Standing Waves and Normal Modes

Equation of Standing Wave:

• Let us consider, at any point u and time t, there are two waves, one moving to the left and the other moving to the right. The wave travelling in the positive direction of the x-axis is given as,

y1(u, t) = a sin (ku – ωt),

and that moving in the negative direction of the x-axis is given as,

y2(u, t) = a sin (ku + ωt),

As per the principle of superposition, the combined wave is represented as,

y (u, t) = y1(u, t) + y2(u, t)

= a sin (ku – ωt) + a sin (ku + ωt)

= (2a sin ku) cos ωt

• Here, the term called (2a sin ku) provides amplitude of the oscillation of all the element of wave at a position ‘u’ and we also notice that the wave which is represented by this type of equation need not be described as a travelling wave, or as a waveform that does not move to either sides. The expression represents the standing wave, wherein the point of minimum, maximum, and the null remains at only one position throughout its propagation.
  
• The amplitude is said to be zero for all the values of ku that give sin ku = 0. Those values which are given by the ku = nπ, for the values n = 0, 1, 2, 3, …
  
• Substituting the values k = 2π/λ in expression for the amplitude, we get
  \(\begin{array}{l}u=\frac{n\lambda}{2}\end{array} \)
  , for the values of n = 0, 1, 2, 3, …

• The position of the zero amplitudes are called as nodes. The distance of 2λ or say half a wavelength will separate the two consecutive nodes. The amplitude is said to have a maximum value of 2a, that occurs for all the values of ku that give ⎢sin ku ⎢= 1.

• The values are ku = (n + ½) π for all the values of n = 0, 1, 2, 3, … Hence, by substituting the value for k = 2π/λ in given expression, we get the value for
  \(\begin{array}{l}u=\frac{(n+\frac{1}{2})\lambda }{2}\end{array} \)
  for the values of n = 0, 1, 2, 3, … as that of the positions for maximum amplitude. These are called as the Anti-nodes. These anti nodes are said to be separated by λ/2 and located about half a way between the pairs of nodes. Let’s consider a string with length L whose ends are fixed. The two ends of this string are called as nodes.

Nodes and Anti nodes:

• A node is a point along with a standing wave where the wave has minimum amplitude.

• The opposite of a node is an anti node, a point where the amplitude of the standing wave is a maximum. These occur midway between the nodes.

Normal Mode:

A mass on a spring has one natural frequency at which it freely oscillates up and down. A stretched string with fixed ends can oscillate up and down with a whole spectrum of frequencies and patterns of vibration. These special “Modes of Vibration” of a string are called standing waves or normal modes.

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