9.3 Stationary Waves

Stationary waves (also called standing waves) are a fundamental phenomenon arising from the superposition of waves. Unlike progressive waves that travel through a medium, stationary waves store energy in fixed oscillating segments.

Principle of Superposition

The Principle of Superposition states that when two or more waves overlap at a point, the resultant displacement is the algebraic sum of the individual displacements:

$y_{\text{resultant}}=y_1+y_2$

This principle is the foundation for understanding how stationary waves form.

Formation of Stationary Waves

Stationary waves are formed by the superposition of two identical progressive waves traveling in opposite directions through the same medium.

Conditions for formation:

• The two waves must have the same frequency and wavelength.
• The two waves must have the same (or similar) amplitude.
• They must travel in opposite directions (e.g., an incident wave and its reflection).

Mathematical Description

Consider two waves traveling in opposite directions: $y_1=A\sin (kx-\omega t),y_2=A\sin (kx+\omega t)$

By superposition: $y(x,t)=y_1+y_2=[2A\sin (kx)]\cos (\omega t)$

The term $[2A\sin (kx)]$ represents the amplitude at position $x$ — it varies with position but not with time. This is the hallmark of a stationary wave: the amplitude pattern is fixed in space.

Nodes and Antinodes

Spacing Between Nodes and Antinodes

• Distance between two consecutive nodes = $\frac{\lambda }{2}$
• Distance between two consecutive antinodes = $\frac{\lambda }{2}$
• Distance between a node and adjacent antinode = $\frac{\lambda }{4}$

Phase Relationship

All particles within the same loop (between two adjacent nodes) vibrate in phase with each other, but with different amplitudes. Particles in adjacent loops vibrate in antiphase (phase difference = $\pi$ radians).

Graphical Representation

A stationary wave can be visualized as a wave pattern that oscillates between maximum positive and maximum negative displacement without moving left or right. The nodes remain fixed while the antinodes oscillate up and down.

Antinode positions:    ↑   ↑   ↑
                    ___/\___|___/\_
                   /    |   |    \
Node positions:   N     N   N     N

At $t=0$: maximum displacement pattern At $t=T/4$: all particles at equilibrium (zero displacement) At $t=T/2$: maximum displacement pattern (inverted)

Harmonics in Stationary Waves

For a string of length $L$ fixed at both ends, both ends must be nodes. This restricts the allowed wavelengths:

$L=n\frac{{\lambda }_n}{2}⟹{\lambda }_n=\frac{2L}{n},n=1,2,3,\dots$

The corresponding frequencies are: $f_n=\frac{v}{{\lambda }_n}=\frac{nv}{2L}=n{f}_1$

Where:

• $f_1=\frac{v}{2L}$ is the fundamental frequency (1st harmonic)
• $f_n=n{f}_1$ is the $n^{th}$ harmonic
• $v=\sqrt{T/\mu }$ is the wave speed ($T$ = tension, $\mu$ = linear mass density)

Key Differences: Progressive vs Stationary Waves

Example: Guitar String

When a guitar string is plucked, the vibration travels to both fixed ends and reflects back. The incident and reflected waves superpose to form a stationary wave. The string vibrates in one or more loops depending on the frequency, producing the fundamental tone and overtones (harmonics).