Standing Wave on a String

The standing waves are formed by the superposition of two harmonic waves of equal amplitude and frequency travelling through the medium in the opposite direction. The standing waves are also known as stationary waves. These waves are localised and not progressive, hence the name stationary waves.

Standing Waves

Standing waves are characterised by nodes and antinodes. The amplitude of vibration of the particle is maximum at the antinodes and minimum at the nodes. The vibration within a string can produce a variety of patterns. Each pattern corresponds to the vibration that takes place at a particular frequency, and it is called harmonic. The lowest frequency at which the string vibrates to form a standing wave is called the fundamental frequency or the first harmonic. The wavelength and the speed of the wave determine the frequency of each harmonic.

Standing Waves on an Infinite Length String

Let us consider two harmonic waves of the same amplitude and period T and wavelength λ travelling at the same speed in the opposite direction.

y₁ = A sin (kx – ωt)

y₂ = A sin (kx + wt)

Considering the principle of superposition, the resultant can be calculated as

Resultant y = y₁ + y₂

= A sin (kx – ωt) + A sin (kx + wt)

y = 2Asin (kx) cos (ωt)

The equation represents the SHM of the collection of particles. Here, the term 2ASin (kx) is the amplitude of the resultant wave. From the expression of amplitude, it can be concluded that the amplitude of the particles executing SHM depends upon the location of the particles.

Nodes

The amplitude of the wave is zero for all the values of kx that give sin kx = 0. It means for kx = 0, π, 2π …..nπ, the amplitude of vibration of the particles will be zero. Here, n is an integer.

By substituting k = 2π/λ, we get

(2π/λ)x = nπ

⇒ x = nλ /2

Therefore, x = 0, λ /2, λ , 3λ /2……..

These points of zero displacements of the particles are called the nodes.

Antinodes

The amplitude will have a maximum value of 2A for all values of kx that give sin kx = ± 1. This means, for kx = π/2, 3π/2 ——— (n + ½)π, the amplitude of vibration of the particles will be maximum.

By substituting k = 2n+1π/λ, we get

(2π/λ)x = (2n+1)π /2

⇒ x = (2n+1)λ /4

⇒ x = λ /4, λ , 3λ /4,5λ /4……..

These points at which the displacement of the particles is maximum are called antinodes. The nodes and the antinodes in a standing wave are equally spaced, the distance is equal to λ/2, where λ is the wavelength of the wave.

Standing Waves on a String with Two Fixed Ends

Consider a uniform string of length L which is stretched between two fixed ends. A wave that travels in one direction along the string reflects at the end and returns inverted because of the fixed ends. These two identical waves, travelling in the opposite direction, form the standing wave on the string. The length of the string is given as L, so the wavelength of the wave is restricted by the boundary condition.

λ = 2L/n , here n = 1,2,3…..

The standing waves are formed in the string only if the wavelength satisfies the relationship with L. If v is the speed with which the waves travel along the string, then the frequency of the standing wave is

f = v/λ = nv/2L , n = 1,2,3…..

Nodes are formed at the fixed ends. In addition to the nodes, if an antinode exists at the centre of the string, the stretched string is said to vibrate in the fundamental frequency. The lowest resonance frequency, corresponding to n = 1, is the fundamental frequency. The higher frequencies are called the harmonics. Harmonics are the integer multiples of the fundamental frequency.

Standing Wave on a String with One End Fixed

When the string has one of its ends fixed and the other end is free, the node will be formed at the fixed end, and the antinode will be formed at the free end. The simplest standing wave formed in this condition is one-quarter a wavelength long. The next possible standing wave will be formed by adding both node and antinode. The length of this wave will be three-quarters of the wavelength. In general, the wavelength of the wave can be written as

λ = 4L/n , here n = 1,2,3…..

Then, the frequency of the standing wave is restricted to

f = nv/4L

Characteristics of Standing Waves

1. The standing waves are stationary. The disturbance does not travel in any direction.
2. Standing waves have points of zero amplitude called nodes and points of maximum amplitude called antinodes.
3. There will not be any flow of energy across any section of the medium.
4. The distance between the two consecutive nodes or the antinodes is equal to λ/2, whereas the distance between an antinode and its adjacent node is λ/4.
5. The pressure change is maximum at the node and minimum at the antinodes.
6. The medium is divided into a number of segments. All the particles in a particular segment will vibrate in phases. However, the amplitude increases from zero to maximum between the node and the antinode. Particles in neighbouring segments vibrate in opposite phases.
7. The period of oscillation of any particle in the medium is the same as that of the component waves.

Solved Questions

1. The distance between the nearest node and antinode in a stationary wave is

a. λ/2

b. λ/4

c. 1

d. 0

Answer: b. λ/4

2. Which of the following properties is different in a stationary wave and a progressive wave?

a. Frequency

b. Amplitude

c. Phase of the wave

d. Propagation of energy

Answer: d. Propagation of energy

Solution:

The progressive wave propagates energy, while the stationary wave will not propagate energy.

3. For the given stationary wave,

a. 15

b. 7.5

c. 25

d. 8.5

Answer: b. 7.5

Solution:

Comparing the equation

k = π/15

⇒ 2π/λ = π/15 (Since k = 2π/λ)

⇒ λ = 30

The distance between the nearest antinode and node is λ/4 = 30/4 = 7.5

Wave on String Rapid Revision

Standing Wave on String