Wave Function

Wave Motion of Class 11

Wave Function

The disturbance created by a wave is represented by a wave function. For a string, the wave function is a (vector) displacement; whereas for sound waves, it is a (scalar) pressure or density fluctuation. In the case of light or radio waves, the wave function is either an electric or magnetic field vector.

How to express a wave function mathematically?

Let us look at the pulse from two different reference frames. The XY – frame is stationary whereas the X′Y′ frame moves with the velocity of the pulse as shown in the Fig.(14.4).

We assume that the origins coincided at
t = 0. In the moving frame the pulse is at rest, so at any time the vertical displacement y′ at position x′ is given by some function f(x′) that describes the shape of the pulse:

Wave Function

Fig.(14.4)A pulse traveling at speed v relative to the xy frame. In the x′y′ frame of the pulse, it is at rest and the shape is described by f(x′). In the xy frame, the pulse is described by f(x – vt).

y′ = f(x′)(14.1)

In the stationary frame, the pulse has the same shape but is moving at a velocity v which means that the displacement y is a function of both x and t.

The coordinates of the pulse as measured in the two frames are related as:

y′ = y

x′ = x – vt

Thus the equation (14.1) may be modified as

y = f(x – vt)(14.2)

Any given feature (phase) of the pulse, for example, its peak, has a fixed value of x′, which means

x′ = x – vt = constant

The quantity (x – vt) is called the phase of the wave function.

Differentiating w.r.t. time, we get

dx/dt = v

where v is the wave velocity or phase velocity.

It is the velocity at which a particular phase of the disturbance travels through space.

If the wave is travelling along the negative x – axis, the wave function is represented by

y = f(x + vt)(14.3)

In general, the wave motion in one dimension is given by

y = f(x ± vt)(14.4)

IMPORTANT

In order for the function to represent a wave travelling at speed v, the three quantities x, v and t must appear in the combinations (x + vt) or (x – vt). Thus, (x- vt)2 is acceptable, but
(x₂ – v₂t₂) is not.

Example: 14.1

The wave function of a pulse is given by

y = 3/2 + (x - 4t)²

Determine the wave velocity of the pulse and indicate the direction of propagation of the wave.

Solution

On comparing the given expression with y = f(x – vt), we get the velocity of the wave:
v = 4m/s

and, the wave propagates along the positive x – axis.