Pluck the main string of a sitar, or the thickest string of a guitar, and look at it from the side. The string blurs into a smooth spindle. The spindle is widest at the middle and pinched to nothing at the two ends. Yet a disturbance on a string travels, as the chapter Propagating Waves on a String showed. Nothing on the plucked string seems to travel at all.
Standing waves
A standing wave is a special interference pattern. It forms when two identical waves travel in opposite directions along the same string and add by the principle of superposition. The two waves have the same amplitude, the same frequency and the same wavelength.
A common way to get the second wave is reflection. A wave on a string reflects from a boundary, such as a tied end. The reflected wave then interferes with the original, incident wave. The chapter Superposition and Reflection showed both halves of this story. Overlapping waves simply add their displacements, and a boundary sends a new wave back along the string. The reflected wave has the same frequency and the same wavelength as the incident wave.
A standing wave is the pattern formed by the superposition of two identical waves travelling in opposite directions along the same medium. The two waves have the same amplitude, frequency and wavelength.
The mathematical description of a standing wave
To analyse the pattern, consider two sinusoidal waves on one string. They have the same amplitude \(A\), the same frequency and the same wavelength, and they travel in opposite directions. As in the chapter Introduction to Waves, \(k = 2\pi/\lambda\) is the wave number and \(\omega = 2\pi f\) is the angular frequency.
- Wave 1, travelling to the right: \(y_1(x, t) = A\sin(kx - \omega t)\)
- Wave 2, travelling to the left: \(y_2(x, t) = A\sin(kx + \omega t)\)
By the principle of superposition, the net displacement of the string, \(y_{\text{string}}\), is the algebraic sum of the two displacements:
\[ y_{\text{string}}(x, t) = A\sin(kx - \omega t) + A\sin(kx + \omega t) \]
The identity for the sum of two sines turns this sum into a product. The derivation below takes it one step at a time.
\[ y_{\text{string}} = A\sin(kx - \omega t) + A\sin(kx + \omega t) \]
Why. the principle of superposition. At every point and at every instant, the displacements of the two waves add.
\[ \sin P + \sin Q = 2\sin\left(\dfrac{P + Q}{2}\right)\cos\left(\dfrac{P - Q}{2}\right) \]
Why. the identity for the sum of two sines. Take \(P = kx - \omega t\) and \(Q = kx + \omega t\).
\[ \dfrac{P + Q}{2} = kx, \qquad \dfrac{P - Q}{2} = -\,\omega t \]
Why. in the sum \(P + Q\) the \(\omega t\) terms cancel. In the difference \(P - Q\) the \(kx\) terms cancel.
\[ y_{\text{string}} = 2A\sin(kx)\cos(-\,\omega t) \]
Why. put step 3 into step 2, and multiply by the common amplitude \(A\).
\[ y_{\text{string}}(x, t) = [2A\sin(kx)]\cos(\omega t) \]
Read it. the cosine is an even function, \(\cos(-\theta) = \cos\theta\). This is equation (11).
Put \(x = 0\) in equation (11). Then \(\sin(kx) = 0\), and \(y = 0\) at every instant. So the point \(x = 0\) never moves. Lesson 3 uses this fact for a string tied at \(x = 0\).
Reading equation (11)
The structure of equation (11) is quite different from that of a travelling wave. In a travelling wave, \(x\) and \(t\) appear together in one combination, \(kx - \omega t\). In equation (11) the position \(x\) and the time \(t\) are separated into two different functions. One function multiplies the other.
1. The amplitude function
The first function is \(A_x = 2A\sin(kx)\). It depends only on the position \(x\). It defines a stationary envelope. The envelope sets the maximum amplitude of oscillation for the particle at each position. A particle where \(\sin kx = 1\) swings with amplitude \(2A\). A particle where \(\sin kx = 0\) does not swing at all.
Step through the chips of Fig 1.1. At \(\cos\omega t = 1\) the string fills the upper half of the envelope. At \(0.5\) every point is at half of its largest height. At \(0\) the whole string is flat for an instant. The last two chips repeat the first two, turned upside down. In every snapshot the string is the same shape, \(\sin kx\), only scaled. The dots at \(0\), \(2\), \(4\) and \(6\,\text{m}\) never leave the rest line.
2. The time function
The second function is \(\cos(\omega t)\). It depends only on the time \(t\). It describes the simple harmonic motion shared by all the oscillating particles on the string. Every particle oscillates with the same angular frequency \(\omega\), the angular frequency of the two travelling waves. The particles differ only in their amplitudes, which the envelope sets.
The wave profile does not travel along the \(x\) axis. It oscillates in a fixed position, and so it is called a standing wave. The older name, a stationary wave, says the same thing.
Two waves of equal amplitude and equal wavelength travel along one string, one to the right and one to the left. What will their sum, the string itself, do?
In the figure above each wave has an amplitude of \(1\,\text{cm}\), a wavelength of \(4\,\text{m}\) and a period of \(4\,\text{s}\). The dashed waves slide past each other in opposite directions. The solid string does not slide at all. The dot at \(x = 2\,\text{m}\) reads \(0.00\,\text{cm}\) at every instant. The dot at \(x = 3\,\text{m}\) swings between \(+2\,\text{cm}\) and \(-2\,\text{cm}\), twice the amplitude of either wave.
Where \(\sin kx\) is negative, the amplitude function \(2A\sin kx\) is negative too. The size of the swing is \(|2A\sin kx|\). The minus sign carries a meaning of its own: those particles move opposite to the particles where \(\sin kx\) is positive. Lesson 2 builds on this sign.
Now return to the plucked sitar string. Its two ends are tied, so the waves on it reflect again and again. The waves travelling each way add to a standing wave. The spindle that the eye sees is the envelope of that standing wave. The travelling waves are really there, but the eye sees only their sum, and the sum does not travel.
In the standing wave \(y = [2A\sin(kx)]\cos(\omega t)\), which statement about the factor \(2A\sin(kx)\) is correct?
Two waves, \(y_1 = 3\sin(\pi x/6 - 50\pi t)\) and \(y_2 = 3\sin(\pi x/6 + 50\pi t)\), travel on one string. Here \(y\) is in mm, \(x\) in cm and \(t\) in s. Find the amplitude of oscillation of the particle at \(x = 1\,\text{cm}\), in mm.
For the same pair of waves, \(y_1 = 3\sin(\pi x/6 - 50\pi t)\) and \(y_2 = 3\sin(\pi x/6 + 50\pi t)\), with \(t\) in s, with what frequency does the particle at \(x = 1\,\text{cm}\) oscillate?
Which statements are true of the standing wave \(y = [2A\sin(kx)]\cos(\omega t)\)?
A standing wave is made of two travelling waves. Why does the pattern itself not travel?
In the sum, \(y = [2A\sin kx]\cos\omega t\), the position and the time are in separate functions. The shape \(\sin kx\) contains no time, so its zeros and its peaks stay at fixed places. The factor \(\cos\omega t\) only makes the whole shape grow, shrink and flip. The two travelling waves carry the pattern equally in opposite directions, so the pattern goes neither way.
- Writing the amplitude of a standing wave as \(2A\) for every particle.The amplitude of the particle at \(x\) is \(|2A\sin kx|\). Only some particles reach \(2A\), and some never move.
- Reading the coefficient of \(t\) in \(\cos(\omega t)\) as the frequency.The coefficient is the angular frequency \(\omega\). The frequency is \(f = \omega/2\pi\).
- Thinking that the frequency of a particle depends on its position.Every particle shares the time function \(\cos\omega t\). Position changes the amplitude only.