Physics WAVE MOTION

Wave :

If you drop a little pebble in a pond of still water, the water surface gets disturbed. The disturbance does not remain confined to one place, but propagates outward along a circle. If you continue dropping pebbles in the pond, you see circles rapidly moving outward from the point where the water surface is disturbed. It gives a feeling as if the water is moving outward from the point of disturbance. If you put some cork pieces on the disturbed surface, it is seen that the cork pieces move up and down but do not move away from the centre of disturbance. This shows that the water mass does not flow outward with the circles, but rather a moving disturbance is created. Similarly, when we speak, the sound moves outward from us, without any flow of air from one part of the medium to another. The disturbances produced in air are much less obvious and only our ears or a microphone can detect them. These patterns, which move without the actual physical transfer or flow of matter as a whole, are called waves.

The waves we come across are mainly of three types:

(a) Mechanical waves
(b) Electromagnetic waves and
(c) Matter waves

Mechanical waves are most familiar because we encounter them constantly; common examples include water waves, sound waves, seismic waves, etc. All these waves have certain central features : They are governed by Newton's laws, and can exist only within a material medium, such as water, air, and rock. The common examples of electromagnetic waves are visible and ultraviolet light, radio waves, microwaves, x-rays etc. All electromagnetic waves travel through vacuum at the same speed `c`, given by `c = 299792458 m//s` (speed of light)

Wave Motion :

A wave-motion is the transmission of energy from one place to another through a material or a vacuum. Wave motion may occur in many forms such as water waves, sound waves, radio waves and light waves, but waves are basically of only two types:

(a) Transverse waves

(b) Longitudinal waves

`text(Wave Dimension :)`

`text(One Dimensional Waves)`
Wave confined to travel either to the right or left along on straight line are one-dimensional wave, e.g. wave produced on a string.

`text(Two Dimensional Waves)`
Waves that propagate over a surface are two-dimensional waves, e.g. vibration of the surface of a drum head.

`text(Three Dimensional Waves)`
Three- dimensional waves propagate in all directions, e.g. a sound wave.

Transverse Waves :

The oscillation is at right angles to the direction of propagation of the wave. Example of this type are water waves and most electromagnetic waves.
Figure (a) shows how a small segment of a string moves as a transverse pulse passes. As the leading edge of the pulse reaches it, the segment moves perpendicular to the string's equilibrium position. Its displacement reaches a maximum as the peak of the pulse passes and then returns to its equilibrium position after the pulse has moved on. It is obvious from the figure (b), that the particles on the leading edge are moving upward while those on the trailing edge are moving downward.

Longitudinal Waves :

In a longitudinal wave, particles of the medium are displaced in a direction parallel to energy transport. The oscillation is along the direction of propagation of the wave. An example of this type is sound waves.

Let us consider a one-dimensional wave travelling along the `x`-axis. The oscillations may not be simple harmonic in nature. It can proved that only those functions of `y` such that

`y=f(x,t)` which satisfy the differential equation :`(partial ^2y)/(partial t^2)= c^2 * (partial^2 y)/(partial x^2)` represent a solution to the wave equation. Here `c` is the wave speed.

`f(x,t)=f(ax pm bt)` is the general solution of the above differential equation.
where`c=b/a`

The `(+)` plus sign between `ax` and `bt` Implies yhe wave is travelling in the negative direction. The `(-)` negative sign between `ax` and `bt` implies the wave is travelling In the positive `x` direction. further ` (x, t)` must be finite everywhere at all times.

Describing Waves

Two kinds of graph may be drawn-displacement-distance and displacement-time.
A displacement-distance graph fro a transverse mechanical wave shows the displacement y of the vibrating particles of the transmitting medium at different distance x from the source at a certain instant i.e. it is like a photograph showing shape of the wave at that particular instant.
The maximum displacement of each particle from its undisturbed position is the amplitude of the wave. In
the figure, it is OP or OQ.

`text(Wavelength)` `(lambda) :` It is generally taken as the distance between two successive crests or two successive through. To be more specific, it is the distance between two consecutive points on the wave which have same phase.
`lambda=v/f`

`text(Time Period)` `(T) :` The time period for a point on the string is the time taken to complete one cycle of its periodic motion. It is exactly the same time that it takes for one wavelength to pass the point.

`text(Frequency)` `(f) :` The number of vibrations of point on the string that occur in one second or the number of wavelengths that pass a given point in one second.
`f=1/T`

`text(Wave velocity)` `(v) :` Since in one period T the wave advances by one wavelength A, therefore, the wave velocity is
`v=(lambda/T)=lambdaf=omega/k`

`text(Amplitude)` `(A) :` The maximum displacement of a particle on the medium from the equilibrium position.

Sound Waves

Sound wave