# Standing Waves in a String

We have talked about the reflection of a wave pulse in a string when it arrives at a boundary point (either a fixed end or a free end).

Now let’s look at what happens when a sinusoidal wave is reflected by a fixed or free end of a string. The waves (incident and reflected) interfere with each other.

The general term interference is used to describe the resultant of two or more waves passing through the same region at the same time.

When two waves of the same frequency and amplitude travel in opposite directions at the same speed their superposition gives rise to a new type of wave called stationary waves or standing wave.

Suppose that the two waves of same amplitude and frequency travelling in opposite directions at same speed are

Y1 = A sin (ωt − kx) and

Y2 = A sin (ωt + kx)

(As Y2 is the displacement due to a reflected wave from a free boundary a phase change of π will not take place)

Then, by principle of superposition,

Y = Y1 + Y2

= A [Sin (ωt − kx) + Sin (ωt + kx)]

Y = 2A Cos kx Sin ωt

From this it is clear that-

(a) As this equation satisfies the wave equation-

$\displaystyle \frac{d^2 y}{dx^2} = \frac{1}{v^2}\frac{d^2y}{dt^2}$ ; It represents a wave

(b) As it is not of the form F (ax ± bt), the wave is not travelling and it is called standing wave.

(c) the amplitude of the wave = 2A cos kx

It is not constant but varies periodically with position (x) and not with time as in case of beats, which we will see later.

Amplitude (A) is maximum when –

cos kx = ± 1 or kx = 0, π, 2π

or x = 0, λ/2, λ, 3λ/2 …. [as k = 2π/ λ ]

The points where A is maximum are called antinodes i.e. at antinodes A = 2A

A is minimum when

cos kx = 0 or kx = π/2 , 3π/2

or, x = λ/4 , 3λ/4

The points where amplitude is minimum are called nodes i.e. at node A = 0.

(d) Distance between two consecutive nodes or antinodes is λ/2 and the distance between a node and its adjacent antinode is λ/4.

(e) Nodes are always at rest, and the displacement at antinodes is always a maximum.

(f) All points lying between a node and an antinode are in same phase and are out of phase with the points lying between its neighbouring node and antinode.

### Properties Of Standing Waves

A standing wave has the following properties :

Maximum displacement of antinodes is ± 2A. All points (except nodes) pass their mean position twice in one time period.

(g) Since antinodes have always maximum displacement, their velocity is also maximum compared to other points and velocity at nodes is zero.

(h) Standing waves can be transverse or longitudinal. In case of longitudinal stationary waves nodes are points of maximum pressure (minimum displacement) because phase difference of π/2 between pressure wave and displacement wave. And antinodal points have minimum pressure (maximum displacement). In organ pipes longitudinal stationary waves are obtained.

(i) In stationary waves, amplitudes of different points are not equal.

Anode = 0 ; Aantinode = maximum = 2 A.

For other points As increases from 0 to A and then decreases for A to 0.

(j) As in stationary waves nodes are permanently at rest; so no energy can be transmitted across them. I.e. energy of one region (segment) is confined in that region. However this energy oscillates between elastic PE and KE of the particles of the medium.
When all the particles are at their extreme positions the KE is minimum. while elastic PE is maximum and when all the particles (simultaneously) pass through their mean position KE will be maximum while elastic PE is a minimum. The total energy confined in a segment (Elastic PE & KE) always remains the same

(k) In standing wave if the amplitude of component waves are not equal then. Amin≠ 0 i.e. node will not be permanently at rest and so some energy will pass across the node and wave will be partially standing.