الموجية

Introduction A wave is a disturbance from an equilibrium state that moves or propagates with time from one region to of space to another.

Examples Dropping a stone into the water produces a disturbance which spreads out horizontally in all directions along the surface. A source of sound produces a fluctuation in pressure in the surrounding atmosphere, and this disturbance is propagated to distant points.

Light, radio waves, x-rays, and γ rays are all examples of electromagnetic waves. A characteristic of all waves is the ability to transport energy from one region of space to another.

Propagation of a Disturbance All mechanical waves require (1) some source of disturbance, (2) a medium that can be disturbed, and (3) some physical mechanism through which elements of the medium can influence each other.

A pulse traveling down a stretched rope.

A traveling wave or pulse that causes the elements of the disturbed medium to move perpendicular to the direction of propagation is called a transverse wave.

Stretched spring.

A traveling wave or pulse that causes the elements of the medium to move parallel to the direction of propagation is called a longitudinal wave.

The motion of water elements on the surface of deep water in which a wave is propagating is a combination of transverse and longitudinal displacements. Elements at the surface move in nearly circular paths. Each element is displaced both horizontally and vertically from its equilibrium position.

Various forms of waves

Consider a pulse traveling to the right on a long string,

Consequently, an element of the string at x at this time has the same y position as an element located at (x – vt) had at time t = 0:

We can represent the transverse position y for all positions and times, measured in a stationary frame with the origin at O, as

Similarly, if the pulse travels to the left, the transverse positions of elements of the string are described by

The function y, sometimes called: the wave function, and depends on the two variables x and t. it is written: y(x, t)

The wave function y(x, t) represents the y coordinate, the transverse position of any element located at position x at any time t. the wave function y(x), sometimes called the waveform,

Example A pulse moving to the right along the x axis is represented by the wave function where x and y are measured in centimeters and t is measured in seconds. Plot the wave function at t = 0, t = 1.0 s, and t = 2.0 s.

Solution this function is of the form: y = f (x - vt). The wave speed is: v = 3.0 cm/s. The maximum value of y is given by : A = 2.0 cm. Representing y by letting (x t = 0.)

The wave function expressions are:

We now use these expressions to plot the wave function versus x at these times. For example, let us evaluate y(x, 0) at x = 0.50 cm:

Sinusoidal Waves

The point at which the displacement of the element from its normal position is highest is called the crest of the wave. The distance from one crest to the next is called: the wavelength The wavelength is the minimum distance between any two identical points (such as the crests) on adjacent waves.

The period T is the time interval required for two identical points (such as the crests) of adjacent waves to pass by a point. The period of the wave is the same as the period of the simple harmonic oscillation of one element of the medium.

The same information is more often given by the inverse of the period, which is: the frequency f. The frequency of a periodic wave is the number of crests (or troughs, or any other point on the wave) that pass a given point in a unit time interval.

The frequency of a sinusoidal wave is related to the period by: The most common unit for frequency, is second-1, or hertz (Hz). The corresponding unit for T is seconds.

Because the wave is sinusoidal, we expect the wave function at this instant to be expressed as: y(x, 0) = A sin ax, where A is the amplitude and a is a constant to be determined. At x = 0, we see that y(0, 0) = A sin a(0) = 0,

The next value of x for which y is zero is Thus,

For this to be true, we must have

If the wave moves to the right with a speed v, then the wave function at some later time t is

the wave speed, wavelength, and period are related by the expression

Substituting

We can express the wave function in a convenient form by defining two other quantities: the angular wave number k (usually called simply the wave number)

angular frequency

we generally express the wave function in the form where is the phase constant,. This constant can be determined from the initial conditions.