18.1.1 Interference of Water Waves

Introduction

Water waves in a ripple tank provide one of the clearest and most visual demonstrations of two-source interference. Because water waves have relatively large wavelengths (millimetres to centimetres), the interference pattern is easy to observe directly.

The Ripple Tank Experiment

A ripple tank is a shallow transparent tray filled with water, illuminated from above so that the wave pattern is projected onto a screen below.

Setup

1. Two small vibrating dippers (point sources) are attached to the same motor so they vibrate in phase at the same frequency.
2. Each dipper acts as a coherent point source of circular water waves.
3. The two sets of circular waves spread outward and overlap across the surface of the water.

Observation

When the two sets of waves overlap, a characteristic interference pattern is produced:

• Bright lines (antinodal lines / constructive interference): regions where crests meet crests or troughs meet troughs. The water surface oscillates with maximum amplitude.
• Dark lines (nodal lines / destructive interference): regions where a crest from one source meets a trough from the other. The water surface remains relatively calm (minimum amplitude).

The pattern consists of alternating nodal and antinodal lines radiating outward from the midpoint between the two sources.

Principle of Superposition

The interference pattern arises from the Principle of Superposition:

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

Constructive and Destructive Interference

Constructive Interference

Occurs when two waves arrive at a point in phase (crest meets crest, trough meets trough).

• Resultant amplitude is maximum (sum of individual amplitudes).
• Phase difference condition: $\Delta \varphi =0,2\pi ,4\pi ,\dots$ radians
• Path difference condition: $\Delta x=n\lambda (n=0,1,2,3,\dots )$

Destructive Interference

Occurs when two waves arrive in anti-phase (crest of one meets trough of the other).

• Resultant amplitude is minimum (zero if amplitudes are equal).
• Phase difference condition: $\Delta \varphi =\pi ,3\pi ,5\pi ,\dots$ radians
• Path difference condition: $\Delta x=\left(n+\frac{1}{2}\right)\lambda (n=0,1,2,3,\dots )$

Conditions for a Stable Interference Pattern

For a clear, stable interference pattern to be observed, the two sources must be:

1. Coherent — same frequency and a constant phase difference (ideally zero, i.e., in phase). If the phase difference changes randomly, the pattern shifts continuously and washes out.
2. Monochromatic — single wavelength (or at least a narrow range). Multiple wavelengths produce overlapping patterns that blur the fringes.
3. Similar amplitudes — for maximum contrast between bright and dark regions. If one wave has much larger amplitude than the other, the destructive interference point will not reach zero, reducing fringe visibility.

Path Difference and Fringe Location

For a point P on the water surface, let:

• $r_1$ = distance from source $S_1$ to point P
• $r_2$ = distance from source $S_2$ to point P
• Path difference: $\Delta x=|r_2-r_1|$

Worked Example

Problem: Two coherent water wave sources produce waves of wavelength $\lambda =2\text{ cm}$. Point P is $10\text{ cm}$ from source $S_1$ and $14\text{ cm}$ from source $S_2$. What type of interference occurs at P?

Solution: $\Delta x=14-10=4\text{ cm}=2\lambda$

Since $\Delta x=2\lambda =n\lambda$ with $n=2$, this satisfies the constructive interference condition. Point P is on an antinodal line — maximum amplitude.

Summary

• The ripple tank experiment visually demonstrates two-source interference using water waves.
• Coherent sources (same frequency, constant phase difference) are essential for a stable pattern.
• Constructive interference: $\Delta x=n\lambda$; Destructive interference: $\Delta x=(n+\frac{1}{2})\lambda$.