explain the formation of a stationary wave using a graphical method, and identify nodes and antinodes

What is a stationary wave? 🎸

A stationary wave is a wave pattern that does not travel through a medium. Instead, it stays in one place, with points that oscillate in place called nodes (zero displacement) and points that oscillate with maximum amplitude called antinodes (maximum displacement).

How do stationary waves form? 📐

Imagine a string fixed at both ends, like a guitar string. When you pluck it, a wave travels to the right. At the same time, a wave travels to the left from the other end. When these two waves meet, they superimpose. The resulting displacement at any point \(x\) and time \(t\) is the sum of the two travelling waves:

\$y(x,t)=A\sin(kx-\omega t)+A\sin(kx+\omega t)\$

Using the trigonometric identity for the sum of sines, this simplifies to:

\$y(x,t)=2A\sin(kx)\cos(\omega t)\$

Notice that the spatial part \$\sin(kx)\$ and the temporal part \$\cos(\omega t)\$ are separated. The spatial part determines where the nodes and antinodes are, while the temporal part tells us how the amplitude at those points changes over time.

Finding Nodes and Antinodes 🧭

The nodes occur where the sine term is zero:

\$\sin(kx)=0 \;\;\Rightarrow\;\; kx = n\pi \;\;\Rightarrow\;\; x = n\frac{\pi}{k} = n\frac{\lambda}{2}\$

So nodes are spaced by half a wavelength \$\lambda/2\$. Antinodes occur where the sine term is ±1:

\$\sin(kx)=\pm1 \;\;\Rightarrow\;\; kx = \frac{\pi}{2} + n\pi \;\;\Rightarrow\;\; x = \left(n+\tfrac12\right)\frac{\lambda}{2}\$

Thus antinodes are halfway between nodes.

Graphical Method (Step‑by‑Step) 📊

1. Draw two sine waves of equal amplitude and wavelength, one moving right (\$A\sin(kx-\omega t)\$) and one moving left (\$A\sin(kx+\omega t)\$).
2. At a fixed time, add the vertical displacements of the two waves at each point along the string.
3. Mark the points where the sum is zero – these are the nodes.
4. Mark the points where the sum is maximum – these are the antinodes.

Repeating this at different times shows that the nodes stay fixed while the antinodes oscillate in place.

Example: Guitar String 🎶

For a guitar string of length \$L\$ fixed at both ends, the allowed standing wave patterns are:

Each mode has \$n\$ nodes (including the fixed ends) and \$n-1\$ antinodes.

Exam Tips for A‑Level Physics 📚

• Remember the key equation: \$y(x,t)=2A\sin(kx)\cos(\omega t)\$. The \$\sin(kx)\$ part gives the node/antinode positions.
• When asked to draw a stationary wave, show the fixed ends as nodes and label the antinodes in between.
• Use the relation \$k=\frac{2\pi}{\lambda}\$ and \$v=\lambda f\$ to convert between wavelength, frequency, and wave speed.
• In questions about harmonics, note that the \$n^{\text{th}}\$ harmonic has \$n\$ nodes and \$n-1\$ antinodes.
• Practice sketching the superposition of two waves to reinforce the concept of constructive and destructive interference.