In post 18.12 I introduced the idea of a standing wave as a vibration that could exist on a string tethered at both its ends. In the picture above, a string of length L is tethered at both ends, A and B. The fundamental wave (shown in red) then has its greatest amplitude of vibration in the centre. I also argued that the string could support further, harmonic, vibrations in which the ends didn’t move but further fixed points, called nodes, along the length of the string. The first harmonic (blue) has a node in the centre of the string. The green harmonic has two nodes in addition to those at A and B; the yellow harmonic has three. These results were obtained intuitively by thinking about possible ways for the tethered string to vibrate.
Now I want to think of a standing wave in a different way – as a travelling wave that is reflected and interferes with itself (post 18.10). This allows mathematical analysis of standing waves that establishes that there really will be nodes in the harmonics.
We’ll think about the one-dimensional sinusoidal wave represented by
where t represents time, x distance along the wave, A its amplitude, ω its angular frequency and κ = 2π/λ where λ is its wavelength. A, ω and λ are defined in post 18.10. Note that κ = 2π/λ is analogous to ω = 2π/T, where T is the time period of the wave (post 18.10) in space instead of in time. The reason for representing the wave by equation 1 is explained in post 19.12.
Now if this wave is reflected back on itself, with no loss of energy (see post 19.8), the reflected wave can be represented by
This reflected wave is identical to the original wave but is travelling in the opposite direction, so –x is replaced by x.
The original wave and the reflected wave will combine to give a wave that is represented by ψ1 + ψ2; this is explained in post 18.10. To calculate what the resultant wave looks like we need to know that the sine of the angle (α + β) is given by
sin(α + β) = sinα.cos β + cosα.sinβ. (3)
For more information about sines and cosines see post 16.50. Equation 3 is proved in the appendix.
The resultant wave is given by
When we apply equation 3 to this result, we get
The result is that
So when κx = π/2, 3π/2, 5π/2, 7π/2…,ψ is always zero. From the definition κ = 2π/λ (see the second paragraph above) this means that the wave function is zero when x = λ/4, 3λ/4, 5λ/4, 7λ/4…, at all values of t – that is at all times.
These points are the nodes. The mathematical analysis in this post shows that the nodes in a standing wave must be a distance 2λ/4 = λ/2 apart. Now let’s look at the picture again. The yellow wave has a wavelength of L/2 with nodes at 0, L/4, L/2, 3L/4 and L. So, the result that was previously obtained intuitively (in post 18.12) is consistent with the mathematical analysis. The red, blue and green waves are also consistent with the mathematical results.
When we meet an idea for the first time, it is often simpler to think about it intuitively. We can then follow this with a more detailed analysis to see if it confirms our intuition. I was taught about standing waves by a mathematical analysis like the one in this post. However, I believe it is often better to have an intuitive idea about how things might work before getting involved in a mathematical analysis.
19.12 The wave equation
19.9 Electromagnetic waves
19.8 Wave energy
18.17 Euler’s relation, oscillations and waves
18.14 Wave shapes
18.12 Vibrating strings
18.11 Motion in a circle, the simple harmonic oscillator and waves
To show that sin(α + β) = sinα.cos β + cosα.sinβ.
Usually this result is proved using a complicated geometric construction. It is much easier to derive it from Euler’s relation (post 18.17). Then we can write that
Applying Euler’s relation to this result gives
When we perform the multiplication on the right-hand side of this equation, we obtain the result that
Comparing the real parts (post 18.16) of equations 5 and 6 gives the result
cos(α + β) = cosαcosβ – sinαsinβ.
Comparing imaginary parts (post 18.16) gives
sin(α + β) = sinα.cos β + cosα.sinβ.