# What is Wave & Wave Functions ?

Wave: We are all familiar with water waves and waves in a stretched string. There are also sound waves, as well as light waves, radio waves and electromagnetic waves.

A wave is actually a disturbance which propagates energy and momentum from one place to the other without the transport of matter.

Waves can be one, two or three dimensional according to the number of dimensions in which they propagate energy.

Waves moving along strings are one dimensional, surface waves or ripples on water are two dimensional, while sound or light waves travelling radially out from a point source are three dimensional waves.

### Types of Wave :

Waves are of two types :
(a) Mechanical waves: The waves which require some material medium for their propagation are called mechanical waves. Sound waves, seismic waves, waves in strings and springs are examples of mechanical waves. Elasticity and density of medium play an important role in propagation of mechanical waves.

(b) Non-mechanical waves: The waves which do not essentially require any material medium for their propagation are called non-mechanical waves. All electromagnetic waves such as γ – rays, x – rays, radio waves, light etc. are non-mechanical.

Any wave whether mechanical or non-mechanical, can be divided into two groups:

(i) Transverse waves: In this case the oscillations are at right angles to the direction of wave motion or energy propagation. Waves in strings are transverse. These are propagated as crests and troughs.

(ii) Longitudinal waves: Particles of the medium oscillate in the direction of wave motion. They are propagated as compression and rarefaction and are also known as pressure waves. Waves in springs and sound waves in air are example of longitudinal waves.

#### Note :

(a) All non-mechanical waves are transverse.

(b) In gases and liquids mechanical waves are always longitudinal e.g. sound waves in air and water.

(c) In solids mechanical waves can be either transverse or longitudinal depending on the mode of excitation. The speeds of the two waves in the same solid are also different.

Illustration : Why transverse mechanical waves can not be propagated in liquids and gases ?

Solution: To transmit a transverse mechanical wave the medium must be elastic so as to provide a restoring force when acted on by shearing stress. But liquids and gases can not sustain shear stress to provide restoring force, but rather start flowing when acted on by shearing stress.

### Wave functions , wave profile & forms of wave functions

A function of one variable such as x(t) is sufficient to describe the motion of a point mass moving along a line

e.g. $x = u t + \frac{1}{2} a t^2$

To give a mathematical description of a wave pulse functions which depend on two variables such as x and t are required.

Wave functions : Functions which can mathematically represent a moving wave pulse are called wave functions.

Let us find the general form of wave functions by studying the example of a wave pulse moving with speed v from left to right in a stretched string.

We choose a co-ordinate system such that the positions of points on the string are plotted along the x-axis and the displacements of the points from their equilibrium position along y-axis.

The shape of the disturbance is called its wave profile . The wave function will therefore be a function of both x and t.

If we could measure the displacement y at one instant of time , say t = 0, we would see that it is some function of x say f(x).

Now if the pulse or wave travels along the string with speed v, without change in shape , then the displacement of the point x at time t, y (x, t) is the same displacement that occurs at the pint (x − vt) at time t = 0.

i.e. y (x , t) = f(x − v t)

The functional forms of f will be different for waves of different shapes but the argument of the wave function i.e. the variable inside it will be (x − vt) for all waves moving with speed v along the x-axis from the left to the right.

Similarly ,  the wave function of a pulse moving from the right to the left along the x-axis will be of the form Y = g (x + vt).

In general if the wave function has the form

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