Wave Function

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?

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.